dynamical and thermodynamical properties of a ... - unibo.it · dynamical and thermodynamical...

UNIVERSITA DI BOLOGNADIPARTIMENTO DI FISICA

Dottorato di ricerca in FISICA

CICLO XVIII

Dynamical and

thermodynamical properties

of a 2D Coulomb system

and applications to beam physics

Carlo Benedetti

Ph.D. Thesis

Bologna, 2006

Relatore: Coordinatore del dottorato:

Prof. Giorgio Turchetti Prof. Roberto Soldati

Settore disciplinare FIS/01

Contents

Introduction 1

1 The 2D Coulomb system 7

1.1 Derivation of the model . . . . . . . . . . . . . . . . . . . . . 71.2 The model of 2D Coulomb Oscillators . . . . . . . . . . . . . 141.3 Mean field approach . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.1 Derivation of the Poisson-Vlasov equations . . . . . . 161.3.2 Self consistent stationary distributions . . . . . . . . . 181.3.3 Collective dynamics: the moments method . . . . . . . 23

1.4 Landau’s kinetic theory . . . . . . . . . . . . . . . . . . . . . 391.4.1 Derivation of the Fokker-Planck equation . . . . . . . . 401.4.2 The 2D Coulomb cross section . . . . . . . . . . . . . . 431.4.3 Scaling law for the relaxation time to Boltzmann equi-

librium . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2 The simulation code 59

2.1 General features of the code . . . . . . . . . . . . . . . . . . . 592.2 The electric field computation . . . . . . . . . . . . . . . . . . 642.3 The parallelization . . . . . . . . . . . . . . . . . . . . . . . . 702.4 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . 722.5 Future developments . . . . . . . . . . . . . . . . . . . . . . . 76

3 Numerical studies 79

3.1 The collisional relaxation process in KV systems . . . . . . . 793.1.1 Scaling with N of the relaxation time . . . . . . . . . . 85

3.2 Statistical properties of Coulomb collisions . . . . . . . . . . . 863.3 Relaxation in time-dependent systems . . . . . . . . . . . . . . 903.4 The collisional relaxation process in anisotropic Gaussian sys-

tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.5 Interplay between collisional effects and non-linear phenomena

in the Montague resonance . . . . . . . . . . . . . . . . . . . . 95

3.5.1 Acceleration of the relaxation process near to the res-onance . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.5.2 How collisions interact with the resonance . . . . . . . 983.5.3 Dynamic crossing of the Montague resonance: the role

of collisions . . . . . . . . . . . . . . . . . . . . . . . . 105

Conclusions 111

A The longitudinal coherence hypothesis 113

B The 2D Debye length for a charged plasma 115

C The drift coefficient for the KV distribution 117

D The moments method with collisions 119

Bibliography 123

Introduction

The statistical mechanics of systems with long-range interactions1 out of theequilibrium is nowadays a topic of active research and of great interest.[1] Inthe description of physical systems a fundamental distinction must be madebetween systems whose components interact via short or long range forces.In the first case each elementary component of the system interacts onlywith its (nearest) neighbours while in the second case each component isinfluenced by all the remaining components of the system. This difference inthe nature of the interaction induces a considerable complexity in the formaltreatment of interacting systems with long range forces, compared to the oneswhere short range forces are involved, and leads also to deep differences intheir dynamical evolutions and properties. The description of the relaxationtowards the equilibrium in systems where short range forces are involved isbased on the Boltzmann approach [3] who first derived a kinetic equationfor Hamiltonian N -body system in his theory of gases. It has been shownthat the Boltzmann equation can be rigorously derived from the Liouvilleequation by using the BBGKY hierarchy and the derivation is exact in theGrad limit. [4]In the case of long range forces a complete and satisfactory treatment, as theprevious one, is still lacking. This is due to the fact that the phenomenologyof systems where long range forces are involved is somewhat more compli-cated since these systems, as we have already pointed out, are dominated bymean field effects. It is well known for example that systems with long rangeinteractions exhibit non standard thermodynamical properties at equilibrium(inequivalences between canonical and microcanonical descriptions, negativespecific heat, temperatures jumps)[2].As far as we are interested in the description of these systems out of the

1To define long range interactions let’s consider the potential energy U of a particleplaced in the center of a sphere of radius R in d dimension, where matter is homogeneouslydistributed. Assuming the interacting potential decays as 1/rα for r large and, neglectingthe contribution of a sphere of radius ε around the particle, we have U ∼ Rd−α. Theinteraction is long range if α ≤ d, in this case U diverges if R → ∞ and so the effect ofthe boundaries can not be neglected.[2]

2 CONTENTS

statistical equilibrium, the mean field approach (Vlasov equation) providesan useful instrument which is accurate on temporal scales short enough com-pared to the relaxation time. The Vlasov equation is then the main instru-ment to study collective effects and instabilities of the system. However, dueto the complexity of the mean field equations, only a numerical investigationof their solutions is in general possible. To this purpose a wide class Vlasov-PIC codes or Vlasov-fluid codes have been developed but their numericalreliability needs to be checked.[5, 6]Concerning the kinetic theory of systems with long range interactions, Lan-dau (1936) and Chandrasekhar (1942) first proposed kinetic equations respec-tively in case of plasmas and stellar systems.[7, 8] In the Landau’s approach,to whom we will refer in this thesis, collisions are introduced in a mean fieldframework modelling their effects as a stochastic process. The basic assump-tions of this approach are that the change in the momentum of a generictest particles can be thought as a sum of many small independent binarycollisions yielding to a macroscopic diffusion process in velocity space. Lateron, Lenard[11] and Balescu[12] developed a more precise (but also more for-mal) kinetic theory that allows the removal of some divergences which werepresent in the original Landau’s approach.In this thesis we present some analytical and numerical studies of a twodimensional Coulomb oscillators system[13] which is a model to describe abeam of charged particles, protons or ions, in a circular accelerator (storagering). In these machines very long bunches (the longitudinal dimension ismuch longer then the transversal ones) are supposed to circulate for a highnumber of turns (105÷6) and the hypothesis of a continuous circulating cur-rent, rendering the problem 2D, is fulfilled. Dealing with a 2D model hassome advantages, in fact in this case the mean field 2D Poisson-Vlasov equa-tions have analytical solutions (unlike the 3D case). As a consequence thenumerical schemes to solve the 2D mean field equations (PIC codes) can bebenchmarked against the analytical solutions. This comparison is necessaryto control the effect of numerical noise (due to round off and truncation er-rors), which causes a linear growth of the invariants of the motion[14]. Inthe 2D case it’s also possible to give an analytical description of the coherentmodes of the system [9, 10] and determine their stability. This is usuallydone by solving the linearized Poisson-Vlasov equation using the method ofcharacteristics.[37, 45] In this work we propose an alternative method to solvethe Vlasov equation based on the phase space moments chain. This methodcould offer some advantages in terms of flexibility and simplicity comparedto the traditional approaches.A key feature of our 2D Hamiltonian model is that it takes into accountthe small angle and hard collisions between charged particles. This allows

CONTENTS 3

us to explore the relaxation process to the thermodynamic equilibrium andthe validity limits of the mean field approximations. We notice that colli-sional effects (intra beam scattering) can play an important role in the newgeneration of high intensity accelerators at moderate energies such as thestorage rings for neutron spallation sources (SNS)[15], for high energy den-sity physics (SIS100 of the FAIR project at GSI)[16] and for the drivers ofheavy ion fusion (HIF)[17], since very small losses can be tolerated in orderto avoid activation of the beam pipe. The inclusion of the collisional effectsin our model is achieved by integrating Hamilton’s equation of motion forthe system of N Coulomb oscillators by using symplectic algorithms[18]. Acrucial point in the solution of the equations of motion, which affects consid-erably the performance of the simulation code and so the number of particlethat can be simulated, is the computation of the electric field acting on eachparticle. The method adopted here, based on a multipolar expansion of thefar field performed after a hierarchical space splitting, has an (optimal) com-putational complexity N logN whereas the full direct computation requiresN2 calculations[19, 20, 21]. The accuracy we obtain in the modulus of theelectric field is 10−3 ÷ 10−4 compared to the full direct calculation. Thecode has been parallelized with MPI, nevertheless N -body simulations areextremely demanding in terms of cpu time required. As a consequence, thenumber of particles that can be simulated is smaller than the “physical” oneand to keep reasonably short the simulation time, an increase of the spacecharge intensity with respect to the physical one is adopted to speed up thesimulations (enhancement of the Coulombian forces). In this case, the knowl-edge of a scaling law for the relaxation time is necessary in order to infer thecorrect behaviour of the real system from the simulated data.[22, 23]Scaling law for the relaxation time have been obtained by using the Landau’stheory. To analyze the kinetic equations of Landau’s theory in 2D one dif-ficulty is related to the Coulomb cross section for a binary collision process,which, unlikely the 3D case, has not an explicit analytic expression. In theasymptotic region, corresponding to large values of N and leading to thecontinuum limit, we have found an analytic approximation which allowed usto prove a 1/N scaling of the cross section. We have then proved that forany phase space distribution of the system the relaxation time scales linearlywith N ; the slope, which depends on the cut-off of the Coulomb potential,can be adjusted to reproduce the result of the N bodies Hamiltonian. Eventhough the order of magnitude is the same as the Debye radius, a significantdifference is found. It is worthwhile to notice that in the N → ∞ limit themean field theory is recovered.We have also studied, starting from the simulated data, the statistical prop-erties of the Coulomb collision process.[24] This is necessary in order to check

4 CONTENTS

the validity of the soft collisions assumption which is implicitly assumed inthe the Landau’s theory. A preliminary investigation shows that the distri-bution of the momentum changes due to collisions has a power law decayingqueue (due to hard collisions events), therefore its contribution should notchange, at least qualitatively, the picture emerging from the Landau’s theory.From a dynamical point of view the separation between the collisionlessregime, characterized by a time scale of O(1) (compared to the characteristicdynamical time of the system) on which a “violent relaxation” towards ameta-stable equilibrium occurs, and the collisional one, characterized by thethermodynamic relaxation occurring on a time scale of O(N), seems to bea general feature of systems where long range forces are involved. Exam-ples of this behaviour can be found in two-dimensional vortices[25, 26, 27],plasmas[28], stellar systems[8, 29] and classical spin systems (HMF) [30, 31].However, in particular regions of the parameters space, it’s possible to havean interplay between collisional effects and non-linear phenomena. We haveinvestigated this phenomenon in the case of the Montague resonance, whichis a relevent example for the beam physics. [32, 33, 34] We have shownthat collisions play a role in the trapping of particles into the resonance andthis causes an acceleration of the equipartition process in anisotropic systems.

The thesis is organized into three chapters, the outline is the following:

• in the first chapter we present the 2D Coulomb oscillators model and wediscuss its relations with the beam dynamics. First the system is stud-ied in the mean field approximation (Poisson-Vlasov equation). Thestationary solutions of the Vlasov equations are briefly outlined and amethod (the moments method) to solve the linearized Vlasov equationand determine the collective modes of the system and their stability ispresented. In the second part we will deal with the Landau’s kinetictheory for the 2D Coulomb system. A scaling law for the collisionalrelaxation time as a function of the particles number is proposed.

• in the second chapter we discuss, in detail, the features of the numer-ical code MPI–DYNAM which provides the numerical integration ofthe Hamilton equations of motion for the 2D system of Coulomb oscil-lators. After discussing the general features of the code (initial condi-tions, symplectic integration and data post processing), we present thequad-tree algorithm for the electric field calculation (computationalcomplexity N logN). In the last part of this chapter we show theperformances of the parallel version of the code. Several tests havebeen carried out in order to check the reliability of the code. Compar-

CONTENTS 5

isons have been made with Vlasov-PIC codes commonly used in beamphysics.

• in the last chapter we will present some numerical results obtainedwith the MPI–DYNAM numerical code. In particular we will analyze indetail the relaxation towards the thermodynamic equilibrium for the 2DCoulomb oscillator system and we will check numerically the validity ofthe scaling law proposed in chapter one. The analysis of the relaxationprocess has been extended to isotropic/non-isotropic systems and totime-dependent systems. Then we will present the statistical propertiesof a Coulomb collision process starting from the simulated data. Finallywe will show an example of the interplay between collisional effects andnon linear phenomena in the case of the Montague resonance.

6 CONTENTS

Chapter 1

The 2D Coulomb system

The 2D Coulomb oscillators system is a model that allows us to investigatecollective and collisional effects in a charged particles beam confined by ex-ternal linear forces.The dynamical evolution of systems where long range forces are involvedcan be studied in the mean field approximation by using the Poisson-Vlasovtheory. This approximation is adequate as far as collisional effects are negli-gible. In order to study the long term evolution of these systems (relaxationto Boltzmann equilibrium and transport properties) collisional effects haveto be taken into account. This can be done numerically by direct integrationof the Hamilton equations of motion of the system and analytically by usingthe kinetic theory first proposed by Landau, in the case of plasmas, and byChandrasekhar, in the case of stellar systems.

1.1 Derivation of the model

In this section we will present the 2D Coulomb oscillators model, in partic-ular we will discuss its relations with the transverse dynamics of a chargedbeam.Let’s consider a mono-energetic infinite continuous beam ofNp charged parti-cles per unit length, travelling along the z axis (see figure 1.1). Each particlehas charge e and mass mp and is subjected to a confining potential. Weassume that the transverse displacements and velocities are small for all theparticles

vx,i, vy,i vz,i ∀i , (1.1)

so that the trajectories remain close to the z axis. The Hamiltonian for thesystem of interacting particles in the non-relativistic approximation is given

8 The 2D Coulomb system

z

x

y

v 0

v

v

v

v

0

0

0

0

Figure 1.1: Coasting beam of Np particles per unit length traveling alongz-axis.

by

H =∑

i

(P2

i

2mp+ V (conf)(ri)

)+

1

4πε0

∑

i>j

e2

‖ri − rj‖, (1.2)

wherePi = mpvi (1.3)

and the confining potential is

V (conf)(r) =1

2Kxx

2 +1

2Kyy

2 (1.4)

where we assume Kx, Ky are positive constant (constant focusing case)1. TheHamilton equations of motion for the transverse degrees of freedom of thei-th particle are

xi = Px,i/mp

yi = Py,i/mp

Px,i = −Kxxi +e2

4πε0

∑

i6=j

xi − xj

d3ij

Py,i = −Kyyi +e2

4πε0

∑

i6=j

yi − yj

d3ij

(1.5)

1In general Kx, Ky may depend on z, in particular if Kx(z) = Kx(z + L), Ky(z) =Ky(z + L) we have the periodic focusing case and L is the length of the focusing cell(FODO cell).

1.1 Derivation of the model 9

and for the longitudinal one we have

zi = Pz,i/mp

Pz,i =e2

4πε0

∑

i6=j

zi − zj

d3ij

(1.6)

where dij =√

(xi − xj)2 + (yi − yj)2 + (zi − zj)2.Assuming a strong coherence in the longitudinal motion of the beam werequire that all the particles have the same longitudinal velocity

zi = v0 ∀i (1.7)

this condition guarantees that the longitudinal particles density doesn’t changein time and since it is completely uniform along z we have

pz,i = 0 ∀i (1.8)

which is consistent with (1.7). From now on we will consider only thetransversal dynamics of the system.Denoting by Rb the mean transversal dimension of the beam, the (mean)particles density n and the (mean) specific length ` are given by

n =Np

πR2b

` = n−1/3 . (1.9)

We divide the whole beam into a collection of small cylinders of height `

z

Rb

l

k=0k=1

k=−1

k=−2

.... k=0k=−1k=−2 k=1 k=2 ........

z

z

Figure 1.2: Decomposition of the beam into a collection of equivalent cylin-ders of height `.


(see figure 1.2), the cylinders are denoted by an integer label

k ∈ (· · · ,−2,−1, 0, 1, 2, · · · ) (1.10)

the number of particles on each cylinder is

N = Np ` = (NpRb)2/3π1/3 . (1.11)

By the longitudinal coherence hypothesis, the dynamics of the particles in-side a given cylinder (e.g. k = 0) is representative of the dynamics in anyother cylinder, as a consequence the knowledge of the behaviour of the par-ticles inside the cylinder k = 0 (the “reference cylinder”) gives a complete

description of the transverse dynamics of the beam. We denote by r(k)i the

transversal displacement from z-axis of the i-th particle inside the k-th cylin-der and by z

(k)i = k` its longitudinal position in the beam: we assume, for

sake of simplicity, that all the particles in the same cylinder have the samelongitudinal position. Actually, this hypothesys is not strictly necessary (seeAppendix A). The transverse electrostatic force acting on the i-th particle inthe k = 0 cylinder due to all the remaining particles (see the r.h.s. of (1.5))is given by

Felecti =

e2

4πε0

∞∑

k=−∞

N∑

j=1

j 6=i if k=0

r(0)i − r

(k)j(

‖r(0)i − r

(k)j ‖2 + (k`)2

)3/2. (1.12)

In order to simplify expression (1.12) we can arrange the particles labels in

all the cylinders with k 6= 0 in such a way that r(k)i depends “weakly” on k

(see figure (1.3)), namely

r(k)j ' r

(0)j (1.13)

inserting (1.13) in (1.12) and exchanging the summation order we obtain

Felecti ' e2

4πε0

N∑

j=1, j 6=i

(r

(0)i − r

(0)j )

∞∑

k=−∞

1(‖r(0)

i − r(0)j ‖2 + (k`)2

)3/2

.

(1.14)Let’s consider now the second summation in (1.14)

S(‖r(0)i − r

(0)j ‖) ≡

∞∑

k=−∞

1(‖r(0)

i − r(0)j ‖2 + (k`)2

)3/2, (1.15)


an estimation is given by the following integral

S(‖r(0)i − r

(0)j ‖) =

1

‖r(0)i − r

(0)j ‖3

∞∑

k=−∞

(1 +

(k`)2

‖r(0)i − r

(0)j ‖2

)−3/2

'

' 1

`‖r(0)i − r

(0)j ‖2

∫ ∞

−∞(1 + u2)−3/2du =

2

`‖r(0)i − r

(0)j ‖2

. (1.16)

The approximated expression (1.16) is a good estimation of (1.15) when

‖r(0)i − r

(0)j ‖ ≥ `, but if ‖r(0)

i − r(0)j ‖ ` then (1.16) is an underestimation

of the right result. In this limit the leading term in the series (1.15) is thek = 0 term, so we should take (see also figure (1.4))

S(‖r(0)i − r

(0)j ‖) '

2

`‖r(0)i − r

(0)j ‖2

if ‖r(0)i − r

(0)j ‖ > `

2

1

‖r(0)i − r

(0)j ‖3

if ‖r(0)i − r

(0)j ‖ < `

2.

(1.17)

For sake of simplicity, we will consider from now on

S(‖r(0)i − r

(0)j ‖) ' 2

`‖r(0)i − r

(0)j ‖2

(1.18)

r r r1

1

3

4

5

4

5

4

33

5

1

k=−1 k=0 k=1

reference cylinder

1 1 1

(−1)(0)

(1)

22

2

Figure 1.3: We arrange particles labels in such a way that r(k)i depends

“weakly” on k.


and we will discuss later on the implications of this approximation. In thiscase the electrostatic force acting on the i-th particle is simply given by

Felecti ' e2

4πε0

2

`

N∑

j=1, j 6=i

r(0)i − r

(0)j

‖r(0)i − r

(0)j ‖2

. (1.19)

Let’s rewrite now the equations for the transverse motion of the particles inthe reference cylinder using the curvilinear abscissa

ds ≡√v2

x,i + v2y,i + v2

z,i dt ' vz,i dt = v0dt (1.20)

as the independent variable instead of time, we have

d2xi

ds2= −ω2

x0 xi +ξ

2Ex,i

d2yi

ds2= −ω2

y0 yi +ξ

2Ey,i

for i = 1, · · · , N (1.21)

with

Ei =2

N

N∑

j=0, j 6=i

r(0)i − r

(0)j

‖r(0)i − r

(0)j ‖2

(1.22)

0.00 2.002.0

10.0

0.50 1.00 1.50

4.0

6.0

8.0

x/l

log(

S(x

))

Figure 1.4: S(x) as a a function of x/` for ` = 0.1. Black curve: exact (1.15).Red curve: approximation (1.17) (top line). Green curve: approximation(1.17) (bottom line).


where xi, yi are the components of r(0)i and the following parameters have

been introduced:- the (bare) phase advances per unit length

ωx0 =

√Kx

mpv20

, ωy0 =

√Ky

mpv20

(1.23)

- the perveance

ξ =e2Np

2πε0mpv20

(1.24)

The phase advances per unit length are related to the confining force strength,moreover

νx0 =ωx0

2πand νy0 =

ωy0

2π(1.25)

(the bare tunes) are the number of transversal (horizontal and vertical) oscil-lations per unitary longitudinal displacement along the beam without spacecharge. The perveance is related to the total charge per unit length in thebeam, space charge effects are determined by its value.Equations (1.21) can be derived from the following N -body Hamiltonian

HN =N∑

i=1

(p2

x,i + p2y,i

2+

1

2ω2

x0 x2i +

1

2ω2

y0 y2i

)− ξ

N

∑

i<j

log(rij) (1.26)

where px,i = dxi/ds, py,i = dyi/ds and rij = ‖r(0)i − r

(0)j ‖.

We observe that formally (1.26) describes a 2D system of particles confinedby linear forces interacting via a potential of the form

φ(r) ∝ − log r , (1.27)

where r is the interparticle distance. The potential φ(r) satisfies the Laplace

equation in 2D

∇2φ =1

r

d

dr

(rdφ

dr

)= 0 for r 6= 0 (1.28)

so (1.26) describes a 2D Coulomb oscillators system and can be thoughtalso a system of N interacting charged wires.

To summarize we recall that in this section we have established a connec-tion between a 3D coasting beam with Np charged particles per unit length (ξ


being the perveance) in the constant focusing case (ωx0, ωy0 being the barephase advances per unit length), and a 2D system of N (given by (1.11))Coulombian Oscillators. This analogy is strongly based on the longitudinalcoherence assumption. In fact we have seen that the dynamics of the 2DCoulomb system, described by the Hamiltonian (1.26), is equivalent to thedynamics of the particles in the reference cylinder of the original beam, andsince the particles motion inside the reference cylinder is representative ofthe particles behaviour in any other cylinder, we have that (1.26) provides apicture of the transverse dynamics of the whole original beam. To be moreprecise the analogy between the 2D and the original 3D model is complete

as far as collective effects are concerned. Regarding collisional effects the twosystems are only qualitatively equivalent because of the approximation wemade in (1.18). This approximation implies that the interaction potential inour 2D model is softened compared to the original one, so the “collisionality”of the 2D model underestimates the one of the 3D model. This approxima-tion can be avoided in order to make complete the analogy between thetwo models2, nevertheless dealing with a pure 2D Coulomb system can offersome advantages, in terms of reliability of the results, when we study it froma numerical point of view (see chapter 2).

1.2 The model of 2D Coulomb Oscillators

Let’s rewrite again the equations of motion and the Hamiltonian functionfor the 2D Coulomb system of N interacting bodies (ri ≡ (xi, yi) being thecoordinates of particle i)

d2xi

ds2= −ω2

x0 xi +ξ

2Ex,i

d2yi

ds2= −ω2

y0 yi +ξ

2Ey,i

for i = 1, · · · , N (1.29)

where the (normalized) electric field acting on particle i is given by

Ei =2

N

N∑

j=0, j 6=i

ri − rj

‖ri − rj‖2(1.30)

2We notice that in this case the 2D system would not be Coulombian anymore

because the condition (1.28) doesn’t hold.

