Volume 40A, number 3 PHYSICS LETTERS 17 July 1972

KINETIC E Q U A T I O N F O R A D E N S E CLA S S ICA L PLASMA

M. BAUS * Faculty of Sciences, University of Brussels, Belgium

Received 25 April 1972

The Balescu-Guernsey-Lenard equation for an homogeneous classical dilute electron gas is extended to higher densities.

In the traditional derivation of the Balescu-Guern- sey-Lenard equation [ 1] two difficulties are immedia- tely apparent: 1) one neglects the triple correlations by considering their equilibrium magnitude and 2) one eliminates a small-distance divergence using Landau's argument [2] by introducing a short-distance cut-off evaluated in equilibrium. In non-equilibrium situations these approximations clearly become unsatisfactory. Moreover, neglecting the triple correlations amounts to neglecting part of the collective effects which are responsible for the system's intrinsic large and small distance cut-offs. Indeed, part of the triple correlation contribution corresponds to a screening between the colliding particles produced by the nearby field parti- cles just as the screening produced by the distant field particles is taken into account through the Vlassov terms appearing in the second member of the B.B.G.K.Y. hierarchy. This triple correlation effect is important for distant colliding particles for which it will introduce a cut-off at the mean free path, but also for colliding particles which are close because they now move in the force field of nearby field particles. Further, triple correlations can certainly not be ne, glected for those charged systems for which the num- ber of particles in a Debye sphere is not large such as dense systems, or systems which have strong correla- tions such as turbulent plasmas and the electron liq- uids encountered in solid statephysics [3].

We now report some results Obtained using as guide the recent progress made in dense neutral gas theory by Pomeau [4]. The system of equations formed by the first three members of the B.B.G.K.Y. hierarchy for the correlation functions [5] is solved under the conditions we can 1) neglect four particle correlations

* Charg6 de recherches du Fonds National Beige de la Recherche Scientifique.

because we do not want to retain correlations be- tween field particles, 2) neglect the term nonlinear in the binary correlations in order to be able to solve the system of equations, 3) use the adiabatic hypothe- sis, 4) neglect interactions between the colliding par- ticles but not between a colliding and a field particle because we assume the particles to be weakly coupled to each other but not to the medium as a whole, i.e., collective effects dominate, 5) take the long time limit for a stable system. The resulting kinetic equation for a homogeneous electron gas reads then:

af(p;t)_ f f d6ofdk IVkl2 k" 3 bt n dp _~ 2rr a87r3 le(k,w)l 2 0 - p

r F. t . × {Re Pk(v ,co) Im Fk(o , 6o)-RePk(o,co) Im k(o ,co)}

(1)

In eq. (1) Re and Im denote respectively the real and imaginary part and moreover:

ek(O ,CO) =[ik "O--iog+Ck(O ,60)]-1 f(p: t) (2a)

Fk(o,w ) = [ik.o-iw+ Ck(o,o~)]-I ik .-~ f(p;t) (2b)

e(k, w) = 1 + n f dp V k Fk(O, oo) (2c)

where Ck(o, oo) is a finite frequency collision operator for an inhomogeneous system but linearized around f(p; t). The remaining notations are standard: n is the number desity, p = mu, V k is the Fourier transformed Coulomb potential, etc. The kinetic eq. (1) conserves the total number, momentum and kinetic energy of the particles. An H-theorem can be proven for (1) each time the hermitian part of Ck(~, co) is negative definite, i.e. when the system is stable, and moreover the H-quan-

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Volume 40A, number 3 PHYSICS LETTERS 17 July 1972

tity becomes stationary for a maxwellian. Eq. (1) clearly reduces to the Balescu-Guernsey-Lenard equa- tion when we substitute a priori +0 for Ck(O,w ) in (2). However for Ck(o , w) 4 : 0 the behaviour of the func- tions defined in (2) will be very different from their Balescu-Guernsey-Lenard counterpart. Indeed, the ef- fective potential (see (2c)) is now no longer evaluated on the basis of a reversible Vlassov equation but con- tains now correlational effects (compare with ref. [3]) whereas (2a -b ) indicate the sharp wave-particle reso- nance of the Balescu-Guernsey-Lenard equation is broadened. Moreover individual particle and transport modes at finite k and w can now be excited in the sys- tem. Needless to say that in order for eq. (1) to be closed we have to evaluate (2) which can be done ex- plicitly only by model considerations. To get some in- sight into (1) we have used a zero frequency, infinite wavelength, Fokker-Planck model [6] for Ctc(o, co) in which case we can show explicitly that eq. (1) con- serves the non-negative character o f f ( p ; t) whereas the k-integral now converges for large Ik [ proving the con- jecture, recently made by the author [7], that colli- sions with field particles for intermediate range ira-

pact parameters will modify completely the binary collision analysis of the small distance behaviour of the Balescu-Guernsey-Lenard equation. Finally, with this model the dependence of (1) on the plasma expan- sion parameter becomes non-analytic. Details about the statements made here will be published elsewhere.

We thank R. Balescu, A. Grecos and P. R6sibois for their comments.

References

[ 1] D. Montgomery and D. Tidman, Plasma kinetic theory (Mc. Graw-Hill, N.Y. 1964).

[2] L.D. Landau, Phys. Z. Soviet Un. 10 (1936) 154. [3] K.F. Berggren, Phys. Rev. A1 (1970) 1783;

S. Ichimaru, Phys. Rev. A2 (1970)494; Phys. Fluids 13 (1970) 1560.

[4] Y. Pomenau, Phys. Rev. A3 (1971) 1174, Phys. Lett. 27A (1968) 601.

[5] E.A. Frieman and R. Goldman, J. Math. Phys. 8 (1967) 1410.

[6] J.P. Dougherty, Phys. Fluids 7 (1964) 1788. [7] M. Baus, Ann. Phys. 62 (1971) 135.

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