1.2 The model of 2D Coulomb Oscillators 15

and

HN =

N∑

i=1

(p2

x,i + p2y,i

2+

1

2ω2

x0 x2i +

1

2ω2

y0 y2i

)− ξ

N

∑

i<j

log(rij) (1.31)

where rij = ‖ri − rj‖. We have seen in the previous section that (1.29)describes, under some assumptions, the transverse dynamics of a costingbeam of Np charged particles per unit length. We recall also that the numberof particles in the 2D model is determined by the following relation

N = N∗ ≡ (NpRb)2/3π1/3 (1.32)

where Rb is the mean transverse size of the beam. The parameters ξ, ωx0/ωy0,respectively the perveance of the beam and the bare phase advances per unitlength, are related to the total charge in the system and to the confiningforce strength.Obtaining a complete and general picture of the dynamics of the 2D Coulombsystem described by (1.31) is certainly a difficult task and requires the com-bined use of different analytical and numerical tools. For instance the collec-tive behaviour of the system can be studied in the mean field approximationby using the Poisson-Vlasov theory. In this case we need to fix ξ, ωx0/ωy0

and the initial distribution in phase space for the system (see section 1.3).The value of N is related to the collisionality level of the system. Even iffor any given real beam the “collisionality” of the corresponding 2D model isfixed by equation (1.32), we can examine the behaviour of the 2D system indifferent regimes by changing N and keeping fixed ξ (so the current in theoriginal beam doesn’t change). In particular if we consider N → ∞ we are inthe collisionless regime, in this limit the mean field approximation (Poisson-Vlasov theory) becomes exact. On the other hand, if we take a value of Nmuch smaller than N ∗ we have an enhancement of the collisional effects (ac-celeration of the relaxation process) compared to the case N = N ∗. This isthe standard situation when we study the system by computer simulations.Numerical integration of equations (1.29) is extremely demanding in termsof CPU time required, so we are able to simulate only systems where N is(much) smaller than N ∗. For example assuming Np ∼ 1011÷12 part/m andRb ∼ 10 mm we have N ∗ ∼ 105÷6 and usually N ∼ 104. In this case theknowledge of scaling laws is necessary in order to infer the behaviour of thesystem (avoiding making the simulation) when an higher number of particlesis considered (see the Landau’s theory in section 1.4).


1.3 Mean field approach

In this section we will present shortly the mean field approach for the 2DCoulomb system. The Coulomb oscillators model is a system of N interactingbodies, in the mean field approach we neglect the direct particle-particleinteraction. Each particle does not “see” its immediate neighbours but onlythe smooth collective field of the particles distribution as a whole. Thismethod is adequate in order to study the collective behaviour of the systemand its stability properties.

1.3.1 Derivation of the Poisson-Vlasov equations

Let’s consider the electric field acting on a test particle (the i-th) in the 2Dmodel (see (1.30))

E ≡ Ei =

N∑

j=1, j 6=i

Eij, Eij =2

N

ri − rj

‖ri − rj‖2. (1.33)

All the particles of the system contribute to the force acting on the testparticle because the Coulomb force is a long range interaction. We can splitthe sum (1.33) into two parts

E =∑

j, rij<R∗

Eij +∑

j, rij>R∗

Eij ≡ E(collis.) + E(mf) (1.34)

the first term is the contribution due to “nearby” particles and represents thecollisional part of the interaction. In fact the (few) particles in the imme-diate neighborhood of particle i-th are seen as discrete point charges whichwill effectively change the test particle trajectory in very short distances.The close encounters with these neighbors can be described as collisions be-cause they are responsible for rapid fluctuations in the motion of the testparticle. The second term in (1.34), due to “distant” particles, is the mean

field contribution. E(mf) is the sum of many small two-body contributionso we expect it is a smooth function of the coordinates. This term can bedescribed in terms of a smooth space-charge potential φ(mf)(r) related to themean particles density ρs via Poisson’s equation

E(mf) = −∂φ(mf)

∂r∇2φ(mf) = −4πρs , (1.35)

where ρs satisfies the following normalization condition∫dxdy ρs(x, y) = 1 . (1.36)

1.3 Mean field approach 17

The length R∗ separates between the near (collisional) and far (mean field)regions. We expect R∗ is of the the same order of magnitude of the Debyelength (ΛD)3. The expression for ΛD is (see Appendix B)

Λ2D ∼

⟨p2

x + p2y

⟩

4 ξR2 , (1.37)

where⟨p2

x + p2y

⟩/2 is the (non-dimensional) “temperature” of the system and

R the size of the system.If the collisional part of the interaction can be neglected the total Hamil-tonian (HN) of the 2D Coulomb systems becomes a sum of single particleHamiltonian functions

HN =N∑

i=1

H(ri,pi; s) , (1.38)

where

H(r,p; s) =p2

x + p2y

2+

1

2ω2

x0x2 +

1

2ω2

y0y2 +

ξ

2φ(mf)(r; s) . (1.39)

We notice that in general φ(mf) is a function of the time (s) because it has tobe computed self-consistently starting from the actual particles distribution.We need now to know the evolution equation for the mean particles densityρs. To do this let’s consider the Liouville equation for the 2D Coulombsystem

∂ρN

∂s+ [ρN , HN ] = 0 (1.40)

whereρN = ρN(r1,p1, r2,p2, · · · , rN ,pN ; s) (1.41)

is the distribution in the 4N -dimensional phase space (normalized to 1) and[ · , · ] are the classical Poisson’s brackets. The spatial particles density (withthe correct normalization) is given then by

ρs(r; s) =1

N

(∫ρN (r,p1, r2,p2, · · · , rN ,pN ; s) dp1dr2dp2 · · ·drNdpN+

+

∫ρN(r1,p1, r,p2, · · · , rN ,pN ; s) dr1dp1dp2 · · ·drNdpN +

+ · · · +

+

∫ρN(r1,p1, r2,p2, · · · , r,pN ; s) dr1dp1dr2dp2 · · ·dpN

).(1.42)

3Some authors [40] assert that the mean interparticle distance is the appropriate lengthscale which separates between the two regions. Our numerical experiments suggest thatthis is not true (see chapter 3).


We can simplify (1.40) by considering that the Hamiltonian of the 2D Coulombsystem in the mean field approximation is a sum of N single particle Hamil-tonian functions. In this case, since we are dealing with identical particles,the solution of the Liouville equation can be written as

ρN =N∏

i=1

ρ(ri,pi; s) (1.43)

where ρ(ri,pi; s) is the single particle phase space distribution (normalizedto 1). Inserting (1.43) and (1.38) in (1.40) and using (1.35) and (1.42) weobtain the Poisson-Vlasov equations

∂ρ

∂s+ [ρ,H] = 0 (Vlasov)

H =p2

x + p2y

2+

1

2ω2

x0x2 +

1

2ω2

y0y2 +

ξ

2φ(mf)

∇2φ(mf) = −4πρs = −4π

∫ρ(r,p; s)dp (Poisson) .

(1.44)

1.3.2 Self consistent stationary distributions

Equations (1.44) describe the (self-consistent) evolution of an initial singleparticle phase space distribution. From the behaviour of ρ we know also theevolution of the spatial density distribution ρs and we can also analyze thedynamics of a generic test particle in the mean field potential. Obviously theset of Poisson-Vlasov equations (PVe) has many solutions. From an analyt-ical point of view we need to find out distribution functions which can beanalyzed mathematically without excessive difficulties and which representphysically interesting cases.A particular class of self-consistent distributions is the one of the station-ary distributions which can be constructed if we know a set (I1, I2, · · · ) ofintegrals of the motion for the system

dIjds

=∂Ij∂s

= [Ij, H] = 0 ∀j . (1.45)

Any arbitrary function of these integrals satisfies the Vlasov equation, in fact

ρ = f(I1, I2, · · · ) ⇒dρ

ds=∑

j

∂f

∂Ij

dIjds

≡ 0. (1.46)

Let’s consider in detail some examples of stationary self-consistent distribu-tions.


The KV distribution

The KV (from Kapchinskij-Vladimirskij) distribution[35, 37] is probably themost popular stationary solution of the PVe. A KV system has a constantparticles density within an elliptical region (Ax, Ay being the semiaxes). Sincethe particles density is constant then in the inner region the space charge forceis linear and the related potential is quadratic

ρs =

1

πAxAy

x2

A2x

+y2

A2y

≤ 1

0x2

A2x

+y2

A2y

> 1(1.47)

and

φ(mf) = − 2x2

Ax(Ax + Ay)− 2y2

Ay(Ax + Ay)for

x2

A2x

+y2

A2y

≤ 1 (1.48)

where we have imposed φ(mf)∣∣0

= 0 and ∂φ(mf)

∂x

∣∣0

= ∂φ(mf)

∂y

∣∣0

= 0. The totalpotential is

W =1

2ω2

x0x2 +

1

2ω2

y0y2 +

ξ

2φ(mf)(r) ≡ 1

2ω2

xx2 +

1

2ω2

yy2

where we have introduced the depressed phase advances per unit length(ωx, ωy)

ω2x = ω2

x0 −2ξ

Ax(Ax + Ay)ω2

y = ω2y0 −

2ξ

Ay(Ax + Ay). (1.49)

We see that the space charge reduces the confining force strength. To quan-tify in a more precise way the defocusing effect we can introduce the tunesdepression:

σx =ωx

ωx0

σy =ωy

ωy0

, (1.50)

for a weak space charge σx, σy → 1 and in case of a strong space chargeσx, σy → 0.The single particle Hamiltonian decomposes into the sum of two independentterms (no coupling between x and y planes)

H =p2

x

2+

1

2ω2

xx2 +

p2y

2+

1

2ω2

yy2 ≡ Hx +Hy, (1.51)

then Hx and Hy are both integrals of motion. In a KV system the motion ofa generic test particle is particularly simple: each particle performs harmonic


oscillations in x and y, the oscillation frequencies are respectively ωx, ωy.It is not difficult to prove that the analytical expression of the single particlephase space distribution of a KV system is given by

ρKV = Cδ(Hx + THy − Ex) (1.52)

where δ(·) is the Dirac “function” and

C =

√T

2π2AxAyEx =

1

2ω2

xA2x T =

ω2xA

2x

ω2yA

2y

. (1.53)

Geometrically (1.52) represents an ellipsoidal surface in the 4-dimensionalphase space (x, y, px, py). The projection of (1.52) on the phase planes(x, y), (x, px), (y, py), · · · yields always an uniformly filled ellipse.Let’s define the r.m.s. emittances of the system

ε2x =⟨x2⟩ ⟨p2

x

⟩− 〈xpx〉2 ε2y =

⟨y2⟩ ⟨p2

y

⟩− 〈ypy〉2 . (1.54)

The emittances εx, εy are related respectively to the horizontal and to thevertical spread of the system. These parameters are important in beamdynamics because they provide a quantitative basis to describe the qualityof the beam. A beam of good quality is characterized by “small” values of theemittances (i.e. small width and small divergence from the longitudinal axis).For example, if there is halo formation around the beam core we observe agrowth of the emittances. The quality of a beam surrounded by an halo isconsidered bad because the particles in the halo will probably be lost.For a KV system, by using (1.52), we obtain

⟨x2⟩

=A2

x

4

⟨p2

x

⟩=ω2

xA2x

4〈xpx〉 = 0

⟨x2⟩

=A2

y

4

⟨p2

y

⟩=ω2

yA2y

4〈ypy〉 = 0

〈xy〉 = 〈xpy〉 = 〈ypx〉 = 〈pxpy〉 = 0

(1.55)

then the r.m.s. emittances are

εx =ωxA

2x

4εy =

ωyA2y

4. (1.56)

We can also define the temperature associated with each degree of freedomof the system (as the mean value of the incoherent part of the kinetic energy)

kBTx ≡ ε2x〈x2〉 kBTy ≡

ε2y〈y2〉 . (1.57)


For a KV system we have

kBTx =1

4ω2

xA2x kBTy =

1

4ω2

yA2y kBT =

1

2(kBTx + kBTy) (1.58)

and in general the system will not be “equipartitioned”, i.e. kBTx 6= kBTy 6=kBT . The KV is the only known analytical distribution that can be extendedalso to the periodic focusing case. In this case the semiaxes of the ellipticalregion change periodically in time and the periodicity is equal to the one ofthe confining force.

The Maxwell-Boltzmann distribution

The Maxwell-Boltzmann (MB) distribution[37] should represent the natural4

thermodynamic equilibrium state for the Coulomb system. Any arbitraryinitial distribution should relax to the MB if Coulomb collisions are takeninto account. The analytical expression of the distribution is

ρMB = C exp(−H/kBT ) (1.59)

where kBT is the (non-dimensional) temperature of the 2D system and C isthe normalization constant. The oscillation frequencies (ωx, ωy) of a generictest particles in a MB system depend on the oscillation amplitude (ax, ay)being non-linear the space charge forces

ωx = ωx(ax, ay) ≤ ωx0

ωy = ωy(ax, ay) ≤ ωy0 .(1.60)

We have ωx → ωx0 and ωy → ωy0 when the amplitudes are large. Themaximum defocusing effect (maximum tunes depression) is in the center ofthe distribution (ax, ay → 0). No analytical expressions exist in general forthe spatial density profile and for the space charge potential. Only in thelimit of low and high temperatures we have simple analytical expressions. Inparticular if kBT → 0 the density profile tends to become uniform within anellipse whose semiaxes are

Ax =√

2ξωx0

1s

1+

„

ωx0ωy0

«2

Ay =√

2ξωy0

1r

1+“

ωy0ωx0

”2.

(1.61)

4In case of long range forces there is no warranty that the standard methods of theequilibrium statistical mechanics can be utilized (surface effects could be relevant). How-ever, in this case, the Debye shielding mechanism should make the interaction effectivelyshort range.


In the limit for kBT large the contribution of the space charge potentialbecomes negligible and so the spatial density profile becomes Gaussian

ρs 'ωx0ωy0

πkBTexp

(−ω2

x0x2 + ω2

y0y2

2kBT

)(1.62)

To examine more in detail some aspects of this distribution let’s consider thesymmetric case: round beam + ωx0 = ωy0 = ω0, then

ρ(MB)s =

∫ρMB dpxdpy = C ′e−W (r)/kBT (1.63)

where W (r) = 12ω2

0r2 + ξ

2φ(r) and

∇2φ(r) ≡ 1

r

d

dr

(rdφ

dr

)= −4πC ′e−W (r)/kBT . (1.64)

The constant C ′ is fixed by the normalization condition

C ′∫ +∞

0

e−W (r)/kBT 2πrdr = 1. (1.65)

Equation (1.64) with condition (1.65) can be solved numerically by using arecursive method. Denoting ψ = dφ

dr, from (1.64) we obtain the following

system

dφ

dr= ψ

dψ

dr= −ψ

r− 4πC ′e−( 1

2ω2

0r2+ ξ

2φ)/kBT

(1.66)

to be solved with the initial conditions ψ(0) = 0, φ(0) = 0. The solution forsmall values of r is

φ(r) ' −πC ′r2

ψ(r) ' −2πC ′r.(1.67)

We notice that in (1.66) the value of C ′ is not known a priori. We start withan arbitrary value for C ′ and we perform a numerical integration of (1.66)with the appropriate initial conditions. We compute (again numerically) thenormalization integral

I = C ′∫ +∞

0

e−( 12ω2

0r2+ ξ2φ(r))/kBT 2πrdr, (1.68)

and if|I − 1| < ε , (1.69)


0.0 2.50.0

0.4

0.5 1.0 1.5 2.0

0.1

0.2

0.3

r [mm]

ρ s(MB

) (r)

[m

m-2

]ω

0=1.0 rad/m ξ=1.0

0.0 5.00.0

2.5

1.0 2.0 3.0 4.0

0.5

1.0

1.5

2.0

r [mm]

Ε(MB

) (r)

[m

m-1

]

ω0=1.0 rad/m ξ=1.0

Figure 1.5: Left: plot of the spatial profile of the MB distribution (ρ(MB)(r)s )

for kBT=1 (black), 0.1 (red), 0.01 (green), 0.001 (blue) and 0.0 (light blue).Right: self consistent electric field for the same values of the temperature asin the left panel.

(ε is the tolerance) then the value of the normalization constant C ′ is accept-able and the numerical solution of (1.66) gives the correct behaviour of thespace charge potential (whereas (1.63) gives the space charge density profile).On the other hand, if |I − 1| > ε, we have to modify recursively the valueof C ′ until we satisfy condition (1.69). We notice that I is a monotonicallyincreasing function of C ′ and by using a bisection procedure we can rapidlyget the correct result. In figures 1.5 we have a plot of the density profile (leftpanel) and the behaviour of the self consistent electric field as a function ofr for different values of the temperature kBT .

1.3.3 Collective dynamics: the moments method

The moment’s chain.

In this section we will consider a perturbed KV distribution and we will studythe coherent oscillations frequencies of the system and their stability prop-erties. This problem has been already studied by using the characteristicsmethod by Gluckstern (for symetric KV systems)[10] and by Hofmann (fornon-symmetric KV systems)[9] in the constant focusing case. The approachhere presented, still under development, is somewhat more simple and canbe easily extended to the periodic focusing case and to 3D systems. Further-more also the extension to non-KV systems is, in principle, possible.


We want to study the evolution the following phase space distribution

ρ = ρ0 + δρ, (1.70)

where ρ0 is the unperturbed KV distribution. The function ρ satisfies theVlasov equation

∂ρ

∂s+ [ρ,H] = 0 H =

p2x + p2

y

2+

1

2kx0x

2 +1

2ky0y

2 +ξ

2φ. (1.71)

The parameters kx0, ky0 depend on time in the periodic focusing case (kx0(s+L) = kax0(s), ky0(s + L) = ky0(s)), otherwise they are constant (kx0 ≡ω2

x0, ky0 ≡ ω2y0). Retaining only first order terms in the perturbation δρ in

(1.71), we have∂δρ

∂s+ [δρ,H0] + [ρ0, δV ] = 0 (1.72)

where H0 is the unperturbed KV Hamiltonian

H0 =p2

x + p2y

2+

1

2kxx

2 +1

2kyy

2 , (1.73)

with

kx = kx0 −2ξ

Ax(Ax + Ay)ky = ky0 −

2ξ

Ay(Ax + Ay)(1.74)

and Ax, Ay are the the semiaxes of the unperturbed beam5; δV is the per-turbation of the space charge potential related to δρ via Poisson’s equation

δV =ξ

2δφ ∇2δφ = −4πδρs =

∫dpxdpyδρ. (1.75)

In order to study the evolution of the perturbation δρ we have to solve jointlyequations (1.72) and (1.75). Equation (1.72), using (1.73), can be rewrittenas follows

∂δρ

∂s=∂δV

∂x

∂ρ0

∂px+∂δV

∂y

∂ρ0

∂py− px

∂δρ

∂x− py

∂δρ

∂y+ kxx

∂δρ

∂px+ kyy

∂δρ

∂py. (1.76)

We consider now the moments in phase space associated to the distributionδρ defined as

⟨xiyjpk

xply

⟩≡∫dx dy dpx dpy x

ipjxy

kply δρ(x, y, px, py, s) (1.77)

5Ax, Ay are periodic functions of the time in the periodic focusing case: Ax(s) =Ax(s + L) and Ay(s) = Ay(s + L).


where n ≡ i+ j + k + l is the moment order.The time evolution of

⟨xiyjpk

xply

⟩is given by

d⟨xiyjpk

xply

⟩

ds≡∫dxdydpxdpyx

ipjxy

kply

∂δρ

∂s. (1.78)

where ∂δρ/∂s is given by (1.76). Starting from n = 1 (the case n = 0 has noimportance), we can build the hierarchy of the moments equations, which isan infinite set of coupled differential equations

n = 1

˙〈x〉 = 〈px〉˙〈px〉 = −kx 〈x〉 + ξ

2〈δEx〉0

· · · (x→ y) · · ·

n = 2

˙〈x2〉 = 2 〈xpx〉˙〈xpx〉 = 〈p2

x〉 − kx 〈x2〉 + ξ2〈xδEx〉0

˙〈p2x〉 = −2kx 〈xpx〉 + ξ〈pxδEx〉0

· · · (x→ y) · · ·˙〈xy〉 = 〈xpy〉 + 〈ypx〉˙〈xpy〉 = 〈pxpy〉 − ky 〈xy〉 + ξ

2〈xδEy〉0

˙〈ypx〉 = 〈pxpy〉 − kx 〈xy〉 + ξ2〈yδEx〉0

˙〈pxpy〉 = −kx 〈xpy〉 − ky 〈ypx〉 + ξ2〈pyδEx〉0 + ξ

2〈pxδEy〉0

n = 3

˙〈x3〉 = 3 〈x2px〉˙〈x2px〉 = 2 〈xp2

x〉 − kx 〈x3〉 + ξ2〈x2δEx〉0

˙〈xp2x〉 = 〈p3

x〉 − 2kx 〈x2px〉 + ξ〈xpxδEx〉0˙〈p3x〉 = −3kx 〈xp2

x〉 + 3ξ2〈p2

xδEx〉0˙〈xy2〉 = 〈y2px〉 + 2 〈xypy〉˙〈y2px〉 = 2 〈ypxpy〉 − kx 〈xy2〉 + ξ

2〈y2δEx〉0

˙〈xypy〉 = 〈ypxpy〉 +⟨xp2

y

⟩− ky 〈xy2〉 + ξ

2〈xyδEy〉0

˙〈ypxpy〉 =⟨pxp

2y

⟩− kx 〈xypy〉 − ky 〈y2px〉 + ξ

2〈ypyδEx〉0 + ξ

2〈ypxδEy〉0

˙⟨xp2

y

⟩=⟨pxp

2y

⟩− 2ky 〈xypy〉 + ξ〈xpyδEy〉0

˙⟨pxp2

y

⟩= −kx

⟨xp2

y

⟩− 2ky 〈ypxpy〉 + ξ

2〈p2

yδEx〉0 + ξ〈pxpyδEy〉0· · · (x→ y) · · ·

(1.79)

n = 4 · · ·


where the electric field perturbation is given by

δEx = −∂δφ∂x

δEy = −∂δφ∂y

(1.80)

and the mean values 〈 · 〉0 are computed using the unperturbed distributionρ0. The space charge ξ provides the coupling between the equations by meansof terms like 〈xiyjpk

xplyδEx〉0, 〈x

iyjpkxp

lyδEy〉0 which, in general, depend on all

the spatial moments 〈xiyj〉 via the perturbing electric field.Equations (1.79) can be solved if we consider a particular class of perturba-tions δρ which gives a polynomial perturbing potential. In fact, if

δφ(n) ∼n∑

k=0

ckxkyn−k (1.81)

and if the coefficients ck can be expressed self consistently as a linear com-bination of only the spatial moments at order n (like 〈xiyn−i〉) then the mo-ments chain can be truncated and the equations at order n in the hierarchy(1.79) become a closed set of differential linear equations whose eigenmodesdescribe the n-th order choerent modes of the Coulomb system.

Polynomial perturbing potentials.

We need to find out all the types of density perturbation which give polynomial-like potentials as in (1.81), then we have to connect the coefficients of thepolynomial expansion to the spatial moments

⟨xjyk

⟩.

The perturbing potential δφ in the inner region can be split into two contri-butions

δφ = δφEdge + δφBody (1.82)

where the first term is due to the charge at the edge of the distribution,the second one to the charge distribution within the elliptical region (bodycharge). We notice that the total (edge+body) extra charge induced by theperturbation δρ must be zero.

For the edge term we have in general

δφEdge = −∫ 2π

0

dϕ qEdge(ϕ) log[(x− Ax cosϕ)2 + (y − Ay sinϕ)2] (1.83)

where (Ax cosϕ, Ay sinϕ) is a generic point on the edge of the KV chargedistribution and qEdge(ϕ)dϕ the extra charge in this position, (x, y) are the


coordinates of a generic point inside the charge distribution. ExpandingqEdge(ϕ) in Fourier series we have

qEdge(ϕ) =a0

2+

∞∑

k=1

ak cos kϕ+

∞∑

k=1

bk sin kϕ . (1.84)

Inserting (1.84) in (1.83) we obtain (neglecting additive constants) the fol-lowing polynomial expression for δφEdge

δφEdge = 2π

(2

Ax + Ay

)(a1x + b1y) +

+2π

2

(2

Ax + Ay

)2

(a2(x2 − y2) + b22xy) + (1.85)

+2π

3

(2

Ax + Ay

)3

(a3(x3 − 3xy2 − 3

4(A2

x − A2y)x) + b3(3x

2y − y3 − 3

4(A2

x − A2y)y)) +

+ · · · .

For example, the quadratic perturbing potential

δφ(2)Edge = 2π

(2

Ax + Ay

)2

b2 xy (1.86)

is generated by the following edge charge distribution

q(2)Edge(ϕ) = b2 sin 2ϕ (1.87)

where the coefficient b2 can be related to the mean value 〈xy〉, namely

〈xy〉 =

∫ 2π

0

q(2)Edge(ϕ)Ax cosϕAy sinϕ dϕ =

π

2AxAy b2 (1.88)

and finally (1.86) can be rewritten as

δφ(2)Edge =

16 〈xy〉AxAy(Ax + Ay)2

xy . (1.89)

By considering the perturbing potential (1.89) the moments chain (1.79)truncates and the relevant part is represented solely by the last four equa-tions at order n = 2 (skew quadrupole mode).To summarize in table 1.1 we show, for the orders n = 1, 2, 3, 4, a classifi-cation of the edge charge perturbations and the corresponding polynomialpotentials. Apart from the order n, the perturbations are characterized also


order (n) q(n)Edge(ϕ) spatial profile δφ

(n)Edge

1 - Even a1 cosϕ

4πa1

Ax+Ayx

1 - Odd b1 sinϕ

4πb1Ax+Ay

y

2 - Even a2 cos 2ϕ

4πa2

(Ax+Ay)2(x2 − y2)

2 - Odd b2 sin 2ϕ

8πb2(Ax+Ay)2

xy

3 - Even a3 cos 3ϕ

16πa3

3(Ax+Ay)3(x3 − 3xy2 − 3(A2

x−A2y)

4x)

3 - Odd b3 sin 3ϕ

16πb33(Ax+Ay)3

(3x2y − y3 − 3(A2x−A2

y)

4y)

4 - Even a4 cos 4ϕ

8πa4

(Ax+Ay)4(x4 + y4 − 6x2y2 − (A2

x − A2y)(x

2 − y2))

4 - Odd b4 sin 4ϕ

32πb4(Ax+Ay)4

(x3y − xy3 − A2x−A2

y

2xy)

Table 1.1: Edge charge perturbation for n =1,2,3,4, and corresponding poly-nomial potential in the internal region. The plots describe the spatial profileof the charge distribution (black curve unperturbed cross section, red per-turbed).


by the symmetry with respect to the angular variable ϕ, where the evenmodes (cos kϕ) have the x axis as symmetry plane.

For the body charge term the calculations are in general more complicated(at the moment is still a work in progress). However it’s not difficult to showthat for a body charge perturbation which is constant within the unperturbedprofile (n

(2)Body = const), the corresponding perturbing potential is quadratic

δφ(2) = −2π n

(2)Body

Ax + Ay

(Ayx2 + Axy

2) (1.90)

and the perturbation n(2)Body can be related to the spatial moments 〈x2〉 and

〈y2〉. The total body charge in this case is

QBody = πAxAy n(2)Body (1.91)

and since the total extra charge must be zero, then the total edge charge

QEdge =

∫ 2π

0

qEdge(ϕ)dϕ = πa0 (1.92)

is non vanishing and we have

QBody +QEdge = 0 ⇒ a0 = −QBody

π= −AxAy n

(2)Body . (1.93)

It’s not difficult to show that (1.90) is the only relevant contribution atorder n = 2, body charge can not produce a skew quadrupolar perturbation.Furthermore it’s easy to prove that body charge perturbations play no roleat order n = 1.

Coherent modes - the constant focusing case

Let’s discuss in some details the coherent modes of the 2D Coulomb systemin the constant focusing case.

• Even & odd dipolar mode (n = 1).In this case we have only edge charge terms (body charge contributions areabsent). An even perturbation is characterized by

q(1)Edge = a1 cosϕ with δφ

(1)Edge =

4πa1

Ax + Ayx , (1.94)


and the relevant equations are the first two at level n = 1 in the momentshierarchy (1.79)

˙〈x〉 = 〈px〉˙〈px〉 = −ω2

x 〈x〉 + ξ2〈δEx〉0 .

(1.95)

The coefficient a1 can be related to 〈x〉, in fact

〈x〉 =

∫ 2π

0

Axa1 cos2 ϕdϕ = πAxa1 ⇒ a1 =〈x〉Axπ

. (1.96)

The perturbing electric field is then

δEx = −∂δφ

(1)Edge

∂x= − 4 〈x〉

Ax(Ax + Ay). (1.97)

Inserting (1.97) in (1.95) and using the definition of ω2x we get

¨〈x〉 = −ω20x 〈x〉 . (1.98)

We see that this mode corresponds to a horizontal oscillation of the system,the oscillation frequency is ω0x. A similar result holds in the odd case for thevertical degree of freedom.

• Even quadrupolar mode (n = 2).In this case both edge and body charges have to be taken into account, theiranalytical expressions are

q(2)Edge =

a0

2+ a2 cos 2ϕ q

(2)Body = n

(2)Body (constant) . (1.99)

The a0 term assures the total charge conservation if we have (see (1.93))

a0 = −AxAyn(2)Body . (1.100)

The perturbing potential reads

δφ(2) = δφ(2)Edge + δφ

(2)Body =

4πa2

(Ax + Ay)2(x2 − y2) +

2πa0

Ax + Ay

(x2

Ax+y2

Ay

)

(1.101)where the coefficients a0 and a2 are related to the spatial moments 〈x2〉 and〈y2〉, in fact

〈x2〉 =∫ ∫

Bodyx2n

(2)Bodydxdy +

∫ 2π

0A2

x cos2 ϕ (a0

2+ a2 cos 2ϕ) dϕ

〈y2〉 =∫ ∫

Bodyy2n

(2)Bodydxdy +

∫ 2π

0A2

y sin2 ϕ (a0

2+ a2 cos 2ϕ) dϕ

(1.102)


and we obtain

〈x2〉 = π4a0A

2x + π

2a2A

2x

〈y2〉 = π4a0A

2y − π

2a2A

2y .

(1.103)

Defining⟨x2⟩

=A2

x

4η1

⟨y2⟩

=A2

y

4η2 , (1.104)

the perturbing potential can be cast in the following form

δφ(2) =η1(2Ax + Ay) + η2Ay

Ax(Ax + Ay)2x2 +

η1Ay + η2(Ax + 2Ay)

Ay(Ax + Ay)2y2 . (1.105)

The relevant equations in this case are the first six at level n = 2 in thehierarchy (1.79). Inserting in these equations the perturbing electric fieldcoming from (1.105), after a little algebra we obtain

η′′1

η′′2

=

−4 − 1 + 2δ

(1 + δ)2σ2 − σ2

(1 + δ)2

− σ2δ2

(1 + δ)2−4α2 − (2 + δ)δ

(1 + δ)2σ2

η1

η2

(1.106)

where we have posed

η′′i =ηi

ω2x

, (1.107)

and we have defined

α =ωy

ωx, δ =

Ax

Ay, σ2 =

2ξ

AxAyω2x

. (1.108)

Denoting by λ1, λ2 the eigenvalues of the matrix in (1.106), the coherentfrequencies are given by

ωi = ωx

√−λi . (1.109)

If λi < 0 then ωi is real and the mode is stable (oscillatory), otherwise themode is unstable (exponential growth).In the symmetric case

ω2x0 = ω2

y0 = ω20 ω2

x = ω2y = ω2 Ax = Ay = R (1.110)

which impliesα = δ = 1 , (1.111)


0.0 1.00.0

4.0

0.2 0.4 0.6 0.8

1.0

2.0

3.0

ωy /ω

y0

ω/ω

y0

ωy/ω

x=0.48 A

x/A

y=1.54

0.0 1.00.0

3.0

0.2 0.4 0.6 0.8

1.0

2.0

ωy /ω

y0

ω/ω

y0

ωy/ω

x=0.48 A

x/A

y=1.54

Figure 1.6: Examples of coherent frequencies for the second order even (leftpanel) and odd (right panel) modes in the anisotropic case. The blackcurves and red curves (respectively the real and imaginary part of the co-herent frequency) are the Hofmann results, the green and blue diamonds(real/imaginary part) are the moments method prediction.

the coherent frequencies are

ω21 = 2(ω2

0 + ω2) ω22 = ω2

0 + 3ω2, (1.112)

and we can recognize the usual envelope modes.[37]The oscillation frequencies in the asymmetric case are shown in figure 1.6(left panel), the agreement with Hofmann calculations is very good. It’s notdifficult to show that these modes are always stable for any space chargestrength and for any anisotropy level in the system.

• Odd quadrupolar mode (n = 2).The edge charge term

q(2)Edge = b2 sin 2ϕ , (1.113)

is the only contribution in this case. We recall that (see equation (1.88))

〈xy〉 =π

2AxAy b2 (1.114)

and the perturbing potential is given by (1.89). The relevant equations arethe last four at level n = 2 in the hierarchy (1.79). Defining

〈xy〉 =AxAy

4η1 〈pxpy〉 =

AxAy

4ωxωyη2 , (1.115)


and using the perturbing electric field associated to the skew quadrupolepotential (1.89), we get

η′′1

η′′2

=

−1 − α2 − 1 + δ2

(1 + δ)2σ2 2α

2α+δ2 + α2

(1 + δ)2

σ2

α−1 − α2

η1

η2

(1.116)

where the parameters α, δ, σ2 are the same as in (1.108). Again the coher-ent frequencies are computed starting from the eigenvalues of (1.116) using(1.109).For symmetric systems the frequencies are

ω21 = 0 ω2

2 = ω20 + 3ω2. (1.117)

The first one is zero because the system is rotationally symmetric and forisotropic systems the angular rotation has no restoring force. The secondfrequency is identical to the ω2 of the isotropic even case (see (1.112)).In the asymmetric case the solutions for the even quadratic mode are alwaysstable but this is no longer true for the odd mode. In fact we can show thatif √

Ax

Ay

<ωy0

ωx0

< 1ωy

ωx

> 1 (1.118)

or

1 <ωy0

ωx0

<

√Ax

Ay

ωy

ωx

< 1 (1.119)

one of the coherent frequencies in the odd case becomes imaginary, as aconsequence a small initial perturbation of the equilibrium position 〈xy〉 =〈xpy〉 = 〈ypx〉 = 〈pxpy〉 = 0 will growth exponentially. As we can see from(1.118) and (1.119), the instability requires a strong anisotropy between hor-izontal and vertical planes in order to be activated. The quantity 〈xy〉 isrelated to the tilting angle (ψ) of the charge distribution

ψ ∼ 4 〈xy〉A2

x − A2y

(1.120)

as a consequence the growth of 〈xy〉 causes a rotation of the charge profile.This x − y coupling is responsible for an emittance exchange between thehorizontal and the vertical degrees of freedom. We can show that (assuming


0.0 1.010^-2

10^-1

1

10

0.2 0.4 0.6 0.810^-2

10^-1

1

10

ωy/

ωy0

ω/ω

y0

ωy/ω

x=0.48 A

x/A

y=1.54

Figure 1.7: Examples of coherent frequencies for the third order even modein the anisotropic case. The black and red curves (respectively the real andimaginary part of the coherent frequency) are the Hofmann calculations, thegreen and blue diamonds (real/imaginary part) are the moments methodprediction.

initially εy < εx)

εx(s) + εy(s) = εx(0) + εy(0)

εy(s) ' εy(0)(1 + c exp(µs)) ,

(1.121)

where c > 0 is a constant and the exponential growth rate µ is given by

µ = 2 |Imω1| , (1.122)

where ω1 is the unstable coherent frequency.In figure 1.6 (right panel) we show a sketch of the behaviour of the coherentfrequencies for the odd case, again we find a good agreement with Hofmanncalculations.

• Even & odd high order modes (n = 3, 4).At the moment we don’t have an expression for the body charge terms corre-sponding to high order multipolar modes (sextupolar, octupolar, etc.), nev-ertheless we can give as well an estimation of the coherent modes frequenciesby using only the edge charge terms. Obviously we expect to obtain a subset of all the correct modes because, ignoring body charges, we are neglecting


0.0 1.010^-2

10^-1

1

10

0.2 0.4 0.6 0.810^-2

10^-1

1

10

ωy /ω

y0

ω/ω

y0

ωy/ω

x=0.48 A

x/A

y=1.54

0.0 1.010^-2

10^-1

1

10

0.2 0.4 0.6 0.810^-2

10^-1

1

10

ωy /ω

y0

ω/ω

y0

ωy/ω

x=0.48 A

x/A

y=1.54

Figure 1.8: Examples of coherent frequencies for the fourth order even (leftpanel) and odd (right panel) modes in the anisotropic case. The black and redcurves (respectively the real and imaginary part of the coherent frequency)are the Hofmann’s results, the green and blue diamonds (real/imaginarypart) are the moments method prediction.

some degrees of freedom.The results for the isotropic case are shown below

n = 3 ω21,2 = 1

2ω2

0 + 92ω2 ± 1

2

√ω4

0 + 6ω20ω

2 + 57ω4

n = 4 ω21,2 = 1

2ω2

0 + 192ω2 ± 1

2

√ω4

0 − 2ω20ω

2 + 145ω4

(1.123)

the agreement with Gluckstern/Hofmann results is complete.For the anisotropic case the results are shown in figure 1.7 for the sextupolarcase (even) and in figure 1.8 for the octupolar case (even and odd). Apartfrom the fact that we can not reproduce all the curves, the differences be-tween the two approaches are small (at maximum 15 %) and only in thestrong space charge regime (ωy/ωy0 < 0.2).

Coherent modes - the periodic focusing

Apart from its relative simplicity compared to the Gluckstern/Hofmann ap-proach, one of the advantages of the moments method is related to the factthat it can be easily extended to the periodic focusing case. We recall thatin this case the confining force is a periodic function of the “time” s andL is the period (i.e. the periodicity length). Typically for kx0(s) we choose


a piecewise constant function according to the following schema (FODO cell)

K

K

F

D−

L

Kx0(s)

slF

lO

lD

lO

and ky0(s) = −kx0(s).

The cell structure, concerning the x degree of freedom, is constituted by afocusing element (F), a drift (O), a defocusing element (D) and a second driftelement (O). For the y plane the focusing and defocusing elements are ex-changed. The global effect of the cell is confining in both x and y directions.The cell is characterized by the following parameters (referring to the plotabove):

• KF , KD ⇒ focusing/defocusing quadrupolar gradients;• `F , `D, `O ⇒ respectively the length of the focusing/defocusing/drift ele-ments. We notice that L = `F + `D + 2`O.The confining efficiency of a FODO cell is measured by the bare phase ad-vances ϕx0, ϕy0. In fact, we recall that the position of a particle can beexpressed as

x(s) = ax(s) cos(ψx(s))

y(s) = ay(s) cos(ψy(s))(1.124)

where ax(s), ay(s) are the amplitudes of the motion and ψx(s), ψy(s) are thephases of the particle. After the transit through a FODO cell the particle’sphases are modified according to

ψx(s+ L) = ψx(s) + ϕx0 ψy(s+ L) = ψy(s) + ϕy0 , (1.125)

i.e. the cell phase advances are the net increments of the x and y phases ofthe test particle (we can also consider the phase advances per unit lengthby considering ωx0 = ϕx0/L, ωy0 = ϕy0/L). The space charge reduces theconfining efficiency and so we have

ϕx0 → ϕx < ϕx0 ϕy0 → ϕy < ϕy0 , (1.126)

where ϕx and ϕy are the depressed phase advances.For a KV system in the periodic focusing case the unperturbed momentsof the charge distribution (and in particular the semiaxes Ax, Ay) depend


periodically on s with the same periodicity of the confining lattice. Forexample the unperturbed second order moments of the distribution are

〈x2〉0(s) = βx(s)εx 〈xpx〉0(s) = −αx(s)εx 〈p2x〉0(s) = − εx

βx(s)(1 + α2

x(s))

〈y2〉0(s) = βy(s)εy 〈ypy〉0(s) = −αy(s)εy 〈p2y〉0(s) = − εy

βy(s)(1 + α2

y(s))

〈xy〉0(s) = 〈xpy〉0(s) = 〈ypx〉0(s) = 〈pxpy〉0(s) = 0(1.127)

where the periodic functions αx(s), αy(s), βx(s), βy(s) are the optical func-tions of the confining cell (for a detailed analysis see M. Reiser [37]) and εx,εy are the unperturbed r.m.s. emittances of the system.Let’s consider now the behaviour of the system in case of a skew quadrupolarperturbation (see (1.89)). We recall that we have to take into account thelast four equations at level n = 2 in the hierarchy (1.79) and the unper-turbed values 〈·〉0 have to be computed starting from (1.127). Defining (forsimplicity)

η1 = 〈xy〉 η2 = 〈xpy〉 η3 = 〈ypx〉 η4 = 〈pxpy〉 (1.128)

after a little algebra we get

η1

η2

η3

η4

=

0 1 1 0

−ky0(s) +2ξ

(Ax(s) + Ay(s))20 0 1

−kx0(s) +2ξ

(Ax(s) + Ay(s))20 0 1

− 8ξ(〈xpx〉0(s) + 〈ypy〉0(s))(Ax(s) + Ay(s))2Ax(s)Ay(s)

−kx(s) −ky(s) 0

η1

η2

η3

η4

(1.129)where Ax(s) = 2

√〈x2〉0, Ay(s) = 2

√〈y2〉0. We rewrite (1.129) in the follow-

ing compact form

~η = M(s)~η , M(s) = M(s+ L) . (1.130)

The linear system (1.130) can be studied (numerically) by using the Floquet’stheory. Let’s consider the following four initial conditions

~η1(0) =

1000

~η2(0) =

0100

~η3(0) =

0010

~η4(0) =

0001

(1.131)


then we determine the evolution of ~η i from s = 0 to s = L

~ηi(0) → ~ηi(L) ≡ ~ηi∗ . (1.132)

We construct the transition matrix T collecting (as columns) the four vectorsη∗i . By using T we can compute the evolution of an arbitrary initial condition~v(0) up to a time nL (i.e. after n periods, where n is an integer) by taking

~v(n) = T n ~v(0) . (1.133)

The coherent frequencies of the system are related to the eigenvalues of thetransition matrix T . In particular, denoting by λi the eigenvalues of T , wehave that a coherent mode is unstable if

λmax ≡ maxi

Re(λi) > 1 . (1.134)

As in the constant focusing case, the skew quadrupolar instability is respon-sible for an emittance exchange between the x and y planes. The emittancesevolve according to

εx(n) + εy(n) = εx(0) + εy(0)

εy(n) ' εy(0)(1 + c exp(µn)) ,

(1.135)

where c > 0 is a constant and the exponential growth rate µ is given by

µ = 2 lnλmax . (1.136)

In figure 1.9 we show the stability chart for the skew quadrupolar coherentmode. Keeping fixed the confining force parameters and the perveance, wehave considered different working points corresponding to different values ofthe initial r.m.s. emittances (εx, εy) and we have investigated the stabilityproperties of the coherent mode for each point. The yellow and red circles arerespectively the stable and unstable points. The black curve in 1.9 separatesbetween the zones

I) ϕx0 < ϕy0 and ϕx < ϕy (above the black line) (1.137)

and

II) ϕx0 < ϕy0 and ϕx > ϕy (below the black line) . (1.138)

As we can see, there is a very good matching between

1.4 Landau’s kinetic theory 39

0.05 0.750.05

0.75

0.15 0.25 0.35 0.45 0.55 0.65

0.15

0.25

0.35

0.45

0.55

0.65

εx

[mm mrad]

ε y [

mm

mra

d]

0.45 0.55 0.2

0.3

0.47 0.49 0.51 0.53

0.22

0.24

0.26

0.28

εx

[mm mrad]

ε y [

mm

mra

d]Figure 1.9: Left: stability chart in the emittances plane for the skewquadrupolar mode. The yellow points are stable, the red ones unstable.The FODO cell parameters are KF = 11.7 m−2, KD = 12.3 m−2, `F = `D =0.2 m, `O = 0.3 m (⇒ L = 1 m), the perveance is ξ = 2. Right: detail ofthe left plot.

yellow zone (stable) ⇔ zone Iand

red zone (unstable) ⇔ zone II.

There is therefore a strong numerical evidence that in the periodic focusingcase the condition which determines the instability of the skew quadrupo-lar mode is given by (1.138) (region II). We notice that condition (1.138) isequivalent to the one for the instability of the skew quadrupolar mode inthe constant focusing case (see (1.119)).

1.4 Landau’s kinetic theory

In this section we will discuss in some details the Landau’s kinetic theory fora 2D Coulomb system. By means of this approach we can include the effectsof Coulomb collisions in a mean field framework. The kinetic theory allows usto obtain a deeper understanding of the dynamics of the system, in particularwe can investigate the relaxation process towards the thermodynamic equi-librium as a function of the system parameters (number of particles, spacecharge and confining force strength).The Landau’s theory is a phenomenological approach to the problem of colli-


sional effects in the dynamics of N -body systems. Landau (1936) and Chan-drasekhar (1942) first developed the kinetic theory for systems of interactingparticles via long range forces respectively in case of plasmas and stellar sys-tems. Later on, Lenard and Balescu developed a more precise (but also moreformal) kinetic theory that allows the removal of some divergences whichwere present in the original Landau’s theory.

1.4.1 Derivation of the Fokker-Planck equation

In the Landau’s theory binary Coulomb collisions are included in a meanfield approach as a Wiener stochastic process. Let’s consider the momentumchange of a test particle on the time interval ∆s

∆p = −∂H∂r

∆s+ ∆c p, (1.139)

the first contribution is due to mean field and the second one is the randomterm due to collisions. Assuming collisions are

- instantaneous (i.e. binary, multiple collisions effects are neglected),

- frequent,

- soft,

we can write the following diffusion equation (Poisson-Vlasov-Fokker-Planckequation) for the single particle phase space distribution

∂ρ

∂s+ [ρ,H] = −

2∑

i=1

∂

∂pi(Kiρ) +

1

2

2∑

i,j=1

∂

∂pi∂pj(Di,jρ) , (1.140)

where H is the (mean field) single particle Hamiltonian. The drift term K

and the diffusion coefficient Dij are given by

K =

⟨∆cp

∆s

⟩Dij =

⟨∆cpi ∆cpj

∆s

⟩(1.141)

where the average 〈·〉 is taken over the realization of the stochastic process.The equations of motion for the test particle in presence of a stochastic termare (Langevin equations)

dr

ds= p

dp

ds= −∂H

∂r+ K +

√DQ(s)

(1.142)


Lab. frame CM frame

p

pout

Θθ

p

out

p

’

’

u

u

u

uout

−out

−

Figure 1.10: Kinematic of a 2D binary collision in the laboratory frame (leftpanel) and in the center of mass frame (right panel).

where D is the diffusion matrix and Q(s) ≡ (Qx(s), Qy(s)) is a Wienerstochastic process

〈Qi(s)〉 = 0 〈Qi(s)Qj(s′)〉 = δijδ(s− s′). (1.143)

To obtain an analytic expression for the drift and diffusion terms let’s con-sider a binary collision process: we denote by p the initial momentum of thetest particle and by p′ the one of a generic field particle (see figure 1.10);

p(out),p′(out) are the momenta after the process. The momentum change ofthe test particle in the collision is given by p(out) − p. The drift term is

K = N

∫dp′ρ(r,p′, s)‖p− p′‖

∫ π

−π

dθdσ

dθ(p(out) − p), (1.144)

where N is the number of particles in the system, dσdθ

is the differential crosssection for a 2D Coulomb collision in the laboratory frame and θ is the anglebetween p(out) and p. It is convenient to perform the integration over thescattering angle after moving to the center of mass frame. The coordinateschange is

P = p + p′

u = 12(p − p′)

. (1.145)

For the test particle

p =P

2+ u p(out) =

P

2+ u(out), (1.146)


where

u(out) =

(cos Θ − sin Θsin Θ cos Θ

)u (1.147)

and Θ is the scattering angle in the center of mass frame. Finally we have

p(out) − p = u(out) − u =

(cos Θ − 1 − sin Θ

sin Θ cos Θ − 1

)p − p′

2. (1.148)

Denoting by dσdΘ

the differential cross section in the center of mass frame andrecalling that

dσ

dθdθ =

dσ

dΘdΘ and

dσ(Θ)

dΘ=dσ(−Θ)

dΘ(1.149)

inserting (1.148) in (1.144), we get

K = −N2

∫dp′ρ(r,p′, s)(σ0 − σ1)‖p − p′‖(p − p′) , (1.150)

where σk are the moments of the differential cross section in the center ofmass frame which depend on u = ‖u‖ = 1

2‖p − p′‖

σk =

∫ π

−π

dΘdσ

dΘcosk Θ . (1.151)

The expression for the diffusion coefficient is

Dij = N

∫dp′ρ(r,p′, s)‖p − p′‖

∫ π

−π

dθdσ

dθ(p(out) − p)i (p

(out) − p)j,

(1.152)after a little algebra we obtain

D = 4N

∫dp1ρ(r2,p1, s)‖p2 − p1‖× (1.153)

×∫ π

−π

dΘdσ

dΘsin2 Θ

2

sin2 Θ2u2

1 + cos2 Θ2u2

2 (sin2 Θ2− cos2 Θ

2)u1u2

(sin2 Θ2− cos2 Θ

2)u1u2 cos2 Θ

2u2

1 + sin2 Θ2u2

2

,

where u1, u2 are respectively the x and y components of u. In a real binarycollision process the total kinetic energy is conserved so that the mean ki-netic energy per particle remains constant. We notice that in the Landau’s


approach the stochastic force is such that the mean kinetic energy of a testparticle does not change as a consequence of collisions. We recall that

(∂ρ

∂s

)

coll

= −2∑

i=1

∂

∂pi(Kiρ) +

1

2

2∑

i,j=1

∂

∂pi∂pj(Di,jρ) , (1.154)

and the variation of the mean value of the kinetic energy for a test particleis given by

(d

ds

⟨p2

x + p2y

2

⟩)

coll

=1

4

∫dr dp dp′ ρ(r,p, s) ρ(r,p′, s)×

×(σ0 − σ1)‖p − p′‖(p2 − p′2) . (1.155)

where p2 = ‖p‖2 and p′2 = ‖p′‖2. In the integral (1.155) the term

ρ(r,p, s) ρ(r,p′, s)(σ0 − σ1)‖p − p′‖ (1.156)

is symmetric under the change p ↔ p′; on the contrary the term

p2 − p′2 (1.157)

is anti-symmetric under the same transformation. As a consequence theintegral (1.155) vanishes

(d

ds

⟨p2

x + p2y

2

⟩)

coll

= 0 . (1.158)

1.4.2 The 2D Coulomb cross section

To specialize the drift and the diffusion coefficients in case of 2D Coulombcollisions, we need the expression for the differential cross section in thecenter of mass frame (dσ/dΘ) for a 2D Coulomb scattering process. Let’sconsider the motion of a test particle in the potential Vint which representsthe collisional part of the Coulombian interaction in two dimensions (see1.3.1)

Vint(r) = − ξ

Nlog

(r

ΛD

)θ(ΛD − r) (1.159)

where θ(x) is the Heaviside step function and ΛD is the Debye length (seeAppendix B). We denote by b the impact parameter of the test particle, byΘ the scattering angle and by E = u2 and L = bu respectively its energy


u

u , E , L

Ο

Ο

out

outu = u

Λ D

b ∆φ∆φ

Figure 1.11: Collision in the CM frame.

and angular momentum in the center of mass frame (see figure 1.11). Thedifferential cross section for this process is defined as

dσ

dΘ≡ − db

dΘ. (1.160)

Using polar coordinates and letting w ≡ 1/r the inverse radial distance andby φ the polar angle, the energy conservation for the scattering process reads

L2

(dw

dφ

)2

+ L2w2 +ξ

Nlog(wΛD)θ(wΛD − 1) = E . (1.161)

The inversion point of the radial motion is wmax = b−1 in the case of freepropagation (b > ΛD). For a scattering process (b < ΛD) the inversion pointwmax is the solution of the nonlinear equation

b2w2max +

ξ

NElog(ΛDwmax) = 1 (1.162)

and wmax b < 1 < wmaxΛD. The scattering angle is Θ = π − 2∆φ where ∆φis obtained by integrating (1.161) over w in the interval [0, wmax]. Splittingthe integral into [0,Λ−1

D ] and [Λ−1D , wmax] and setting w′ = w/wmax in the

second integral we obtain

∆φ = arcsin

(b

ΛD

)+

∫ 1

1/(ΛDwmax)

dw′√

1 − w′2 − α logw′α ≡ ξ

NE b2 w2max

.

(1.163)The integral can be evaluated numerically after removing the square rootsingularity at w′ = 1 with the change of variable v′ =

√1 − w′. The removal

of the remaining singularity is hard and the convergence is linear with theinverse of the number of integration points. Simple asymptotic expressionshold for ξ/(NE) 1 and ξ/(NE) 1.


Case ξ/(NE) 1. In this case we replace α with ξ/(NE) and we split theintegration interval into two regions [b/ΛD, 1] and [1/(ΛDwmax), b/ΛD]. Thefirst integral is evaluated by changing the integration variable in equation(1.163) according to v′ =

√1 − w′ and expanding the integrand up to first

order in ξ/(NE). To evaluate the second one we replace the integrand with(1 − b2/Λ2

D)−1/2 and multiply it by the integration interval Λ−1D (w−1

max − b)which is approximated (using equation (1.162)) by b ξ/(2NEΛD) log(ΛD/b).Finally we obtain

Θ =ξ

NE

−2

∫ √1−y

0

log(1 − w2)

w2 (2 − w2)3/2dw − y log y√

1 − y2

=

ξ

NEarccos(y)

(1.164)where y = b/ΛD. Inverting (1.164) we obtain

b = ΛD cos

(π

2

Θ

Θmax

), (1.165)

where

Θmax =π

2

ξ

NE, (1.166)

and the differential cross section is given by

dσ

dΘ=π

2

ΛD

Θmax

sin

(π

2

|Θ|Θmax

)Θ ∈ [−Θmax,Θmax] . (1.167)

Finally we can compute an approximated expression for the moments (σk)of the differential cross section in the center of mass frame which enters inthe integral for the drift coefficient (see (1.151)). We have

σ0 − σ1 = 4 ΛDπ − 2

π2Θ2

max . (1.168)

Case ξ/(NE) 1. In this limit the hard sphere cross section is recovered.We approximate the solution of (1.162) with ΛDwmax ' 1 + (1 − b2/Λ2

D) ≡1+δ, where δ 1. Changing the variable v′ = 1−w′ in (1.163) and retainingonly the first order in the small parameter δ we obtain

cos

(Θ

2

)= sin

(arcsin

(b

ΛD

)+ 2

NE

ξ

b

ΛD

(1 − b2

Λ2D

)1/2)

=

=b

ΛD+ 2

NE

ξ

b

ΛD

(1 − b2

Λ2D

)(1.169)


0.0 2.00.0

2.0

0.5 1.0 1.5

0.5

1.0

1.5

Θ/Θmax

(dσ/

dΘ)Θ

max

/ΛD

0.0 1.00.0

1.0

0.5

0.5

Θ / π

(dσ

/dΘ

)/Λ

D

Figure 1.12: Left: scaled cross section (dσ/dΘ)Θmax/ΛD as a function ofΘ/Θmax computed for different values of ξ

NE; diamonds ξ

NE= 1, dotted line

ξNE

= 10−1, dashed line ξNE

= 10−2, stars ξNE

= 10−3, solid line asymp-

totic expression for ξNE

small. For ξNE

≤ 10−4 the perturbative and exactresult differ less than 10−4 and cannot be distinguished on the plot. Right:scaled cross section (dσ/dΘ)/ΛD as a function of the angle Θ for the firstorder perturbative expansion in ξ

NE; filled diamonds exact, empty diamonds

perturbative for ξNE

= 1; stars exact, solid line perturbative for ξNE

= 0.1;

empty circles exact, dashed line perturbative for ξNE

= 0.001.

and after inverting

b = ΛD cosΘ

2

(1 − 2

NE

ξsin2 Θ

2

). (1.170)

The differential cross section is

dσ

dΘ=

ΛD

2sin

|Θ|2

Θ ∈ [−π, π] (1.171)

and its moments satisfy

σ0 − σ1 =8

3ΛD . (1.172)

In figure 1.12 we compare the exact (numerical) behaviour of the cross sec-tion with the asymptotic expressions above presented.

By using (1.168) and (1.172) we can construct a global approximation for


-4.0 4.0-5.0

1.0

-2.0 0.0 2.0

-4.0

-3.0

-2.0

-1.0

0.0

log(α )

log(

σ 0-σ1)

Figure 1.13: Behaviour of σ0 − σ1. The black plot is the exact curve, the redone is the approximated expression (1.173).

the quantity σ0 − σ1, its expression is

σ0 − σ1 =

4ΛDπ − 2

π2Θ2

max forξ

NE< c

8

3ΛD for

ξ

NE> c .

(1.173)

where c =(

83(π−2)

)1/2

and Θmax is defined in (1.166) . The common endpoint

is chosen in order to make the approximation continuous. The numericalanalysis shows that (1.173) provides a good estimation of the exact (numer-ical) value of σ0 − σ1 (see figure 1.13).The moments difference (1.173) (and consequently the drift term K (1.150))depends linearly on the cut-off distance ΛD. We notice however that theseparation between the collisional and the mean field contribution of theCoulombian interaction we made in section 1.3.1 is only qualitatively cor-rect: there could be a numerical multiplicative factor (not known!) betweenthe appropriate value of the cut-off distance (which enters in (1.173)) andthe Debye length ΛD. This inaccuracy in the calculation of the collisionaloperator is particularly important in 2D Coulomb systems because of thelinear dependence of the drift term from the cut-off distance. As a conse-quence the numerical reliability of the Landau’s theory in this case needs tobe accurately checked. This is not true for 3D systems where the drift termdepends on the logarithm of the cut-off distance. In general, in this context,


the uncertainty related to the exact value of the cut-off doesn’t cause seriousproblems.

1.4.3 Scaling law for the relaxation time to Boltzmann

equilibrium

As a consequence of Coulomb collisions any initial distribution in phase spacerelaxes to the Maxwell-Boltzmann distribution (ρMB). Finite N effects selectthe MB distribution among all possible stationary solutions of the Poisson-Vlasov equation. To define the collisional relaxation time for the system let’sconsider first a simplified expression for the Fokker-Planck equation wherethe drift and the diffusion terms are given by

K = −β p Dij = 2β kBT δij , (1.174)

where β is a constant, kBT is the temperature of the system and δij is theKronecker symbol. By studying the Langevin equations associated to theFokker-Plank equation (see (1.142)), it is not difficult to show that the arbi-trary initial distribution approaches exponentially the MB equilibrium. Thecharacteristic time scale of the process (i.e. the collisional relaxation time) isgiven by

τrelax ∼ 1/β . (1.175)

Resuming on a general case where K and D are given by (1.150) and (1.152),defining a “drift coefficient” in the following way

β ∼ −〈K · p〉〈 ‖p‖2〉 (1.176)

where the average 〈·〉 is taken over the distribution ρ, we expect that

τrelax ∼ 1/β . (1.177)

Scaling with N in symmetric systems. In this section we will restrictour analysis to symmetric systems (i.e. ωx0 = ωy0 = ω0 + round spatialdistributions). The drift term (1.150) can be cast in the following form

K = −β(r, p)p (1.178)


where the coefficient β(r, p) is given by

β(r, p) = N

∫ ∞

0

p′dp′ρ(r, p′, s)×

×∫ π

0

dφ (p2 + p′2 − 2 p p′ cosφ)1/2 (σ0 − σ1)

(1 − p′

pcosφ

)(1.179)

(for symmetric systems the phase space density ρ depends only on r = ‖r‖and p = ‖p‖).We will show that for an arbitrary distribution in the limit for N large

β ∼ O

(1

N

), (1.180)

then, assuming (1.177), we obtain the following scaling law for the relaxationtime:

τrelax proportional to N .

The proof of (1.180) is the following. We split the integration domain[0,+∞[×[0, π] in (1.179) into two regions were the two asymptotic expres-sions (1.173) hold:

• Region I:ξ

NE≤ c so that u2 ≥ ε

4

• Region II:ξ

NE> c so that u2 <

ε

4

where ε ≡ 4ξ

Nc. Solving the equation

u2 ≡ 1

4(p2 + p′

2 − 2 p p′ cosφ) =1

4ε (1.181)

with respect to p′, we find two roots p′ = p±(φ) where

p±(φ) = p cosφ± (ε− p2 sin2 φ)1/2. (1.182)

For p <√ε the roots are real and of opposite sign p+ > 0, p− < 0. For

p =√ε the root p+ = 2 p cos φ is positive if 0 ≤ φ ≤ π/2 whereas p− = 0.


Region I Region I

Region IIRegion II

0 πφ0 φ π

p’p’Region II

Region I

0 π

p’

p< ε p= ε p> ε

φφ*

Figure 1.14: Regions I and II for different values of p.

For p >√ε the roots are both real positive in the intervals [0, φ∗], both real

negative in the interval [π− φ∗, π], complex in the interval [φ∗, π−φ∗] where

sinφ∗ =√ε/p 0 < φ∗ <

π

2. (1.183)

As a consequence the regions I and II are defined in a different way dependingon p being smaller, equal or larger than

√ε (see figure 1.14):

• Case p <√ε:

- Region I: p′ ≥ p+(φ)

- Region II: 0 ≤ p′ < p+(φ)

• Case p =√ε:

- Region I: p′ ≥ 2p cosφ 0 ≤ φ ≤ π2

p′ ≥ 0 π2≤ φ ≤ π

- Region II: 0 ≤ p′ ≤ 2 p cosφ 0 ≤ φ ≤ π2

• Case p >√ε:

- Region I: 0 ≤ p′ ≤ p−(φ) p′ ≥ p+(φ) 0 ≤ φ ≤ φ∗p′ ≥ 0 φ∗ ≤ φ ≤ π

- Region II: p−(φ) ≤ p′ ≤ p+(φ) 0 ≤ φ ≤ φ∗

We notice that for every value of p we can find a sufficiently large N such thatp >

√ε so that φ∗ < π/2 and p−(φ) > 0. This is the third case discussed

in the previous list. Moreover in the limit N → ∞ we have φ∗ → 0 and


p±(φ) → p. Partitioning the integral for β (see (1.179) into the regions I eII we can write

β = βI + βII . (1.184)

We first examine the behaviour of βII. Using the appropriate expression forσ0 − σ1 given by (1.173) we have

βII =8

3N

ΛD

p

∫ φ∗

0

dφ

∫ p+(φ)

p−(φ)

dp′p′ ρ(r, p′, s)(p2 + p′2 − 2 pp′ cosφ)1/2(p− p′ cosφ) .

(1.185)We perform the following change of variables

v1 =p− p′ cosφ√

ε

v2 =p′ sinφ√

ε

(1.186)

one can easily verify that p′dp′dφ = εdv1dv2 and that the integration domainis changed into a circle of radius 1. The integral becomes

βII =8

3N

ΛD

pε2∫ 1

−1

dv1

∫ √1−v2

1

0

dv2 ρ(r, p′(v1, v2, ε), s) v1

√v21 + v2

2 (1.187)

where p′(v1, v2, ε) =√

(p−√ε v1)2 + ε v2

2.Since we are interested in the N → ∞ limit (i.e. ε → 0) we can expandρ(r, p′(v1, v2, ε), s) in a power series of

√ε near ε = 0. The result is

ρ(r, p′(v1, v2, ε), s) = ρ(r, p, s) −√ε v1

∂ρ

∂p

∣∣∣∣p

+O(ε). (1.188)

Inserting into the integral we obtain

βII =8

3N

ΛD

pε2∫ 1

−1

dv1

∫ √1−v2

1

0

dv2

(ρ(r, p, s) −

√ε v1

∂ρ

∂p

∣∣∣∣p

+O(ε)

)v1

√v2

1 + v22 .

(1.189)At order zero in

√ε the integral vanishes since the integrand is an even

function of v1 whereas the integration domain is symmetric with respect tothe origin. Therefore we can write

βII = −8

3N

ΛD

pε5/2∂ρ

∂p

∣∣∣∣p

∫ 1

−1

dv1

∫ √1−v2

1

0

dv2v21

√v21 + v2

2 + O(ε3).

(1.190)


For N large βII satisfies

NβII = −4π

15

(4ξ

c

)5/2ΛD

p

∂ρ

∂p

∣∣∣∣p

1√N

+O

(1

N

)(1.191)

and it vanishes in the N → ∞ limit.The behaviour of βI is examined in a similar way starting from the expression

βI = 16 ΛDξ2

N

π − 2

p

∫ ∫

Region I

p′dp′ dφρ(r, p′, s)

(p2 + p′2 − 2 pp′ cosφ)3/2(p− p′ cos φ).

(1.192)In this case the required change of coordinates is

v1 = p− p′ cos φ

v2 = p′ sinφ .

(1.193)

We easily check that p′dp′dφ = dv1dv2, p′ =

√(p− v1)2 + v2

2 and the inte-gration domain is mapped into the half plane v2 > 0 deprived of the half diskof radius

√ε centered at the origin. Changing to polar coordinates

v1 = v cosψ

v2 = v sinψ

(1.194)

we obtain

NβI = 16 ΛD ξ2 π − 2

p

∫ +∞

√ε

dv

∫ π

0

dψρ(r, p′(v, ψ), s) cosψ

v(1.195)

where p′(v, ψ) =√

(p2 − v cosψ)2 + v2 sin2 ψ.We split the second integral into two parts according to

∫ π

0

· · ·dψ =

∫ π/2

0

· · ·dψ +

∫ π

π/2

· · ·dψ , (1.196)

in the second one we perform the replacement ψ = π − ϕ renaming thevariable ϕ again ψ, we obtain the final result which reads

NβI = 16 ΛD ξ2 π − 2

p

∫ +∞

√ε

dv1

v×

×∫ π/2

0

dψ [ρ(r, p′(v, ψ), s) − ρ(r, p′(v, π − ψ), s)] cosψ . (1.197)


We recall that

p′(v, ψ) =√p2 + v2 − 2 p v cosψ

p′(v, π − ψ) =√p2 + v2 + 2 p v cosψ,

(1.198)

therefore limv→0 p′(v, ψ) = limv→0 p

′(v, π − ψ) = p , as a consequence thefunction within square brackets in (1.197) vanishes at least linearly as v goesto zero. We can conclude that in the limit N → ∞ (implying ε → 0) theintegral in the r.h.s of (1.197) is finite and in particular we have

limN→∞

NβI = 16 ΛD ξ2 π − 2

pF(r, p) (1.199)

where

F(r, p) ≡∫ +∞

0

dv1

v

∫ π/2

0

dψ [ρ(r, p′(v, ψ), s) − ρ(r, p′(v, π − ψ), s)] cosψ .

(1.200)Collecting the two intermediate results (1.191) and (1.199) we finally obtainthat:

β scales as 1/N in the limit for N large.

In a similar way we can prove that the same scaling holds also for the

drift coefficient D.

We notice that since the collisional operator of the Fokker-Planck equationvanishes as 1/N for N → ∞, we recover in this limit the Poisson-Vlasovequation.The proof we have already discussed is based on the hypothesis that thephase space distribution ρ is a smooth function of its coordinates (see for in-stance (1.188)). The KV does not satisfy this hypothesis but still in this casewe can prove (see Appendix C) that for a (space averaged) KV d istribution6

β reaches a constant limit for p→ 0 which scales as N−1 and for p→ ∞ theproduct β p3 tends to a constant which also scales as N−1.

The scaling with 1/N of the β coefficient has been checked also numeri-cally in case of the KV and MB distributions. The results are shown in

6 The space averaged KV distribution is defined as

ρKV (p) ≡ 1

πR2

KV

∫ RKV

0

ρKV (r, p) 2πrdr =1

4π2kBTR2

KV

θ(4kBT − p2)

where RKV is the radius of the system and kBT is the temperature.


0.0 4.00.0

10.0

1.0 2.0 3.0

2.5

5.0

7.5

p2

Nβp

0.0 4.00.0

20.0

1.0 2.0 3.0

5.0

10.0

15.0

p2

Nβp

Figure 1.15: Left: plot of the drift coefficient β multiplied by Np as a functionof p2 for the space averaged KV distribution. The parameters are RKV =1mm, kBT=0.25, ξ=1 and we have chosen ΛD = 1 mm. The diamonds repre-sent the exact result obtained for N = 103; the solid line is the approximateresult obtained with the matched approximations for the cross section. Stars,exact for N = 105; dashed line, approximate. Right: same plots (and sameparameters) as in the left panel for the momentum part of the MB distribu-tion: ρp(p) = 1

2πKBTexp(−p2/2kBT ).

figure 1.15. The calculations have been made utilizing the exact numericalexpression and the approximated analytical formula (1.173) for the momentsof the differential cross section. The scaling with N is clear and no significantdifferences occur when we replace the exact moments with the approximatedones.As a final remark we notice that the kinetic theory of 2D Coulomb sys-tems has been studied independently also by Chavanis[36] using a differentapproach based on the Lenard-Balescu equation. This approach is more pre-cise but also more formal compared to the Landau’s method and permitsthe removal of the divergence at large scales in the drift and in the diffusionterms showing that the Debye length is the natural scale of the collisional partof the Coulombian interaction, without introducing any other ad hoc largescale cut-off. The expression for the drift force as a function of the velocityobtained in [36] agrees remarkably well with our numerical results presentedin figure 1.15. The agreement is good (up to a factor 2(π − 2)/π) also forthe analytical asymptotic expressions of the drift term we have derived inAppendix C.


Scaling law for Gaussian beams. The calculations to determine thecollisional relaxation time of the system are generally rather complicated butin case of a Gaussian phase space distribution we can derive a scaling low forthe relaxation time as a function of all the system parameters. The analyticalexpression of the Gaussian (GA) distribution is

ρGA =e− 1

2

„

x2

〈x2〉+y2

〈y2〉+p2x

〈p2x〉

+p2y

〈p2y〉

«

(2π)2

√〈x2〉〈y2〉〈p2

x〉⟨p2

y

⟩ . (1.201)

This distribution is not a stationary solution of the Poisson-Vlasov equa-tions but its parameters can be chosen in such a way that ρGA is “r.m.s.matched”, i.e. the second order phase space moments don’t change in time.To do this we first construct the stationary KV distribution related to thegiven values of the system parameters (ωx0, ωy0, ξ, εx, εy), then the valuesof the coefficients 〈x2〉 , 〈y2〉 , 〈p2

x〉 ,⟨p2

y

⟩of the suitable r.m.s. matched GA

distribution are given by (1.55)7. The Gaussian distribution is widely used inbeam physics because it describes the real beam profile with good accuracy.Let’s define a (mean) drift coefficient according to (1.176), after a little alge-bra we get

〈K · p〉 =

∫dr dp ρGA K · p ∼ −ΛD

ξ2

N

√〈p2

x〉 +⟨p2

y

⟩√

〈x2〉〈y2〉〈p2x〉⟨p2

y

⟩ , (1.202)

where K is given by (1.150) and we have utilized the approximated expres-sion (1.173) for the moments of the differential cross section. We recall thatthe characteristic length scale of the collisional part of the Coulombian in-teraction scales as the Debye length

Λ2D ∼ kBT

ξ

√〈x2〉〈y2〉, (1.203)

where kBT = (〈p2x〉+

⟨p2

y

⟩)/2 and

√〈x2〉〈y2〉 is the way in which the average

size of the system scales. Since 〈 ‖p‖2〉 = 〈p2x〉 +

⟨p2

y

⟩, by using (1.176) and

(1.177) we obtain

τ 2relax ∼ β

−2 ∼ N2

ξ3

√〈x2〉〈y2〉

⟨p2

x

⟩ ⟨p2

y

⟩. (1.204)

We see that also in case of non-symmetric systems we have a linear depen-dence of the relaxation time from the number of particles.

7For a detailed description of r.m.s. equivalent beams see [37].


Equation (1.204) gives only the scaling properties of τrelax, to obtain a pre-cise value for the relaxation time we need to know its value in a particularcase. Denoting by τ

(0)relax the (“experimental”) relaxation time in the reference

case (corresponding to the parameters ω(0)0x , ω

(0)0y , ξ

(0), ε(0)x , ε

(0)y , N (0)) then the

relaxation time in any other case (ω0x, ω0y, ξ, εx, εy, N) is given by

τrelax = τ(0)relax

f(ω0x, ω0y, ξ, εx, εy, N)

f(ω(0)0x , ω

(0)0y , ξ

(0), ε(0)x , ε

(0)y , N (0))

(1.205)

where

f(ω0x, ω0y, ξ, εx, εy, N) =

(N2

ξ3

√〈x2〉〈y2〉

⟨p2

x

⟩ ⟨p2

y

⟩)1/2

, (1.206)

and 〈x2〉 , 〈y2〉 , 〈p2x〉 ,

⟨p2

y

⟩are given by (1.55). The behaviour of τrelax as a

function of some system parameters is shown in figure 1.16. On the top-leftplot is shown the behaviour of τrelax as a function of the maximum verticaltune depression (σy) for different values of N , we see that τrelax → ∞ asthe space charge strength goes to zero. On the top-right plot we have thedependence of the relaxation time from the horizontal bare tune for differentvalues of the space charge strength, we see that τrelax depends weakly fromνx0. On the last plot we have the behaviour of τrelax as a function of thehorizontal emittance for different values of the space charge strength.


0.50 1.000.00

10.00

0.6 0.7 0.8 0.9

2.0

4.0

6.0

8.0

σy

τ rela

x

5.00 8.000.00

1.50

6.0 7.0

0.25

0.50

0.75

1.00

1.25

νx0

τ rela

x

10 400.00

1.50

20 30

0.25

0.50

0.75

1.00

1.25

εx [mm mrad]

τ rela

x

Figure 1.16: Top-left: τrelax (arbitrary units) as a function of σy (maximumvertical tune depression) for N = 2048 (red), N = 4096 (green) and N =8192 (blue); the bare tunes are ν0x = 6.22, ν0y = 6.21. Top-right: τrelax asa function of the tune ν0x (ν0y = 6.21) for σy = 0.8 (red), σy = 0.7 (green)and σy = 0.8 (blue) with N = 2048. In both cases the r.m.s. emittancesare εx/εy = 30/10 mm mrad. Bottom: τrelax as a function of εx (εy=10mm mrad) for σy = 0.8 (red), σy = 0.7 (green) and σy = 0.6 (blue), withν0x = 6.22, ν0y = 6.21 and N = 2048.

Chapter 2

The simulation code

In this chapter we will present the code MPI – DYNAM which allows usto study collisional effects in the dynamics of a 2D Coulomb system. Thiscode, written in C language and parallelized with MPI, provides the numer-ical integration of the equations of motion for the Coulomb system in theconstant and in the periodic focusing case (FODO cells) by using symplecticmethods. Different types of initial distributions are considered. The coreof the simulation code is the routine for the evaluation of the electric field.This algorithm, based on a multipolar expansion of the far field performedafter a hierarchical space splitting, has an optimal computational complexity(N logN) compared to the full direct calculation (N 2).

2.1 General features of the code

The code MPI – DYNAM (developed by the author starting from 2002 inFORTRAN77 and then translated into C language) provides the numericalintegration of the Hamilton’s equations of motion for a 2D system of N inter-acting bodies (via Coulomb forces) confined by a linear field. The numericalsimulation code allows us to investigate directly the effects of Coulomb colli-sions in the dynamics of the system and, in particular, the relaxation processtowards the thermodynamic equilibrium. We can test the validity limits ofthe mean field theory (where collisions are neglected) as well the predictionsand the general reliability of the Landau’s kinetic theory which is based onseveral assumptions that need to be checked carefully.Let’s rewrite again the equations of motion for the system (see section 1.2).Denoting by ri ≡ (xi, yi) the coordinates of particle i, the equations of mo-

60 The simulation code

tion read

d2xi

ds2= −kx0 xi +

ξ

2Ex,i

d2yi

ds2= −ky0 yi +

ξ

2Ey,i

for i = 1, · · · , N (2.1)

where the (normalized) electric field acting on particle i is given by

Ei =2

N

N∑

j=0, j 6=i

ri − rj

‖ri − rj‖2. (2.2)

The parameters kx0, ky0 depend on s in the periodic focusing case (kx0(s +L) = kx0(s) , ky0(s+L) = ky0(s), L is the period), otherwise they are constant(in this case we have kx0 = ω2

x0 , ky0 = ω2y0). The perveance ξ is related to the

total charge in the system and determines the space charge strength. Thetotal Hamiltonian of the system is

HN =

N∑

i=1

(p2

x,i + p2y,i

2+

1

2kx0 x

2i +

1

2ky0 y

2i

)− ξ

N

∑

i<j

log(rij) (2.3)

where rij = ‖ri − rj‖.

Initial conditions

Equations (2.1) have to be integrated starting from a suitable initial con-dition. Usually we choose the initial particle distribution according to astationary (or r.m.s. matched) solution of the Poisson-Vlasov equation suchas the KV or the Gaussian.To generate a KV particles distribution (for the analytical expression seesection 1.3.2) we use a direct mapping of the 3D unitary cube into the hy-persurface of the 4-dimensional ellipsoid in phase space defined as

x2

A2x

+p2

x

ω2xA

2x

+y2

A2y

+p2

y

ω2yA

2y

= 1 , (2.4)

where Ax, Ay are the theoretical semiaxes of the system and ωx, ωy are thedepressed phase advances. The transformation which gives the coordinates

2.1 General features of the code 61

of a generic test particle in a KV system is

x = Ax√η cos(2πηx)

px = Ax ωx√η sin(2πηx)

y = Ay

√1 − η cos(2πηy)

py = Ay ωy

√1 − η sin(2πηy) ,

(2.5)

where η, ηx, ηy are independent random variables uniformly distributed in[0, 1].The generation of a 4D Gaussian distribution (see section 1.3.2) is based onthe Box-Mueller method. Denoting by 〈x2〉 , 〈p2

x〉 , 〈y2〉 ,⟨p2

y

⟩the (theoretical)

parameters of the Gaussian (i.e. the variances of each degree of freedom),the coordinates of a generic test particle are given by

x =√〈x2〉

√−2 log η1 cos 2πηx

px =√

〈p2x〉√−2 log η1 sin 2πηx

y =√〈y2〉

√−2 log η2 cos 2πηy

py =√⟨

p2y

⟩√−2 log η2 sin 2πηy

(2.6)

where the four parameters η1, ηx, η2, ηy are random variables uniformly dis-tributed in [0, 1].Both in the KV and Gaussian cases a (little) reshaping of the generated dis-tribution could be sometimes necessary in order to compensate partially thesmall oscillations of the centroid of the system (which should be at rest) orthe ripples in the r.m.s. apertures of the charge distribution. These effectsare due to the fact that the moments of the generated distribution and theforces acting on each particle are slightly different from the theoretical onesbecause of the statistical fluctuations related to the finite number of particles.

Symplectic integration

In any N -body simulation code, the numerical integration of the equationsof motion represents certainly one of the crucial points. The reliability of


the results in some cases depends strongly on the numerical schema adopted.Since the 2D system of Coulombian oscillators is Hamiltonian, we have uti-lized mainly symplectic algorithms which automatically guarantee the (ap-proximated) conservation of all the invariant of the motion.We notice that the total Hamiltonian of the system (2.3) can be split intotwo contributions

HN = TN + VN (2.7)

where TN and VN are respectively the kinetic and the potential energy part.Denoting by X a generic point in the total phase space of the system

X = (r1,p1, r2,p2, · · · , rN ,pN) , (2.8)

the solution of the equations of motion (2.1) can be formally written by usingthe following Lie series

X = Sexact(s)x = esDHN x ≡∞∑

k=0

skD(k)HN

x

k!, (2.9)

where x, X are respectively the initial condition and the solution at time s;the operator DHN

is the Lie derivative associated to HN and is defined as

DHNA(X) = [A(X), HN(X)] , (2.10)

where [·, ·] are the Poisson’s brackets. Of course we don’t have an explicit

expression for the evolution operator Sexact(s), but the operator defined as

S(2)(∆) ≡ exp

(∆

2DTN

)exp (∆DVN

) exp

(∆

2DTN

), (2.11)

and which can be explicitly evaluated1, satisfies

Sexact(∆) = S(2)(∆) +O(∆3) . (2.12)

Thus (2.11) gives an approximation of the exact Hamiltonian flux in thetime interval ∆ with an error O(∆3) and since it is a composition of three

1Assuming for sake of simplicity a one dimensional system we have

esDT

(qp

)=

(q + ps

p

),

and

esDV

(qp

)=

(q

p − V ′(q)s

).

2.1 General features of the code 63

symplectic maps, it represents a second order symplectic integrator for theexact Hamiltonian HN . The schema (2.11) is known as symmetric leap frogintegrator and is widely used in N -body simulations.The predictions obtained with the second order integrator have been checkedby comparing them with an higher order method. We notice that startingfrom (2.11) we can construct a fourth-order symplectic schema by consideringthe following chain of second order symplectic integrators

S(4)(∆) = S(2)(α∆) S(2)(−β∆) S(2)(α∆) , (2.13)

where the parameters α and β satisfy

2α− β = 1

2α3 − β3 = 0 .(2.14)

We notice that (2.13) is symmetric under the transformation ∆ → −∆ (timereversal), as a consequence it approximates the exact Hamiltonian flux in thetime interval ∆ with an error which must be an odd power of ∆

Sexact(∆) = S(4)(∆) + C3∆3 + C5∆

5 + · · · . (2.15)

The first condition in (2.14) assures that the net time increment is equal to∆, the second one assures that the error at third order in ∆ vanishes, sofinally we have that S(4)(∆) effectively approximates the exact Hamiltonianevolution with an error O(∆5).In the simulation code both the second and fourth order schemes can beutilized, but usually the simulations are performed by using the leap frogalgorithm which is, for a fixed time step, three times faster then the otherone. We utilize the fourth order integrator when we need to validate theresults obtained with the faster scheme.The choice of the integration time step is a very delicate problem. In factmany time scales are involved in the dynamics of the system. The character-istic time scale of the collective effects (Tdyn) is given by the mean oscillationtime of the particles

Tdyn ∼ 2π√ω0xω0y

. (2.16)

Since we are interested in direct particle-particle interactions and collisionaleffects, also the time scale corresponding to the “mean free time”2 has to betaken into account. Denoting by R the average size of the system, by N the

2Defined as the time required to a generic test particle to cover the mean interparticledistance.


number of particles and by kBT the temperature of the system the mean freetime (Tfree) is given by

Tfree ∼√

R2

NKBT. (2.17)

UsuallyTfree Tdyn , (2.18)

and to investigate correctly collisional effects in order to study the relaxationtowards thermodynamic equilibrium the suitable integration time step ∆should be 100 ÷ 1000 times smaller then Tfree. The choice of a very smallintegration step affects considerably the length of the simulations and makesthem extremely demanding in terms of CPU time required. As a consequencethe number of particles that can be simulated is typically restricted to N ∼103 ÷ 104. On the other hand, if we choose a time step such that

Tfree ∆ ∼ 1

50Tdyn , (2.19)

all collisional effects are practically washed out (because the time step cannot resolve them) and we are investigating the behaviour of the system inthe (quasi) mean field approximation.

Data analysis and post processing

The core of the simulation code usually works in batch mode (simulations arevery long). The simulated data can be analyzed (also in real time) by using asuite of post processing routines which allow us to study the evolution of ther.m.s. quantities (second order moments) of the system such as emittances,temperatures, apertures, and the behaviour of some other higher order mo-ments (for instance, fourth order moments have been utilized to study therelaxation process to the Boltzmann equilibrium). We can also obtain thephase plots (x, y) , (x, px) , (x, py) , · · · of the particles distribution as well thefrequency map of all the particles which is an useful tool to analyze linearand nonlinear resonances of the system.

2.2 The electric field computation

Apart from the integration of the equations of the system, an other crucialpoint which may affect considerably the performances of the code (in termsof velocity) is the computation of the electrostatic forces acting on all the Nparticles. In Vlasov (PIC) codes, the electric field acting on each particle is

2.2 The electric field computation 65

i Ω

C2C1 C3 C4

C5 C C C C9876

Figure 2.1: Partitioning of the space into a hierarchy of domains. The par-ticles inside Ω (yellow domain) are the neighbours of particle i. All theremaining particles are organized into the clusters Cj.

computed starting from the Poisson’s equation (1.35) where we approximatethe charge density distribution by a smooth function (mean field approxima-tion). These codes are based on Poisson’s solvers generally fast and accurate,but if we are interested in studying collisional effects in Coulomb system, thenwe have to take into account the direct particle-particle interaction removingthe mean field assumption.Let’s rewrite again the expression for the electric field acting on particle i(see (2.2))

Ei =2

N

N∑

j=1 ,j 6=i

ri − rj

‖ri − rj‖2, (2.20)

the direct computation of (2.20) for i = 1, 2, · · · , N requires O(N 2) calcula-tions and becomes demanding in terms of CPU time as N increases.In the code MPI–DYNAM the evaluation of the electric field is based on ahierarchical scheme [19, 21, 20]. The force acting on particle i is partitionedinto two contributions: the near field and the far field, according to figure2.1

Ei =∑

j∈Ω ,j 6=i

Eij +∑

j 6∈Ω

Eij , Eij =2

N

ri − rj

‖ri − rj‖2. (2.21)


O

r

r

R

C

ki

D j

jC

C,i

i

j

εs(j)

Figure 2.2: Interaction of particle i with the cluster Cj.

The first term is computed by direct evaluation of binary interactions. Forthe second one we organize the distant particles into a hierarchical clustersstructure (cells Cj), then the electric field generated by each cell Cj on particlei is expressed via a multipolar expansion usually truncated to the quadrupoleterm. For instance, the contribution of cluster Cj onto particle i is given by

Ei, Cj= E

(1)i, Cj

+ E(2)i, Cj

+ E(4)i, Cj

+ · · · , (2.22)

where E(m)i,Cj

is the m-pole term. According to figure 2.2, let’s define

nCj= number of particles inside Cj

RCj= center of cell Cj

ε(j)s = position of s-th particle inside cell Cj (rk ≡ RCj

+ εs)

Di ,Cj= ri − RCj

, (2.23)

and we have

Ei, Cj=

∑

particles in Cj

2

N

ri − rk

‖ri − rk‖2=

=

nCj∑

s=1

2

N

ri − RCj− ε

(j)s

‖ri − RCj− ε

(j)s ‖2

=

=

nCj∑

s=1

2

N

Di ,Cj− ε

(j)s

‖Di ,Cj− ε

(j)s ‖2

. (2.24)


If condition‖Di ,Cj

‖ ‖ε(j)s ‖ (2.25)

holds, we can expand in a Taylor series equation (2.24). After a little algebrawe get the following expression for the m-poles

E(1)i, Cj

= 2Q(1) di ,Cj

‖Di ,Cj‖ , (2.26)

E(2)i, Cj

= 22(Q(2) · di ,Cj

)di ,Cj

− Q(2)

‖Di ,Cj‖2

, (2.27)

E(4)i, Cj

= 24(di ,Cj

· Q(4)di ,Cj

)di ,Cj

− 2Q(4)di ,Cj− Tr

[Q(4)

]di ,Cj

‖Di ,Cj‖3

, (2.28)

where

di ,Cj=

Di ,Cj

‖Di ,Cj‖ , (2.29)

and the multipolar coefficients of the cell Cj are given by

Q(1) =nCj

N(total charge in Cj) , (2.30)

Q(2) =1

N

nCj∑

s=0

ε(j)s (dipole moment) , (2.31)

Q(4) =1

N

nCj∑

s=0

[ε(j)

s

]·[ε(j)

s

]†(quadrupole moment) . (2.32)

(2.33)

Evaluation of multipolar coefficients Q(1), Q(2) and Q(4) is made hierarchicallyby organizing cells into a quad-tree structure (see fiure 2.3). We denote byl = 0, 1, 2, · · · , Lmax the levels of the quad-tree. At the root of the tree (whichis the coarsest refinement level of the spatial greed and corresponds to l = 0)we have the whole space partitioned into 16 cells. In general, each cell atlevel l generates four children cells at the subsequent level l + 1. Then, thenumber of cells at level l is given by

n(l)cells = 22(2+l) l = 0, 1, 2, · · · , Lmax . (2.34)

The maximum refinement level Lmax is fixed by the following condition

n(Lmax)cells = N ⇒ Lmax =

1

2log2N − 2 . (2.35)


l+1

l

. . . . . . . . . . . . . . . . . . .

l+2

. . . . . . . . . . . . . . . . . .

Figure 2.3: Quad-tree structure for the hierarchy of cells Cj.

Once we know the multipolar coefficients at the finest refinement level of thehierarchy we can easily reconstruct recursively the coefficients at any level0 ≤ l < Lmax simply by following the quad-tree branches. For example,denoting by Q

(1)l the monopole coefficient of cell Cj at level l (the red one

in figure 2.3) and by Q(1)l+1(k) (k = 1, 2, 3, 4) the monopole coefficients of the

corresponding four children cells at level l + 1 (the blue ones in figure 2.3),then we have

Q(1)l =

4∑

k=1

Q(1)l+1(k) . (2.36)

In a similar way we determine the dipole and the quadrupole terms.Once we know all the multipolar coefficients at any order, it is not difficult toshow that the computation of the electric field acting on a single particle re-quires O(logN) operations and since we have to repeat this calculation for allthe N particles, then the total computational complexity of the electric fieldcalculation is O(N logN) (to compare with the direct computation whichrequires O(N 2) operations). The performances of this scheme are shown infigure 2.4. We plot the CPU time per particle, corresponding to a singleevaluation of the electric field, as a function of the total number of particles.The red line represents the results obtained with the hierarchical scheme,the green line the ones obtained with the classical O(N 2) algorithm. We


0.00

0.10

10^2 10^3 10^4 10^5 10^6 10^7

0.02

0.04

0.06

0.08

N

TC

PU/N

[m

s]

Figure 2.4: CPU time per particle corresponding to a single evaluation ofthe electric field as a function of the total number of particles. The red linerepresents the results obtained with the hierarchical scheme, the green linethe ones obtained with the classical O(N 2) algorithm.

notice that the tree code becomes faster than the direct computation whenN ≥ 500 ÷ 1000. The calculations have been done on a CPU Intel Xeon 2.6GHz.The accuracy in force evaluation is documented in figure 2.5 where we plot,as a function of N , the mean relative error on the electric field modulus (solidline) defined as

⟨∆E

E

⟩≡ 1

N

N∑

i=1

‖E(approx)i − E

(exact)i ‖

‖E(exact)i ‖

, (2.37)

where the term E(exact)i is computed by direct summation of 2.2. In the same

plot we also show (dashed line) the 99th percentile of the errors distribution((∆E/E) 99%). We see that with a truncation to the quadrupolar term in themultipolar expansion of the far field, the mean error in force calculation isabout 10−3 ÷ 10−4.The accuracy of the method can be improved retaining more terms in theexpansion of the far field term. We adopt in this case the following technique:first we compute the Taylor expansion of the electric (far) field up to order


10^2 10^3 10^4 10^5 10^610^-4

10^-3

10^-2

10^-1

N

<∆E

/E>

, (∆E

/E) 99

Figure 2.5: Mean relative error on the electric field modulus 〈∆E/E〉 (solidline) and 99th percentile of the errors distribution (∆E/E) 99% (dashed line).

10 in the center of each cluster-cell. The Taylor series is reconstructed us-ing the Pade approximants technique. With a [2/3] approximant computedfrom a Taylor truncation of order 5 we reach an accuracy of 10−8 withoutcompromising the performances of the code.

2.3 The parallelization

The simulation code has been parallelized by using MPI (Message PassingInterface) in order to speed-up the simulations when a large number N ofparticles is considered. The parallelization scheme, based on a binary treestructure, is the following. Each processor computes the Coulomb forces act-ing on a sub set of Nsing = N/np particles where np is the number of availableprocessors. Processors are organized into a one dimensional torus and dataexchange among them is realized by using a “send receive” MPI function.The scheme adopted for communications, which is organized into severalsteps, is the following: at communication step number one the processor Pk

sends data related to the forces acting on its Nsing particles to processor Pk+1

and receives data from Pk−1. In general, at step number j, the processor Pk

sends all the data collected at step j − 1 to Pk+2j−1 and receives data from

2.3 The parallelization 71

# proc 0 1 2 3

com

m. s

tep

s 1

2

Figure 2.6: Parallelization scheme. Data exchange among the processorsrequires log2 np communication steps. In the plot we have np = 4, twocommunication steps are required.

N/np 1 2 4 8 161000 1.07 0.66 0.46 0.37 -2000 2.51 1.48 0.98 0.74 0.634000 5.19 3.05 1.99 1.48 1.348000 12.3 7.06 4.43 3.14 2.8316000 26.9 15.7 9.45 6.78 5.6432000 69.6 39.6 23.0 16.1 11.9

Table 2.1: Performances of the code. Execution times (in seconds) as afunction of N and np for a short simulation corresponding to 100 integrationtime steps.

Pk−2j−1 . The communication scheme is shown in figure 2.6. Completion ofdata exchange among the processors requires log2 np steps (obviously thenumber of processors must be a power of 2).In table 2.1 we show the performances of the code as a function of N (parti-cles number) and np (number of processors). The values (in seconds) are theexecution times for a short simulation corresponding to 100 integration timesteps. The tests have been performed on an IBM Cluster (28 bi-processorsIntel Xeon, CPU clock 2.8 GHz, 1Gbytes RAM, 3 Gbits/sec fast ethernet in-terconnection) at the Department of Physics, University of Bologna (Italy).


0 500.0

2.0

10 20 30 40

0.5

1.0

1.5

s [m]

<x2 >

, <

y2 > [

mm

2 ]

Figure 2.7: Plot of 〈x2〉 (black) and 〈y2〉 (red) as a function of the times. The diamonds are the prediction obtained with the moments method atorder two.

2.4 Numerical tests

The code has been tested accurately during the last two years by comparingits predictions with the ones obtained by using more traditional PIC codesand/or with some theoretical results (e.g. moments method). We have chosen4 different situations and in all cases the results obtained with the code MPI–DYNAM were in very good agreement with the reference data.

Envelope oscillations - constant focusing

Let’s consider a KV system in the constant focusing case, the parameters are

ωx0 = 1.095 rad/m ωx/ωx0 = 0.8ωy0 = 1.0 rad/m ωy/ωy0 = 0.6

ξ = 2.0 ,

and the unperturbed semiaxes of the charge distribution are Ax0 = 2.5794mm and Ay0 = 1.5087 mm. We perturb the stationary charge profile modi-fying the semiaxes of the initial distribution according to

Ax0 → Ax = Ax0(1 + ηx) Ay0 → Ay = Ay0(1 + ηy) , (2.38)

2.4 Numerical tests 73

0 50010

30

100 200 300 400

15

20

25

s [m]

r.m

.s. e

mitt

ance

s [m

m m

rad]

Figure 2.8: Montague resonance, low space charge. Plot of εx (black) andεy (red) as a function of the time s. The diamonds are the results obtainedwith the PIC code ORBIT.

and we choose ηx = 0.1, ηy = −0.05. In this case the charge distributionis no longer stationary: the semiaxes change periodically in time (mismatchoscillations). In figure 2.7 we show the results of a simulation performedwith the MPI–DYNAM code (the number of particles is N = 20000 and theintegration step is ∆ = 0.05 m). The solid lines are the behaviour of thesecond order spatial moments: 〈x2〉 (black) and 〈y2〉 (red); the diamondsare the predictions obtained with the moments method at order two. Theagreement is excellent.

The Montague resonance - constant focusing

The Montague resonance is a (non linear) space charge emittance couplingwhich occurs when the bare phases advances of the system are nearly equal.In a simulation, when we are in the vicinity of the resonance condition(ωx0 = ωy0), the r.m.s. emittances don’t remain constant. They tend toequalize and the (dynamical) “equipartition” level depends on the spacecharge strength.

– Case I: low space charge.Let’s consider an initial Gaussian distribution, the parameters are


0.0 15.010.0

30.0

3.0 6.0 9.0 12.0

15.0

20.0

25.0

s [m]

r.m

.s. e

mitt

ance

s [m

m m

rad]

10

15

20

25

30

0 2.5 5 7.5 10 12.5 15

turns

ε rms [

mm

mra

d]

εx

εy

Figure 2.9: Montague resonance, high space charge. Left panel: plot of εx(black) and εy (red) obtained with the code MPI–DYNAM. Right panel:results obtained with the PIC code MICROMAP.

ωx0 = 0.9968 rad/m ωy0 = 1.0 rad/m ωy/ωy0 = 0.9829 (in the center)εx = 30 mm mrad εy = 10 mm mrad .

The vertical tune depression ωy/ωy0 is referred to the value at the centerof the charge distribution (maximum tune depression). In figure 2.8 weshow the behaviour of the emittances in a simulation with N = 50000 and∆ = 0.075 m, the solid lines are respectively the horizontal (black) and thevertical (red) r.m.s. emittances (εx, εy); the diamonds are the results ob-tained with the PIC code ORBIT (see the S. Cousineau web page at ORNLhttp://www.ornl.gov/∼cp3/CERNBenchmark/Results). Also in this casethe agreement is very good.

– Case II: high space charge.The parameters are the same as in Case I but now we increase by a factor5 the space charge strength taking ωy/ωy0 = 0.9. As before we plot the be-haviour of the r.m.s. emittances as a function of the time s. In figure 2.9 (leftpanel) we have the results obtained with the code MPI–DYNAM (the num-ber of particles is N = 12000 and the integration step is ∆ = 5 · 10−4 m); onthe right panel we have the results obtained with the PIC code MICROMAP(for more details see the [42]). Also in this case the agreement is good butwe notice that in this case the dynamics of the system is strongly dependenton the parameters, in particular the granularity and/or the statistic fluctu-ations due to the small number of particles utilized can affect considerably

2.4 Numerical tests 75

0 3000.0

1.5

100 200

0.5

1.0

s [m]

r.m

.s.

emitt

ance

s [

mm

mra

d]

0 1000.0

1.5

20 40 60 80

0.5

1.0

s [m]

r.m

.s.

emitt

ance

s [

mm

mra

d]

Figure 2.10: Skew quadrupolar instability in the periodic focusing case. Plotof εx (black) and εy (red) obtained by using the code MPI-DYNAM as afunction of the time s. Left panel: comparison with the PIC code HALODYN(diamonds). Right panel: comparison with the results obtained by using themoments method at order two (diamonds).

the behaviour of the curves in the vicinity of the point where εx ' εy.

Skew quadrupolar instability - periodic focusing case

Let’s consider a KV system in the periodic focusing case (FODO cell). Weconsider a strong anisotropy between the horizontal and the vertical planes sothat the skew quadrupolar coherent mode becomes unstable. The parametersof the system are

KF = 18.8m−2 KD = 12.2m−2

`F = `D = 0.2m `O = 0.3m (L = 1.0m)ξ = 2.0 εx = 1.25 mm mrad εy = 0.2 mm mrad .

In figure 2.10 we show the behaviour of the r.m.s. emittances of the system(εx black, εy in red) obtained with the code MPI–DYNAM (in the simulationN = 20000 particles and ∆ = 1 · 10−3 m). On the left panel the results arecompared with the ones obtained with the PIC code HALODYN (diamonds),on the right plot the comparison has been made taking the predictions ofthe moments method at order two (squares). Again the agreement betweensimulated data and theoretical results is good. Of course we can not expectthat the moments method gives the correct emittances behaviour over a longtime because the instability brings the system rapidly far away from theinitial KV profile where the moments method works.


0 2500.20

0.30

50 100 150 200

0.22

0.24

0.26

0.28

s [m]

r.m

.s.

emitt

ance

s [

mm

mra

d]

Figure 2.11: Octupolar instability in the periodic focusing case. Plot of εx(black) and εy (red) as a function of the time s. The diamonds are the resultsobtained with the PIC code HALODYN.

Octupolar instability - periodic focusing case

We consider again a KV system in a region of the parameter space where anoctupolar instability is excited. The parameters of the system are

KF = 12.1m−2 KD = 11.9m−2

`F = `D = 0.2m `O = 0.3m (L = 1.0m)ξ = 2.0 εx = 0.25 mm mrad εy = 0.25 mm mrad .

In figure 2.11 we show the behaviour of the r.m.s. emittances εx (black)and εy (red) in a simulation with N = 20000 and ∆ = 1 · 10−3 m. Theresults are compared to the ones obtained with the PIC code HALODYN(diamonds). The agreement is quite good considering the extreme sensitivityof the dynamics from the system parameters and from the simulation details.

2.5 Future developments

In future developments of the code MPI–DYNAM we will deal with three-dimensional systems. We recall that our aim is to study mainly collisionaleffects (relaxation to Maxwell-Boltzmann equilibrium) in Coulomb systems

2.5 Future developments 77

and, as a matter of fact, real Coulomb systems (like beams) are essentiallythree-dimensional. However two-dimensional models are important becausethe numeric analysis is somewhat easier in this case, furthermore they areuseful to understand (on a qualitative basis) the dynamical mechanisms ofthe system which we expect will be the same also in the 3D case.Before moving definitively to the 3D case, we will consider an hybrid model.According to the discussion in section 1.1 concerning the derivation of themodel, we modify the (normalized) electric field acting on a generic testparticle according to3

Ei =∑

j 6=i

Eij Eij =

2

N

ri − rj

‖ri − rj‖2‖ri − rj‖ >

`

2

`

N

ri − rj

‖ri − rj‖3‖ri − rj‖ <

`

2,

(2.39)

where ` (see (1.9)) is the mean interparticle distance in the original 3D beam.By using this technique we can mime the effect of a three-dimensional col-lision between point particles still keeping the advantages of a 2D system.The related numerical code is under development.Concerning the pure 3D case, we have already developed a three-dimensionalscalar version of the simulation code. The electric field computation is basedagain on a multipolar expansion of the far field performed after a hierarchicalspace splitting. Again the computational complexity of the force calculationis reduced from N 2 to N logN but the method is less performing than thecorresponding bi-dimensional version. The performances of this scheme areshown in figure 2.12. We plot the CPU time per particle corresponding to asingle evaluation of the electric field as a function of the total number of par-ticles. The red line represents the results obtained with the 3D hierarchicalscheme (oct-tree), the green line the ones obtained with the classical O(N 2)algorithm. The mean error in the modulus of the electric field is shown infigure 2.13. Again with a quadrupolar truncation of the far field the meanerror in force calculation is about 10−3. The tests to validate and improve(in terms of velocity/accuracy) the code are still under consideration.

3We recall that Coulomb force between two point particles behaves as 1/r2 in 3D andas 1/r in 2D.


0.00

0.20

10^2 10^3 10^4 10^5 10^6

0.04

0.08

0.12

0.16

N

TC

PU/N

[m

s]

Figure 2.12: CPU time per particle corresponding to a single evaluation ofthe electric field (3D solver) as a function of the total number of particles.The red line represents the results obtained with the hierarchical scheme, thegreen line the ones obtained with the classical O(N 2) algorithm.

10^2 10^3 10^4 10^5 10^610^-4

10^-3

10^-2

N

<∆

E/E

>

Figure 2.13: Mean relative error on the electric field modulus 〈∆E/E〉 forthe hierarchical schema in 3D with a quadrupolar truncation in the far field.

Chapter 3

Numerical studies

In this chapter we will present some numerical results obtained with thecode MPI–DYNAM concerning the 2D Coulomb oscillators system we havepresented in chapter 1. We will consider the relaxation process in case ofKV systems and we will show that, as predicted by the Landau’s collisionaltheory, the relaxation time towards thermodynamic equilibrium scales asN keeping fixed all the other system parameters. Then we will analyzethe statistical properties of a Coulomb collision process starting from thesimulated data. Later on we will give a short insight into the relaxationprocess in case of time dependent systems. In the last part of this chapterwe will show some examples of the interplay between collisional effects andnon-linear phenomena in the case of the so called “Montague resonance ”.

3.1 The collisional relaxation process in KV

systems

We consider a bidimensional Coulomb oscillators system whose initial parti-cles distribution is the KV (see section 1.3.2). Let’s suppose the system isisotropic, i.e.

ωx0 = ωy0 = ω0 ωx = ωy = ω Ax = Ay = R (3.1)

so the particles are uniformly distibuted within a disc of radius R. We choosethe following parameters for the system

ω0 = 1.0 rad/m ω/ω0 = 0.64 ξ = 2 N = 2000R = 1.84 mm εx = εy =0.542 mm mrad kBT = 0.347.

In figure 3.1 we show the initial particles position, the radial density, momen-tum and energy distributions. We have run a simulation starting from the

80 Numerical studies

-2.50 2.50-2.50

2.50

-1.25 0.00 1.25

-1.25

0.00

1.25

x [mm]

y [

mm

]

0.0 3.00.00

0.20

1.0 2.0

0.05

0.10

0.15

r [mm] ρ s(r

)

0.0 2.50.00

2.00

0.5 1.0 1.5 2.0

0.50

1.00

1.50

p

ρ p(p)

0.0 2.00.00

2.00

0.5 1.0 1.5

0.50

1.00

1.50

E

ρ E(E

)

Figure 3.1: From top to bottom, from left to right: initial particles position,initial radial density, momentum and energy distributions. The histogramsare the simulated data, the green plots are the (theoretical) initial profiles,the red curves are the distributions at thermodynamic equilibrium.

3.1 The collisional relaxation process in KV systems 81

0 80000

25

2000 4000 6000

5

10

15

20

s [m]

χ2 (s)

χ2 test

Figure 3.2: χ2 test for the radial density distribution (red plot) and for themomentum distribution (green plot).

initial condition depicted in figure 3.1, the integration step was ∆ = 5 · 10−5

m. The characteristic time scales of the system are (see section 2.1)

Tdyn ∼ 6 m Tfree ∼ 7 · 10−2 m (3.2)

and the integration time step satisfies

∆

Tfree∼ 10−3 , (3.3)

so Coulomb collisions are completely resolved. In the mean field approxima-tion the initial condition we have chosen should be stationary; actually in thesimulation performed with the collisional code we see that the phase spacedistribution of the system changes in time. We have studied the evolutionof the spatial, momentum and energy distribution of the particles checkingtheir agreement (while the simulation was going on) with the correspondingones in the MB case by using a χ2 test. The values of the (reduced) χ2

test as a function of the time s for the spatial density (red) and momen-tum (green) distributions are shown in figure 3.2. Finally, in figure 3.3, weshow the simulated radial density, momentum and energy distributions aftera time s = 8000 m, corresponding to the end of the simulation. In the sameplot the red curves are the corresponding MB theoretical profiles. It is clear


0.0 3.00.00

0.20

1.0 2.0

0.05

0.10

0.15

r [mm]

ρ s(r)

0.0 2.50.00

2.00

0.5 1.0 1.5 2.0

0.50

1.00

1.50

p ρ p(p

)

0.0 2.00.00

2.00

0.5 1.0 1.5

0.50

1.00

1.50

E

ρ E(E

)

Figure 3.3: From left to right, from top to bottom: radial density, momen-tum and energy distributions at the end of the simulation (s = 8000 m).The histograms are the simulated data, the red curves are the theoreticaldistributions at thermodynamic equilibrium and the green ones the initialKV profiles.


0 80000.50

1.50

2000 4000 6000

0.75

1.00

1.25

s [m]

Xrm

s(s),

Yrm

s(s)

[m

m]

R.M.S. apertures

0 80000.00

1.00

2000 4000 6000

0.25

0.50

0.75

s [m]

ε x(s),

εy(s

) [

mm

mra

d]

R.M.S. emittances

Figure 3.4: Left panel: r.m.s. apertures of the system. The red plot refersto√〈x2〉, the blue one to

√〈y2〉. Right panel: behaviour of the r.m.s.

emittances. The red and blue plots refer respectively to εx and εy.

that the initial KV condition is not stationary: the system relaxes towardsan equilibrium state described by the MB distribution.It is not difficult to show, by using the moments method within the frame-work of the Landau’s theory (see Appendix D), that in the isotropic case thesecond order moments of the distribution don’t change in time (furthermoreno instability exists at this space charge level). In figure 3.4 we show thebehaviour of the r.m.s. apertures and the r.m.s emittances in the simulation.The small ripples are due to statistical mismatch.To obtain a simple quantitative estimation of the relaxation time we need toconsider the following combination of 4th order moments

y(s) =〈p4

x〉 (s) +⟨p4

y

⟩(s)

2(kBT )2− 2 , (3.4)

where kBT is the temperature of the system. By construction y(0) = 0 (KVdistribution) and y(s) → 1 as s → ∞ (MB distribution). It is not difficultto show that the curve y(s) can be fitted by

yfit(s) = 1 − exp(−γs) , (3.5)

where γ−1 measures the relaxation time. In figure 3.5 we show the behaviourof the function − log(1 − y(s)) for the simulation presented before. By cal-culating the best fit line for the data in plot 3.5 we determine the value of γ


0 80000.0

1.5

2000 4000 6000

0.3

0.6

0.9

1.2

s [m]

-lo

g (1

-y(s

))

Figure 3.5: Plot of the function − log(1 − y(s)) for the simulation describedin the text. The black line is the best fit curve.

in (3.5) and the relaxation time is given then by

τrelax ∼ 1

γ∼ 6450 m . (3.6)

We can compare the value obtained “experimentally” to the one obtainedwith the Landau’s theory. By using the expression for the drift coefficientβ(r, p) given in (1.179) and computing its mean value over the (space aver-aged) KV distribution (defined in footnote 6, pag. 53, chapter 1)

β(KV )

=

∫dp dr β(r, p) ρKV (p) , (3.7)

we obtain

β(KV )

= 8 · 10−4 m−1 ⇒ τ(KV )relax ∼ 1200 m . (3.8)

The discrepancy between (3.6) and (3.7) (approximately a factor 5) is notsurprising because of the uncertainty related on the exact value of the cut-offdistance to include in the expression for the 2D cross section (in this case wehave chosen the Debye length). Our numerical experiments suggest (as wewill see also in subsequent sections) that the cut-off distance scales as theDebye length but the precise value should be chosen approximately 5 ÷ 10times smaller than the Debye length. In general we believe that by using theLandau’s theory in the 2D case we can obtain only the order of magnitudeof the relaxation time but not its precise value.


0 100000.00

1.00

2500 5000 7500

0.25

0.50

0.75

s [m]

y(N)

0.0 4.00.00

1.00

1.0 2.0 3.0

0.25

0.50

0.75

s’=s/N

y(N)

Figure 3.6: Left panel: plot of y(N)(s) for different simulations. The numberof particles is N = 1024 (yellow), 2048 (black), 3000 (red), 4096 (green),5000 (blue), 6000 (light blue) and 8192 (purple). Right panel: plot of y(N)

as a function of the rescaled time s′ = s/N .

3.1.1 Scaling with N of the relaxation time

In section 1.4.3 we have demonstrated that for an isotropic system, keep-ing fixed all the system parameters (ω0, ξ , · · · ) and changing the number ofparticles N , the relaxation time scales according to τrelax ∝ N . We haverun different simulations with the same parameters as in section 3.1, takingN = 1024, 2048, 3000, 4096, 5000, 6000, 8192. For each simulation (with adifferent N) we have computed the quantity y(N)(s) defined in (3.4) and wehave measured the relaxation time.In figure 3.6 (left panel) we show the behaviour of y(N)(s) for different val-ues of N . In the right panel we plot y(N) as a function of the rescaled times′ = s/N : we see that all the curves, except partially the one correspondingto N = 1024, are practically superimposed. This suggests that the charac-teristic time scale of the relaxation process is of O(N), as suggested by theproposed scaling law. In table 3.1 we report the values of τrelax(N)×1/N forall the simulations. We show also the values of τrelax(N)× logN/N which isnot asymptotically constant1.The numerical validation of the scaling law τrelax ∝ N also supports indirectlythe fact that the Debye length ΛD is the correct characteristic length-scale ofthe collisional part of the Coulombian interaction. In fact, if we replace ΛD

1The logarithmic correction to the scaling low proposed for stellar systems[38] seemsnot to hold for two-dimensional charged particles systems.


N τrelax(N) [m] τrelax(N) × 1/N τrelax(N) × logN/N1024 2625 2.6 7.82048 6350 3.1 10.33000 8912 3.0 10.44096 12960 3.2 11.65000 15980 3.2 11.86000 19038 3.2 12.18192 26300 3.2 12.5

Table 3.1: Simulated data: relaxation time corresponding to different valuesof N . The statistical uncertainty of these values is between 3 % and 5 %.

with the mean interparticle distance δ =√πR2/N , as suggested by Jansen

in [40], we obtain that the relaxation time theoretically scales as N 3/2, butthis is not consistent with the simulations.We can use the “experimental” value for the relaxation time (see (3.6)) andthe scaling law to estimate the relaxation time in a real beam. We recall thatthe linear particles density and the mean transversal width of a real (pointparticles in 3D) intense beam are respectively

Np ∼ 1011 ÷ 1012 particles/m Rb ∼ 1 ÷ 10 mm , (3.9)

as a consequence (see section 1.2) the corresponding number of particles inthe 2D models is

N∗ = (NpRb)2/3π1/3 ∼ 105 ÷ 106 , (3.10)

and the relaxation time in a real beam is then given by

τ ∗relax ∼ τrelax(N = 2000)N∗

2000∼ 106 cells . (3.11)

This value is certainly comparable with the storage time in a (storage) ringsuch as the SIS100 at GSI [16] and so collisional effects should be taken intoaccount in the design of high intensity accelerators.

3.2 Statistical properties of Coulomb colli-

sions

The main idea of the Landau’s kinetic approach is that collisional effects ina Coulomb system2 can be thought as a Wiener stochastic process super-imposed upon the Vlasov mean field dynamics: denoting by ∆p the total

2We recall that collisions are supposed to be small, independent and frequent.

3.2 Statistical properties of Coulomb collisions 87

momentum change of a test particle in the time interval ∆s we have

∆p = −∂H∂r

∆s + ∆c p, (3.12)

where −∂H∂r

∆s is the mean field contribution and ∆c p is the stochastic (i.e.collisional) one. We want to check this hypothesis starting directly from thesimulated data, in particular we want to investigate the statistical propertiesof the stochastic-collisional term.We have run a simulation starting from an isotropic KV initial condition withthe same parameters as in section 3.1, the number of particles was N = 8192.The simulation length, during which each particle performs, on the average,ten complete oscillations, is much shorter compared to the relaxation timeof the system: as a consequence we expect the charge distribution doesn’tchange significantly during the simulation. We define

ri(sk), pi(sk) i = 1, 2, · · · , N ; k = 0, 1, · · · , K (3.13)

as the simulated position and momentum of particle i at time step sk =k∆s, where ∆s = 2 · 10−3 m is the discretization step. In the mean fieldapproximation each particle performs harmonic oscillations with frequencyω, as a consequence the “noise” B(∆s)i,k due to collisions satisfies

ri(sk+1) = ri(sk) + pi(sk)∆s

pi(sk+1) = pi(sk) − ω2ri(sk)∆s + B(∆s)i,k

i = 1, 2, · · · , N ; k = 0, 1, · · · , K .

(3.14)

In figure (3.7) we show the sample paths of the process B(∆s) for a generictest particle.Starting from the simulated data we have computed the distribution of thevalues of B(∆s), the results are shown in figure 3.8. It is clear that thedistribution is not Gaussian. The plot can be fitted by using the followingexpression

f(b) =

c1 exp(− b2

2σ2

)|b| ≤ b∗

c2 |b|−α |b| > b∗ .

(3.15)


0 100-0.002

0.002

2.5e+04

0

s [m]

B∆s

x

25 50 75

-0.001

0.000

0.001

0 100-0.002

0.002

2.5e+04

0

s [m]

B∆s

y

25 50 75

-0.001

0.000

0.001

Figure 3.7: Sample paths (B(∆s)x , B

(∆s)y ) of a typical collisional process com-

puted starting from the simulated data.

0.0 2.50.5 1.0 1.5 2.010^-3

10^-2

10^-1

10^0

10^1

10^2

10^3

10^4

B(∆s)x X 10-3

N (

b)

Figure 3.8: Distribution of the values of B(∆s)x (normalized to one) computed

from the simulated data. We have shown only the positive branch of thedistribution, the negative one is symmetric. The red and green curves arethe best-fit curves given by (3.15).

3.2 Statistical properties of Coulomb collisions 89

The normalization condition∫f(b) db = 1 and the continuity of f(b) in b = b∗

impose that

1 =√

2πσ2 Erf

(b∗√2σ2

)c1 + 2

b1−α∗

α− 1c2 , c1 exp

(− b2∗

2σ2

)= c2 b

−α∗ .

(3.16)The distribution in (3.15) can be interpreted in the following way: the coreof the distribution is Gaussian and describes the effect of (many) small con-tributions due to distant particles (frequent soft collisions), while the tails,which decrease algebraically, are determined by the (few) nearest neighboursparticles (rare hard collisions). The parameters α, σ, b∗ in (3.15) have beenchosen minimizing the quantity

∆2(σ, α, b∗) =

∫ ∞

−∞|N(b) − f(b; σ, α, b∗)|2db (3.17)

where N(b) is the distribution of B(∆s) (normalized to one) obtained fromthe simulations. The minimization has been done by using a Montecarlotechnique. The best-fit parameters are

α(fit) = 3.99 , b(fit)∗ = 10.4 · 10−5 , σ(fit) = 5.26 · 10−5 . (3.18)

A distribution of the form (3.15) has been found independently by Chavanis.[36,39] More precisely he analyzed the statistics of fluctuations of the force pro-duced at the origin of the reference system by a random distribution of pointparticles in 2D. We recall that the expression of the force at the origin is

F =ξ

N

N∑

i

ri

r2i

(3.19)

and in reference[39] it is shown that the distribution of the values of F isgiven by

f(F) =N2

π2n ξ2 logNexp

(− N2

π n ξ2 logNF 2

)F F ∗

f(F) =n ξ2

N2F 4F F ∗ (3.20)

where

F ∗ ∼(πn ξ2

N2logN

)1/2

log1/2(logN) , (3.21)

and n = N/πR2 is the spatial particles density. From (3.20), considering itsprojection into one plane (for example Fy = 0), we obtain theoretically the


values of the parameters α, σ, b∗ which enter in (3.15). After a little algebrawe get

α(theor) = 3 , (3.22)

b(theor)∗ =

(ξ2 logN

NR2log(logN)

)1/2

∆s = 10.6 · 10−5 , (3.23)

σ(theor) =

(ξ2 logN

2NR2

)1/2

∆s = 5.1 · 10−5 , (3.24)

and we see that the agreement between simulated data (see (3.18)) and the-oretical values is very good as far as b∗ and σ are concerned, in case of α thediscrepancy is significant.

3.3 Relaxation in time-dependent systems

The relaxation process in time-dependent systems is still an open problem.We have considered the collisional relaxation process in case of an initial KVsystem in a FODO lattice. The parameters are

KF = 12.0 m−2 KD = 12.0 m−2

`F = `D = 0.2 m `O = 0.3 m (L = 1.0 m)ξ = 1.0 εx = 0.25 mm mrad εy = 0.25 mm mrad .

We have run three different simulations changing the number of particlesN = 2048, 4096, 8192 (the integration step was ∆ = 5 · 10−5 m). In fig-ure 3.9 we show the evolution of the mean value of the r.m.s. emittances(ε = (εx + εy)/2) as a function of the time (number of FODO cells) forthe three simulations. We see that Coulomb collisions are responsible for acontinuous emittances growth (it seems there is no saturation level) and wecan infer that in this case doesn’t exist an asymptotic stationary (or evenperiodic) equilibrium state as in the constant focusing case. Furthermorethe growth rate of ε is proportional to N (see the table above), accordingto the fact that the collisionality level of the system scales as 1/N even inthe periodic focusing case. The continuous growth of the emittances can bedescribed analytically by using the moments method described in AppendixD.We notice that, in all the simulations presented, we observe also an emittancegrowth which is not related to collisions (see for instance the red curve in theinterval 0 < s < 100, the green curve in 300 < s < 400 and the blue one in1000 < s < 1200). From PIC simulations we know that the working point wehave chosen is in the vicinity of an unstable domain of the parameters space.

3.4 The collisional relaxation process in anisotropic Gaussian systems

91

0 20000.24

0.30

500 1000 1500

0.25

0.26

0.27

0.28

0.29

#FODO cells

(εx+

ε y)/2

N slope slope ×N2048 2.3 · 10−5 0.0474096 1.02 · 10−5 0.0428192 0.56 · 10−5 0.046

Figure 3.9: Evolution of the mean value ε = (εx + εy)/2 of the r.m.s. emit-tances in the periodic focusing case. The colors refer to different values of N :2048 (red), 4096 (green) and 8192 (blue). In the table we show the growthrate of ε as a function of N .

In fact if KF − KD = δ 6= 0, we observe an exponential emittance growthwhich saturates after 50 − 100 FODO cells. This behaviour is due to theexcitation of an unstable collective resonance of the system (octupolar reso-nance). It is clear that the collisional noise plays some role in the excitationof this unstable resonance, indeed (see figure 3.9) this transient disappearsin the limit N → ∞ (the mean field limit) according to PIC simulations.

3.4 The collisional relaxation process in ani-

sotropic Gaussian systems

In this section we will deal with the relaxation process in anisotropic systems.In particular we will give a detailed insight into the collisional equipartitionmechanism. Real beams are often out of Boltzmann equilibrium and arecharacterized by different temperatures in the horizontal and the verticalplanes. In these conditions a wide class of (mean field) instabilities can beexcited and provide a mechanism to exchange energy between the two degreesof freedom. This process is known as dynamical equipartition and generallyoccurs over a time scale of O(1) compared to the characteristic time scale ofthe system. In the absence of instabilities the thermodynamical (collisional)


0 200000

40

time

Temperatures

0 2000010

30

time

Emittances

Figure 3.10: Left: qualitative behaviour of the transversal temperatures ofthe system during the collisional equipartition process (black → horizontal,red → vertical). Right: qualitative behaviour of the r.m.s. emittances (black→ εx, red → εy).

equipartition is the only process which provides energy exchange betweenplanes. It is generally slower compared to the dynamical equipartition andit’s characterized by a time scale of O(N). The presence of two differentregimes during the relaxation towards the final statistical equilibrium seemsto be a common feature in the evolution of systems where long-range forcesare involved. See for example references [25, 26, 27] for two-dimensional vor-tices, [28] for plasmas, [8, 29] for stellar systems and [30, 31, 41] for classicalspin systems (HMF).The qualitative behaviour of the second order moments of the particles dis-tribution (temperatures and r.m.s. emittances) during a typical relaxationprocess, is given in figure 3.10. As a consequence of the energy exchange(due to collisions) between the horizontal and the vertical degrees of free-dom, the two temperatures of the system tend to the same value while thefinal emittances are in general different, their equilibrium values depend onthe confining force strength (in this case we have assumed ωx0 > ωy0).In section 1.4.3 we have derived a scaling law for the relaxation time towardsthe Boltzmann equilibrium in case of a Gaussian charge distribution as afunction of all the system parameters. We recall that, since we are dealingwith a scaling law, to obtain the appropriate value of the relaxation time weneed to know its value in a particular case (the reference case, characterized

by ν(0)0x , ν

(0)0y , ξ

(0), ε(0)x , ε

(0)y , N (0) → τ

(0)relax), then the relaxation time in any

3.4 The collisional relaxation process in anisotropic Gaussian systems

93

0 200010

30

500 1000 1500

15

20

25

# cells

ε x ,εy [

mm

mra

d]

R.M.S. Emittances (νx0

= 8.0)

0 100010

30

250 500 750

15

20

25

# cells

ε x ,εy [

mm

mra

d]


= 4.0)

0 150010

30

375 750 1125

15.0

20.0

25.0

# cells

ε x ,εy [

mm

mra

d]


= 5.0)

0 60010

30

150 300 450

15

20

25

# cells

ε x ,εy [

mm

mra

d]


= 7.25)

0 90010

30

300 600

15

20

25

# cells

ε x ,εy [

mm

mra

d]


= 9.0)

Figure 3.11: Equipartition process, behaviour of the r.m.s. emittances fordifferent values of νx0 and σy. Top: νx0 = 8.0 and σy = 0.8 (red, calibration

curve), σy = 0.7 (green), σy = 0.6 (blue). Middle-left: νx0 = 4.0 andσy = 0.8 (red), σy = 0.7 (green). Middle-right: νx0 = 5.0 and σy = 0.8(red), σy = 0.6 (blue). Bottom-left: νx0 = 7.25 and σy = 0.8 (red), σy = 0.7(green), σy = 0.6 (blue). Bottom-right: νx0 = 9.0 and σy = 0.8 (red),σy = 0.7 (green), σy = 0.6 (blue). The black diamonds are the theoreticalpredictions.


other situation (characterized by ν0x, ν0y, ξ, εx, εy, N) is given by

τrelax = τ(0)relax

f(ν0x, ν0y, ξ, εx, εy, N)

f(ν(0)0x , ν

(0)0y , ξ

(0), ε(0)x , ε

(0)y , N (0))

(3.25)

where

f(ω0x, ω0y, ξ, εx, εy, N) =

(N2

ξ3

√〈x2〉〈y2〉

⟨p2

x

⟩ ⟨p2

y

⟩)1/2

, (3.26)

and 〈x2〉 , 〈y2〉 , 〈p2x〉 ,

⟨p2

y

⟩(the parameters of the Gaussian) are given by

(1.55). We have chosen the following calibration point to fix the scale of therelaxation time

νx0 = 8.0 νy0 = 6.21εx = 30 mm mrad εy = 10 mm mrad

σy =νy

νy0

= 0.8 N = 2048 ,

then we have validated the proposed scaling law by comparing its predictionswith the numerical simulations carried out with the collisional code. In allthe simulations performed we have fixed νx0 = 6.21, N = 2048 (the lineardependence of the relaxation time from the particles number has been inves-tigated in [13]) and the emittances εx = 30 mm mrad, εy = 10 mm mrad.The validity of the scaling law has been checked changing the space chargestrength σy = νy/νy0 and the horizontal bare tune νx0. The results are shownin figure 3.11. By using the relaxation time experimentally obtained in thecalibration case, by using (3.25) and the moments equation (see AppendixD) we have obtained all the relaxation curves presented. The values of νx0

we have chosen for the simulations are in the range 4.0 ÷ 10.0, the values ofσy are in the range 0.6 ÷ 0.8. The agreement between simulated data andanalytical estimations is very good: the proposed scaling law allows us toobtain a reliable description of the collisional equipartition process in a widerange of the parameters space. Unfortunately the simulations require a lotof c.p.u. time, as a consequence we have studied simply the initial behaviourof the relaxation curves.

3.5 Interplay between collisional effects and non-linear phenomena in the

Montague resonance 95

0.0 9.010

30

3.0 6.0

15

20

25

# cells

ε x ,εy [

mm

mra

d]


= 6.22)

0 50010

30

125 250 375

15

20

25

# cells

ε x ,εy [

mm

mra

d]


= 6.22)

Figure 3.12: Left: initial emittances evolution (εx → red, εy → blue) forνx0 = 6.22, νy0 = 6.21, σy = 0.8 and εx/εy = 30/10 mm mrad and N = 2048.Right: emittances evolution obtained by using the collisional code, sameparameters as in the left panel.

3.5 Interplay between collisional effects and

non-linear phenomena in the Montague

resonance

3.5.1 Acceleration of the relaxation process near to the

resonance

We have already investigated the relaxation process in anisotropic Gaussiansystems (equipartition) and we have validated the scaling law proposed onthe basis of the Landau’s collisional theory in the absence of collective (meanfield) effects. Let’s consider now the behaviour of the relaxation processwhen νx0 is similar to νy0. In this case we are in the vicinity of the so calledMontague resonance. In a detailed analytical single-particle analysis, Mon-tague [32] pointed out that the space charge provides a non-linear couplingbetween the horizontal and the vertical degrees of freedom which helps en-ergy exchange and may lead to an emittance transfer between x and y planes.The exchange level depends on the space charge strength and on the distancefrom the resonance condition: if νx0 − νy0 = 0 we have the maximum effect.The typical behaviour of the emittances in case of the Montague resonanceis shown in figures 2.8 and 2.9 in chapter 2.To study the interplay between collisional and non-linear effects we choose a


working point near to the resonance condition:

νx0 = 6.22 νy0 = 6.21εx = 30 mm mrad εy = 10 mm mrad

σy =νy

νy0= 0.8 N = 2048 .

The plot in figure 3.12 (left panel) shows an initial emittances exchange, dueto the effect of the resonance, that saturates in a short time (< 5÷ 10 cells).This behaviour is also present in PIC simulations and, as far as the meanfield dynamics is considered, no other significant phenomena arise for longertime scales [42, 43]: after saturating the emittances remain constant. In thesimulations performed with the collisional code, for time scales longer thenfew cells (see figure 3.12 , right panel), we observe that emittances are notconstant: as a consequence of Coulomb collisions their value tends to the oneat thermodynamic equilibrium.We notice that, if we use the proposed scaling law for the relaxation timein order to describe the emittances evolution in this case and we adopt thecalibration point chosen in the previous section, we obtain a wrong value forthe relaxation time as we can see in figure 3.13: the theoretical calculation(green diamonds in the plot) overestimates the right value of the relaxationtime. Nevertheless we observe that the validity of the scaling law is notcompletely compromised. Taking indeed the case νx0 = 6.22, νy0 = 6.21,σy = 0.8, εx/εy = 30/10 mm mrad and N = 2048 as the new reference point,performing different simulations keeping fixed νx0 and changing the spacecharge strength (σy = 0.6, 0.7, 0.8, 0.9), we have that the scaling law stillallows us to describe correctly the relaxation curves (see figure 3.14). Wethen infer that the acceleration of the relaxation process we have found isnot related to a change in the collisionality level of the system, but is morelikely due to an interaction/interplay of the collisional process with the non-linear dynamics in the vicinity of the Montague resonance. We will describelater on the features of this interplay mechanism.In order to investigate more precisely the acceleration of the relaxation pro-

cess near to the resonance, we have considered several simulations keepingfixed the space charge strength (σy = 0.8) and changing the distance fromthe resonance. We have chosen νx0 in the range 4.0 ÷ 9.5. For each simu-lation, starting from the simulated relaxation curves (εx(s), εy(s)), we havecomputed the following quantity

u(s) = − log

(εx(s) − εy(s)

εx(0) − εy(0)

). (3.27)

At least in the initial part of the simulation, the function u(s) depends lin-early on s and its slope is approximately proportional to the inverse of the



0 50010

30

125 250 375

15

20

25

# cells

ε x ,εy [

mm

mra

d]


= 6.22)

Figure 3.13: Evolution of the emittances (εx → red, εy → blue) in the colli-sional code taking νx0 = 6.22, νy0 = 6.21, N = 2048, εx/εy = 30/10 mm rad.The green diamonds are the theoretical prediction. The analytical predictionoverestimates the relaxation time.

0 50010

30

125 250 375

15

20

25

# cells

ε x,ε y [

mm

mra

d]


= 6.22)

Figure 3.14: Evolution of the emittances in the collisional code in the caseνx0 = 6.22, νy0 = 6.21, N = 2048, εx/εy = 30/10 mm rad, the colors refer toσy =0.6 (red), 0.7 (green), 0.8 (blue) and 0.9 (light blue). The black curvesare the theoretical predictions obtained taking σy = 0.8 as calibration point.


0 10000.0

0.4

250 500 750

0.1

0.2

0.3

s

u’(s

)

3.0 10.00.0

10.0

4.0 5.0 6.0 7.0 8.0 9.0

2.5

5.0

7.5

νx0

acc

eler

atio

n ra

te

acceleration of the relax. process

Figure 3.15: Left: plot of the function u′(s) = u(s)τrelax for simulationswith N = 2048, εx/εy = 30/10 mm mrad, σy = 0.8, νx0 = 5.0 (black), 6.22(red), 6.25 (green), 6.3 (blue), 6.5 (yellow), 8.0 (light blue), 9.5 (purple) andνy0 = 6.21. Right: acceleration rate of the relaxation process compared tothe theoretical case. The blue line is the resonance condition.

relaxation time. If the proposed scaling law describes correctly the behaviourof the relaxation time, the quantity u′(s) = u(s)τrelax still has a linear de-pendence on s but now the slope should not be dependent on the chosensimulation (i.e. on the working point) and we expect that all the curves aresuperimposed. The results are shown in figure 3.15. As we know, if we con-sider working points far from the resonance (e.g. νx0 = 5.0, 6.5, 8.0, 9.5) thescaling holds, but if νx0 − νy0 → 0 (see νx0 = 6.3, 6.25, 6.22) the “experimen-tal” relaxation time becomes shorter and shorter compared to the theoreticalvalue. Still in figure 3.15 (right panel) we show the acceleration rate of therelaxation process as a function of νx0. We see that for νx0 = 6.22 the re-laxation time is 10 times shorter compared to the theoretical expectationwhereas for νx0 = 6.25 it is 5 times shorter.

3.5.2 How collisions interact with the resonance

To understand, at least qualitatively, the basic mechanism which causes theacceleration of the relaxation process in the vicinity of the resonance, it’suseful to consider the tune footprint of the particles. To construct the tunefootprint at a certain time s, we need to consider a simulation segment oflength ∆s centered on s. The length of the segment should be short enough



5.00 6.255.00

6.25

particles in the center

resonant particles

5.25 5.50 5.75 6.00

5.25

5.50

5.75

6.00

νx0

ν y0TUNE FOOTPRINT

6.75 8.005.00

6.25

particles in the center

particles at large amplitudes

7.00 7.25 7.50 7.75

5.25

5.50

5.75

6.00

νx0

ν y0

TUNE FOOTPRINT

Figure 3.16: Tune footprint at the beginning of the simulation for νx0 =6.22 (left panel) and νy0 = 8.0 (right panel). In both cases νy0 = 6.21,εx/εy = 30/10 mm mrad and N =2048. The black line in the left panel isthe resonance condition.

in such a way that the charge distribution doesn’t change significantly dur-ing ∆s and long enough in such a way that a generic test particle performsat least 50 ÷ 100 oscillations. We take the Fourier transform of the parti-cles trajectories in the interval ∆s and finally we associate to each particle(i = 1, 2, · · · , N) its tunes νx,i , νy,i, corresponding respectively to the hori-zontal and the vertical oscillatory motion. In figure 3.16 we show an exampleof the tune footprint at the beginning of the simulation in two cases: near(νx0 = 6.22, left panel) and far (νx0 = 8.0, right panel) from the Mon-tague resonance. We notice that in the first case there are several particles(∼ 3% of the total) whose frequencies are locked into the resonance condition(i.e. νx = νy).Let’s consider the number of particles in the resonance line (or very close

to it) during the simulation. In the mean field approximation (PIC simula-tions) the number of particles locked in the resonance remains constant intime whereas in collisional simulations it increases as we can see from figure3.17. This difference is due to the fact that particles tunes in the collisionalcase change and move over the tune footprint as a consequence of collisions(random motion). In figure 3.18 we show the tune jumps after 100 cells forthe particles initially inside a given region of the tune footprint, respectivelyin the mean field (left panel) and in the collisional case (right panel). Weshow also (in figure 3.19) an example of the random motion of the tune dur-ing the simulation in the collisional case.


0 4000.0

12.0

100 200 300

3.0

6.0

9.0

# cells

% Particles in the resonance

Figure 3.17: Particles locked in the resonance condition as a function of thetime. The black line is the result in the mean field approximation, the redline refers to the collisional case. The parameters of the simulation are thesame as in figure 3.16 (left panel).

We consider now the tune jumps distribution (after a given time, for instance100 cells) in the direction orthogonal to the resonance line

∆i = (~νi(s = 100) − ~νi(s = 0)) · n , (3.28)

where n =(− 1√

2, 1√

2

). The distribution for the case νx0 = 6.22, νy0 = 6.21,

σy = 0.8, εx/εy = 30/10 mm mrad and N = 2048 is shown in figure 3.20(black plot) and is, with a good approximation, Gaussian. The average jumpsamplitude is

∆r.m.s. = 0.09 . (3.29)

The jumps amplitude can be (qualitatively) related to the collisionality levelof the system in fact it decreases increasing N (keeping fixed the total charge)and increases increasing the space charge strength (keeping fixed N). Stillin figure 3.20 (red plot) we show also the jumps distribution for a simulationwith νx0 = 8.0, νy0 = 6.21, σy = 0.8, εx/εy = 30/10 mm mrad and N = 2048.As we can see, in both cases νx0 = 6.22 and νx0 = 8.0, even if the measuredequipartition times are different, the mean jumps amplitude is the same (i.e.the collisionality level is the same). This is a further clue of the fact thatthe acceleration of the relaxation process we observe in the vicinity of theresonance has a “dynamical” genesis.

As a consequence of collisions, particles tunes move in the tunes plane



5.00 6.255.00

6.25

5.25 5.50 5.75 6.00

5.25

5.50

5.75

6.00

νx0

ν y0

TUNE MIGRATION

5.00 6.255.00

6.25

5.25 5.50 5.75 6.00

5.25

5.50

5.75

6.00

νx0

ν y0

TUNE MIGRATION

Figure 3.18: The blue point are the tunes of the particles (after a time of100 cells) initially inside the green circle. The left plot refers to the meanfield case and the right one to the collisional case. The parameters of thesimulation are the same as in figure 3.16 (left panel).

5.00 6.255.00

6.25

5.25 5.50 5.75 6.00

5.25

5.50

5.75

6.00

νx0

ν y0

TUNE FOOTPRINT

Figure 3.19: The green and the blue tracks are the time evolution of thetunes for two different particles in the collisional case. The parameters of thesimulation are the same as in figure 3.16 (left panel).


-0.3 0.30.0

6.0

-0.2 -0.1 -0.0 0.1 0.2

1.0

2.0

3.0

4.0

5.0

∆

jum

ps d

istr

ib.

Figure 3.20: Tune jumps distribution after 100 cells. The black curve refersto the case νx0 = 6.22, νy0 = 6.21, σy = 0.8, εx/εy = 30/10 mm mrad andN = 2048. For the red curve νx0 = 8.0 and the remaining parameters arethe same. The green plot is the Gaussian fit, in this case ∆r.m.s. = 0.09.

0 500-0.1

0.8

125 250 375

0.1

0.3

0.4

0.6

# cells

ν x,i -

νy,

i

Figure 3.21: Behaviour of νx,i − νy,i for three particles in the case νx0 = 6.22,νy0 = 6.21, σy = 0.8, εx/εy = 30/10 mm mrad and N = 2048. The blackcurve is the resonance condition. The red and green plots refer to particleswhich are trapped in the resonance for a certain time.



νx0 Dmin/∆r.m.s. % part. trap. s = 0 % part. trap. s = 100 τ exprelax/τ

theorelax

6.22 0.33 3.6 5.7 0.106.25 0.56 1.3 1.5 0.226.30 0.78 0.1 0.3 0.416.50 2.0 0.0 0.0 1.02

Table 3.2: The injection (and trapping) of the particles in the resonance isrelated to the acceleration of the relaxation process.

and may reach the resonance line and get trapped into the resonance. Thisbehaviour is documented in figure 3.21 where we show the quantity νx,i−νy,i,which describes the distance from the resonance condition for particle i, as afunction of the time for some particles. We see that when a particle becomesresonant (νx,i−νy,i = 0), it remains resonant for a long time (trapping). Thereaching of the resonance depends mainly on:

• the (minimum) distance of the tune footprint from the resonance (Dmin),

• the amplitude of the tunes jumps (∆r.m.s.);

in fact, if∆r.m.s. < Dmin , (3.30)

we expect no particle will reach the resonance line, on the other hand if

∆r.m.s. ∼ Dmin or, a fortiori, ∆r.m.s. > Dmin , (3.31)

the population of the resonance line will increase. The value of Dmin fora given working point depends on νx0 − νy0. In figure 3.22 we show theminimum distance of the tune footprint from the resonance line for νx0 =6.22,6.25, 6.3, 6.5 (in all cases νy0 = 6.21, σy = 0.8, εx/εy = 30/10 mm mrad andN = 2048) and in table 3.2 we compare the values of Dmin with the meanjumps amplitude: we see that the acceleration rate of the relaxation processhas some correlation with the injection and trapping of the particles in theresonance.

Let’s define now the single particle emittances. Denoting by jx,i , jy,i theactions variables associated with the oscillatory motion of particle i, thecorresponding single particle emittances are given by

εx,i = 2jx,i εy,i = 2jy,i . (3.32)

and εx,i, εy,i are related to the r.m.s emittances of the system (εx, εy) asfollows

εx =1

2N

N∑

i=0

εx,i εy =1

2N

N∑

i=0

εy,i . (3.33)


5.25 6.505.00

6.25

6.22

6.25

6.30

6.50

5.50 5.75 6.00 6.25

5.25

5.50

5.75

6.00

νx0

ν y0

TUNE FOOTPRINT

Figure 3.22: Tune footprints corresponding to νx0 =6.22, 6.25, 6.3, 6.5. Theblack dashed lines denote the minimum distance (Dmin) of each tune footprintfrom the resonance line.

The single particle emittances of resonant and non-resonant particles havea different behaviour. In figure 3.23 (left panel) we show the evolution ofthe emittances for a non resonant particle, we see that εx,i and εy,i are notcorrelated, they change randomly and independently. This is no longer truefor a particle locked in the resonance (see figure 3.23, right panel): in thiscase the mean value of εy,i remains constantly larger then the mean value ofεx,i and their variations are always correlated.We are now able to give an interpretation of the acceleration of the equipar-tition process when the working point of the system is in the vicinity of theMontague resonance. Let’s rewrite the expression for the r.m.s. emittancesgiven by (3.33) separating the contribution of the resonant and non-resonantparticles

εx =1

2N

(N∑

non res.εx,i +

N∑

res.εx,i

)

εy =1

2N

(N∑

non res.εy,i +

N∑

res.εy,i

).

(3.34)

The first contribution within brackets is always present and it is related tothe “ordinary” relaxation process, whereas the second one exists only if weare in the vicinity of the resonance (i.e. if particles can reach the resonanceline) and we know that:



0 5000

150

125 250 375

50

100

# cells

single particle emittances

0 5000

150

125 250 375

50

100

# cells

single particle emittances

Figure 3.23: Single particle emittances for a non resonant (left panel) and aresonant (right panel) particle. The black plot refers to εx,i and the red oneto εy,i. The parameters of the system are νx0 = 6.22, νy0 = 6.21, σy = 0.8,εx/εy = 30/10 mm mrad and N = 2048. In the right panel the black and redcrosses are respectively the mean values of εx,i and εy,i.

a. the number of particles in the resonance increases because of the trap-ping mechanism,

b. the (single particle) emittances of the resonant particles satisfy (on theaverage) εy,i > εx,i,

c. the variations of εx,i and εy,i for the resonant particles are always cor-related,

as a consequence we conclude that when we are in the vicinity of the reso-nance, there is a growing number of particles (the resonant particles) whichsystematically and continuously contribute to the growth of the vertical r.m.s.emittance (εy) to the detriment of the horizontal one (εx). So we have that εyincreases faster (and consequently εx decreases faster) compared to the casewhere resonant particles are absent.

3.5.3 Dynamic crossing of the Montague resonance:

the role of collisions

To conclude this overview on the interplay between collisional effects andnon-linear phenomena we will study the behaviour of the 2D Coulomb sys-


5.85 6.4510

30

6.05 6.25

15

20

25

νx

R.M.S. Emittances [mm mrad]

Figure 3.24: Emittances evolution during the dynamical crossing of the reso-nance νx0 = 5.85 → 6.45, νy0= 6.21 for tune ramp over 30 (red), 100 (green),800 (blue), and 2500 (light blue) cells.

tem in case of the dynamical crossing of the Montague resonance.[42, 43] Inthe dynamical crossing the horizontal bare tune (νx0) is varied linearly in therange 5.85 → 6.45 over a certain period of time (corresponding to nT cells),while νy0 remains fixed at 6.21. Again we consider a Gaussian particles dis-tribution, the tune depression (in the center) is σy = 0.9 and the initial r.m.s.emittances are εx = 30 mm mrad and εy = 10 mm mrad. The initial andfinal values for the tune ramp are chosen in order to be sufficiently far fromthe resonance condition. As we know from the mean field theory, during thecrossing of the resonance the two emittances evolve smoothly with a crossingwhen νx0 = νy0(= 6.21). At the end of the crossing εx and εy are exchangedwith respect to the initial condition (see figure (3.24)). In figure (3.25) areshown the final emittances (after the resonance crossing) as a function ofthe tune ramp length. We observe that for nT = 100 (or nT = 200) theexchange is practically complete, for a fast ramp (nT = 30) the exchange isless complete. This behaviour was already observed in PIC simulations [33].Increasing the ramp length (nT from 200 to 2500 cells) again the exchangetends to be incomplete.To explain this behaviour it is useful to have a look at the phase space distri-bution in the plane x, py (see figure (3.26)): far from the resonance (νx0 6= νy0,εx 6= εy) the phase space plot reflects the typical Gaussian shape, near to the



0 250010

30

500 1000 1500 2000

15

20

25

tune ramp length

Final R.M.S. Emittances

Figure 3.25: Emittances at the end of the crossing as a function of the tuneramp length.

resonance condition (νx0 ' νy0, εx ' εy) the phase space shows a four-foldsymmetry which is a direct imprint of the non linear space-charge term inthe Hamiltonian arising in the resonance. Any information regarding theinitial emittances imbalance is kept in the correlated phase space shown infigure (3.26), any perturbation of this structure will affect the exchange pro-cess. Considering time scales where collisions are not important (nT < 200),during the slow crossing of the resonance (nT = 100, 200) the distributionclosely follows an intrinsically matched one which is fully self consistent at allpoints (adiabatic crossing). For a faster crossing of the resonance (nT = 30)the self consistency condition is no more fulfilled and the correlation in phasespace is partially lost rendering the exchange less complete. For slower cross-ing (nT > 400) the collisional effects become relevant. Their effect, whichbecomes the more important the slower is the crossing, tends to destroy thecorrelation in phase space when νx ' νy and this causes a memory loss ofthe initial condition obstructing the exchange process. This explanation issupported by figure (3.26): the correlation in phase space is higher in thecase nT = 100 (center panel) compared to the case nT = 2500 (right panel).As a final remark, we notice that the dynamical crossing of the Montagueresonance has been experimentally studied at CERN-PS from 2002 to 2004.[44] Even if the main features of the process are confirmed by 3D PIC simula-tions, the agreement between theoretical predictions and experiments is not


-20.00 20.00-16.00

16.00

-10.00 0.00 10.00

-8.00

0.00

8.00

x [mm]

p y

x-py phase space

-20.00 20.00-16.00

16.00

-10.00 0.00 10.00

-8.00

0.00

8.00

x [mm]

p y

x-py phase space

-20.00 20.00-16.00

16.00

-10.00 0.00 10.00

-8.00

0.00

8.00

x [mm]

p y

x-py phase space

Figure 3.26: Correlated phase space x, py far from the resonance (left) andin the resonance (νx0 = νy0 = 6.21) for a tune ramp with nT = 100 (center)or nT = 2500 (right).



complete. In particular it has been observed that at the end of the dynamicalcrossing the emittances exchange is highly incomplete, even if the crossing isvery slow. Intra beam scattering (i.e. collisional effects) is suspect to play asignificant role in this context.

Conclusions

In this thesis we have presented a numerical and analytical study of the dy-namical properties of a two dimensional Coulomb oscillators system which isa model to describe a beam of charged particles confined by a linear field ina storage ring. The main results obtained in this work can be summarizedas follows:

• Concerning the study of the dynamics of the system in the mean fieldapproximation, a method (the moment’s method) to solve the linearizedVlasov equation has been presented in order to solve the linearizedVlasov equation. This method allows us to study the coherent modesand their stability for the 2D Coulomb system. This approach couldoffer some advantages in terms of flexibility and simplicity comparedto the traditional “characteristics method”.

• An N -body numerical code which provides the symplectic integrationof the Hamilton’s equations of motion for the 2D Coulomb oscilla-tor system has been developed. The algorithm for the electric fieldcalculation, based on a tree schema, has an (optimal) computationalcomplexity (N logN , where N is the number of particles) compared tothe full direct computation (N 2). The code has been parallelized withMPI. Several tests have been made to verify the numerical reliabilityof the code.

• By using the Landau’s kinetic theory in 2D, a scaling law for the col-lisional relaxation time towards the thermodynamic equilibrium as afunction of the particles number for a generic phase space distributionhas been obtained. It has been shown that the relaxation time of thesystem scales as N keeping fixed all the system parameters. The predic-tions of the theory have been fully confirmed by numerical simulations.Furthermore we have shown that collisional relaxation is comparablewith the storage time in a (storage) ring and so collisional effects shouldbe taken into account in the design of high intensity accelerators.


• Again by using the Landau’s theory a formula describing the relaxationtime as a function of all the system parameters has been obtained inthe case of an anisotropic Gaussian phase space distribution. Also inthis case the theoretical predictions have been confirmed by N -bodysimulations.

• The interplay between collisional effects and non linear phenomena inthe Montague resonance has been studied. We have shown that, inthe case of anisotropic (nonequipartitioned) systems, an accelerationof the equipartition process occurs due to an increment of the particlestrapped in the resonance condition. The trapping mechanism providesan efficient way to transfer energy between the horizontal and the ver-tical degrees of freedom.

• We have studied the statistical properties of a Coulomb collision processstarting from the simulated data and we have determined the spectrumof the momentum changes distribution. We have found that the distri-bution has a power law decaying queue with exponent equal to 4 (dueto hard collisions events) and a Gaussian core (due to soft collisions).The results can be partially interpreted also theoretically and can beuseful to introduce collisional effects, in a completely phenomenologi-cal way, upon a mean field framework bypassing the standard Landau’sapproach.

Appendix A

The longitudinal coherence

hypothesis

In the derivation of the 2D Coulomb oscillators model (see section 1.1) wehave assumed that all the particles in the same cylinder have the same lon-gitudinal position. This hypothesis, which implies a strong longitudinal co-herence, is not strictly necessary and can be released without changing sig-nificantly the results discussed in section 1.1. Let’s assume now that thelongitudinal position of i-th particle inside the k-th cylinder is given by

z(k)i = k`+ η

(k)i , (A.1)

where

|η(k)i | < `/2 . (A.2)

The transverse electrostatic force acting on the i-th particle in the k = 0cylinder due to all the remaining particles (see the r.h.s. of (1.5)) is thengiven by

Felecti =

e2

4πε0

∞∑

k=−∞

N∑

j=1

j 6=i if k=0

r(0)i − r

(k)j(

‖r(0)i − r

(k)j ‖2 + (k`+ η

(0)i − η

(k)j )2

)3/2. (A.3)

Again assuming that r(k)i depends weakly on k we get

Felecti ' e2

4πε0

N∑

j=1, j 6=i

(r(0)i − r

(0)j )S ′(‖r(0)

i − r(0)j ‖) , (A.4)

114 The longitudinal coherence hypothesis

where

S ′(‖r(0)i − r

(0)j ‖) ≡ 1

‖r(0)i − r

(0)j ‖3

∞∑

k=−∞

(1 +

(k`+ η(0)i − η

(k)j )2

‖r(0)i − r

(0)j ‖2

)−3/2

,

(A.5)and finally we have

S ′(‖r(0)i − r

(0)j ‖) ≡ S(‖r(0)

i − r(0)j ‖) +

− 3

‖r(0)i − r

(0)j ‖5

∞∑

k=−∞k`

(1 +

(k`)2

‖r(0)i − r

(0)j ‖2

)−5/2

(η(0)i − η

(k)j )+

+3

2

1

‖r(0)i − r

(0)j ‖5

∞∑

k=−∞

4(k`)2 − ‖r(0)i − r

(0)j ‖2

‖r(0)i − r

(0)j ‖2 + (k`)2

(1 +

(k`)2

‖r(0)i − r

(0)j ‖2

)−5/2

(η(0)i − η

(k)j )2 ,

(A.6)

where S(‖r(0)i − r

(0)j ‖) is defined (1.15). Since we expect that η

(0)i − η

(k)j

are distributed randomly with zero mean, then, on the average, the secondterm in the r.h.s. of (A.6) vanishes. As a consequence the expression for

S ′(‖r(0)i − r

(0)j ‖) coincides with the one for S(‖r(0)

i − r(0)j ‖) up to a small

correction.

Appendix B

The 2D Debye length for a

charged plasma

In this section we will present the Debye theory for a two-dimensional non-neutral plasma[45]. Let’s consider a plasma of charged particles q in thermalequilibrium, we have

∇2φ = −4π q Nρs ρs = ρ0e−(qφ+Vconf )/kBT , (B.1)

where φ is the potential of the plasma, ρs the particles density distribution(normalized to one), N the number of particles, Vconf the external confiningpotential and kBT the temperature. Now we introduce a charge unbalanceq in the origin, the perturbed potential is φ+ δφ and the perturbation δφ isgiven by

∇2δφ = −4π q Nρ0(e−[q(φ+δφ)+Vconf ]/kBT − e−(qφ+Vconf )/kBT ) =

= −4π q Nρs(e−qδφ/kBT − 1) , (B.2)

where we assume that the perturbed system is still in thermal equilibrium.Assuming qδφ/kBT 1 and using the 2D expression for the Laplace operatorwe have

1

r

d

dr

(rdδφ

dr

)' 4π q2Nρs

kBTδφ , (B.3)

and we assume also that the density ρs doesn’t have big changes within thecharacteristic size of the system (∼ R) we have ρs ' 1

πR2 and we can definethe following length scale (Debye length)

Λ2D ≡ kBT

4q2NR2 =

⟨p2

x + p2y

⟩

4ξR2 , (B.4)

116 The 2D Debye length for a charged plasma

where the last equality follows from the definition of perveance we have givenin chapter 1. Finally we have

1

r

d

dr

(rdδφ

dr

)' δφ

Λ2D

(B.5)

and the solution is

δφ = 2qK0

(r

ΛD

)'

−2q log(

rΛD

)r ΛD

2q e−r/ΛDq

r/ΛD

r ΛD

(B.6)

where K0(x) is the modified Bessel function of order zero. We notice thatthe perturbation has a relevant effect only within the Debye radius, thisshows that the Debye’s length ΛD is the characteristic length scale where thediscrete particles effects (i.e. the effects related to the collisional part of theinteraction) become important.For the typical choice of the parameters of the 2D Coulomb system: ξ ∼ 1,ω0 ∼ 1.0 rad/m and εr.m.s. ∼ 0.25 mm mrad, the characteristic dimension ofthe system is R ∼ 1.3 mm and

⟨p2

x + p2y

⟩∼ 0.3, so

ΛD

R∼ 0.27 . (B.7)

In the previous derivation of the Debye’s length for a bidimensional systemwe have made several important assumptions, so we expect that in generalour derivation will be only qualitatively correct, the appropriate value of thelength scale for the collisional part of the interaction might be different

from ΛD.

Appendix C

The drift coefficient for the KV

distribution

The high energy limit. For the space-averaged KV distribution (ρKV (p) =ρ0θ(4kBT − p2), ρ0 = (4π2kBTR

2KV )−1) the behaviour of β in the high en-

ergy regime (p √kBT ) is easy to analyze because the integration domain

[0, 2√kBT ]× [0, 2 π] falls entirely into region I. Still denoting by β the space-

averaged drift coefficient and expanding the integrand in series of p′/p weobtain

β p = 8ΛDξ2

N(π − 2)

∫ 2√

kBT

0

dp′ p′ρKV (p′)

∫ 2π

0

dφp− p′ cosφ

(p2 + p′2 − 2 p p′ cosφ)3/2=

= 8ΛDξ2

N(π − 2)ρ0

∫ 2√

kBT

0

dp′ p′∫ 2π

0

dφ1

p2

(1 + O

(p′

p

))=

= 8 ΛDξ2

N(π − 2)

1

p2

1

πR2KV

+O

(1

p3

), (C.1)

so we have

limp→∞

β(p)p3 = 8 ΛDξ2

N

π − 2

πR2KV

. (C.2)

The low energy limit. The behaviour for p → 0 is more difficult toanalyze because the integration domain is intersected by regions I and II.The contribution of region I (where σ0 − σ1 = ΛD(π − 2)ξ2N−2p−4) to theintegral defining βp is given by

(βp)I = 8ΛDξ2

N(π−2)

∫ 2π

0

dφ

∫ ∞

p+(φ)

dp′ p′ ρKV (p′)p− p′ cosφ

(p2 + p′2 − 2 p p′ cosφ)3/2=

118 The drift coefficient for the KV distribution

= 8ΛDξ2

N(π−2) ρ0

∫ 2π

0

dφ

∫ 2√

kBT

p+(φ)

dp′[p

p′2(1 − 3 cos2 φ) − 1

p′cosφ+O

(p2

p′2

)]=

= 8ΛDξ2

N(π − 2)ρ0

∫ 2π

0

dφ

[p

(1

p+(φ)− 1

2√kBT

)(1 − 3 cos2 φ)+

− log

(2√kBT

p+(φ)

)cosφ+O

(p2

p2+(φ)

)], (C.3)

where the integral on p′ has been performed after expanding the integrandin a series of p/p′. We recall that for p small we have

p+(φ) =√ε

(1 +

p√ε

cos φ

)+O

(p2

ε

), (C.4)

where ε = 4ξ/Nc and c =

(8

3(π − 2)

)1/2

. Inserting expression (C.4) in the

integral (C.3) and retaining only the first order terms in p/√ε we obtain

(βp)I = 4ΛDξ2

N(π − 2)πρ0p

1√kBT

+O

(p2

ε

). (C.5)

The contribution from the second region is given by

(βp)II =4

3ΛD Nρ0

∫ 2π

0

dφ ×

×[∫ p

0

dp′ +

∫ p+(φ)

p

dp′

]p′ (p2 + p′2 − 2 p p′ cos φ)1/2(p− p′ cosφ) . (C.6)

When p′ < p we expand the integrand in a series of p′/p and the result isp′p2−2p′2p cos φ+O(p′3). As a consequence the first integral is O(p4). Whenp′ > p we expand the integrand in a series of p/p′, we obtain

(βp)II =4

3ΛDNρ0

[O(p4) +

∫ 2π

0

dφ

∫ p+(φ)

p

dp′p′ 3

(p

p′(1 + cos2 φ) − cosφ+O

(p2

p′2

))]=

=4

3ΛDNρ0

∫ 2π

0

dφ

(O(p4) +

1

3p3

+(φ) p (1 + cos2 φ) − 1

4p4

+(φ) cosφ+O(p2p2+(φ))

).

(C.7)Using (C.4) in (C.7) we have

(βp)II = O(εp2) (C.8)

and finally from (C.5) and (C.8) we obtain

limp→0

β(p) = ΛDξ2

N

π − 2

πR2KV

1

(kBT )3/2. (C.9)

Appendix D

The moments method with

collisions

Let’s consider the evolution of the second order moments of the distributionaccording to the Fokker-Planck equation presented in (1.140), where we makethe approximation discussed in (1.174) for the drift and the diffusion terms.We obtain (see also [46])

˙〈x2〉 = 2 〈xpx〉˙〈xpx〉 = 〈p2

x〉 − kx0 〈x2〉 + ξ2〈xEx〉 − β 〈xpx〉

˙〈p2x〉 = −2kx0 〈xpx〉 + ξ 〈pxEx〉 − 2β 〈p2

x〉 +D˙〈y2〉 = 2 〈ypy〉˙〈ypy〉 =

⟨p2

y

⟩− ky0 〈x2〉 + ξ

2〈yEy〉 − β 〈ypy〉

˙⟨p2y

⟩= −2ky0 〈ypy〉 + ξ 〈pyEy〉 − 2β

⟨p2

y

⟩+D

(D.1)

where Ex, Ey are the components of the self consistent electric field, D =2βkBT and kx0, ky0 depend on time in the periodic focusing case (kx0(s+L) =kx0(s), ky0(s + L) = ky0(s)), otherwise they are constant (kx0 ≡ ω2

x0, ky0 ≡ω2

y0). Let’s suppose that there are no instabilities and that the phase spacedensity distribution has (and remains with) an ellipsoidal symmetry, it canbe shown that [47]

〈xEx〉 =

√〈x2〉√

〈x2〉 +√〈y2〉

〈yEy〉 =

√〈y2〉√

〈x2〉 +√〈y2〉

(D.2)

〈pxEx〉 =〈xpx〉√

〈x2〉(√

〈x2〉 +√

〈y2〉)〈pyEy〉 =

〈ypy〉√〈y2〉(

√〈x2〉 +

√〈y2〉)

.

(D.3)

120 The moments method with collisions

Equations (D.1) with (D.2) and (D.3) can be utilized to compute theoreticallythe behaviour of the second order moments of the distribution during therelaxation process, provided that instabilities are absent.Let’s consider the emittances of the system

ε2x =⟨x2⟩ ⟨p2

x

⟩− 〈xpx〉2 ε2y =

⟨y2⟩ ⟨p2

y

⟩− 〈ypy〉2 , (D.4)

by using (D.1) we can show that

dε2xds

= ξ(⟨x2⟩〈pxEx〉 − 〈xpx〉〈xEx〉) − 2βε2x +

⟨x2⟩D

dε2yds

= ξ(⟨y2⟩〈pyEy〉 − 〈ypy〉〈yEy〉) − 2βε2y +

⟨y2⟩D .

(D.5)

The terms proportional to ξ in (D.5) are responsible for an emittances changerelated to space charge forces, the remaining terms are related to collisions.Assuming again there are no instabilities and that the phase space distribu-tion has an ellipsoidal symmetry we can use (D.2) and (D.3) in (D.5), weobtain that the terms which contain ξ vanish. We recall that the horizontal,the vertical and the mean temperatures of the system are respectively givenby

kBTx =ε2x〈x2〉 kBTy =

ε2y〈y2〉 kBT =

1

2(kBTx + kBTy) , (D.6)

so the diffusion term D is

D = β

(ε2x〈x2〉 +

ε2y〈y2〉

). (D.7)

By using (D.7) in (D.5), after a little algebra we get

dεxds

=β

2

(kBTy

kBTx

− 1

)εx

dεyds

=β

2

(kBTx

kBTy− 1

)εy .

(D.8)

It is clear that in the isotropic case (without instabilities) we have

kBTx = kBTy = kBT and εx = εy = ε (D.9)

121

and so the r.m.s. emittances do not change. On the other hand, in theanisotropic case the two factors

kBTy

kBTx− 1 and

kBTx

kBTy− 1 (D.10)

have opposite signs and the emittance in the hotter plane tends to decreasewhile the other one tends to increase. We can also obtain a condition de-scribing the behaviour of the mean value of the emittances (ε = (εx + εy)/2).We have that ε decreases if

εyεx<kBTy

kBTxwhen kBTx < kBTy (D.11)

orεyεx>kBTy

kBTxwhen kBTx > kBTy (D.12)

otherwise it increases. From (D.8) we see also that the mean growth (ordecreasing) rate is proportional to the drift term (i.e. to the collisionalitylevel of the system). Furthermore we notice that if εx = εy but the system isnot in thermal equilibrium (kBTx 6= kBTy) then ε can not decrease becauseconditions (D.11) and (D.12) are never fulfilled.Let’s consider now the following function

S = log (εx εy) , (D.13)

it is not difficult to show that as far as collisions are concerned

dS

ds=β

2

(kBTy

kBTx+kBTx

kBTy− 2

)≥ 0 , (D.14)

so S is a non decreasing function and in particular it is stationary onlywhen the system is equipartitioned. By using (D.1) we can obtain also thebehaviour of the temperatures of the system. Defining

kBT =kBTx + kBTy

2∆kBT =

kBTx − kBTy

2, (D.15)

we get

˙kBT = −[kBT

(〈xpx〉〈x2〉 − 〈ypy〉

〈y2〉

)+ ∆kBT

(〈xpx〉〈x2〉 +

〈ypy〉〈y2〉

)]

˙∆kBT = −2β∆kBT +

[kBT

(〈ypy〉〈y2〉 − 〈xpx〉

〈x2〉

)− ∆kBT

(〈ypy〉〈y2〉 +

〈xpx〉〈x2〉

)].

(D.16)

122 The moments method with collisions

The terms within squared brackets in (D.16) are generally small and so thedominant term in the second equation is −2β∆kBT which tends to equalizethe horizontal and the vertical temperatures. Including the expression forthe temperatures into (D.14), we finally obtain

S(s) ' S(0) +1

2log

(kBTx(0) + kBTy(0))2 − (kBTx(0) − kBTy(0)) exp(−4β s)

(kBTx(0) + kBTy(0))2 − (kBTx(0) − kBTy(0))2.

(D.17)

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