on energy conversion from overcritical electric fields ... ·...
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Introduction Framework Results Conclusions
On energy conversion from overcritical electricfields toward thermalized pair-photon plasma
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
July 5, 2012
AB is supported by the Erasmus Mundus Joint Doctorate Program
by Grant Number 2010-1816 from the EACEA of the European Commission
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Contents
1 Introduction
2 Framework
3 Results
4 Conclusions
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Electron-positron pairs produced in a strong electric field E ifE & Ec = m2
ec3/e~ [Sauter (1931), Heisenberg and Euler (1935),Schwinger (1951)]
Ec is far from being reached experimentally [review Di Piazza et al.ArXiv e-prints (2011)]
Blocking of pairs production in astrophysical context as compactstars, hypothetical quark stars, neutron stars is discussed[Usov(1986), Alcock et al. (1986), Belvedere et al. (2012), review Ruffiniet al. (2010)]
Back reaction of pairs on the external field treated in QED in 1+1dimension case for scalar and fermion fields. Results agreed with thesolutions of the relativistic Vlasov-Boltzmann equations [Kluger etal. (1991, 1992)]
The problem can be reduced to the numerical solution of the systemof ordinary differential equations which allow the study pairsannihilation into photons [Ruffini et al. (2003, 2007), Benedetti etal. (2011)]
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Goals
Analysis of energy conversion stored initially in overcritical electricfield to the energy of electron-positron-photon plasma
Study of physical processes (back reaction, plasma oscillations,particle interactions) and associated time scales
Analysis of various forms of energy (kinetic, rest mass, internal,radiative) during energy conversion process
Assumptions
Anisotropic but homogeneous physical system
Axially symmetric momentum space
Applicability of the classical Boltzmann-Vlasov equation
Inclusion of 2-particle QED interactions only
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Cylindrical coordinates in the momentum space (p‖,p⊥, φ)
p‖ · E = p‖ E , p⊥ · E = 0
Using ν (−,+, γ) as label for the kind of particle, usually the distributionfunction fν is used to define its number density
nν =
∫d3p fν
Because we have no dependence on φ, we prefer using a new distributionfunction Fν such that
ρν =
∫ +∞
−∞dp‖
∫ +∞
0
dp⊥ Fν , fν =Fν
2π εν p⊥
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Boltzmann equations for electrons and positrons
♠ ∂F±∂t± e E
∂F±∂p‖
=∑
q
(η∗q± − χ
q± F±
)+ S
where the rate of pair production is [Ruffini et al. (2010)]
S(p‖, p⊥,E ) = − |e E |m3
e(2π)2ε p⊥ log
[1− exp
(−π(m2
e + p2⊥)
|e E |
)]δ(p‖)
Boltzmann equation for photons
♠ ∂Fγ
∂t=∑
q
(η∗qγ − χq
γ Fγ
)Initial condition
♠ E (t0) = E0 ≥ Ec , Fν(p‖, p⊥, t0) = 0
For each of these initial condition, we perform two different runs calledcollisionless and interacting depending if collision term are taken intoaccount or not
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Bulk parallel momentum
Figure: Evolution of E and 〈p‖〉± when theinitial condition is E0 = 10 Ec .
for electrons and positrons
〈p‖〉ν =1
nν
∫d3p fν p‖
The actual electricfield is obtained from theenergy conservation law∑
ν
ρν =E 2
0 − E 2
8π
Both E and 〈p‖〉±oscillate with shiftedphase, because of theback-reaction of pairs onto the external field
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
We distinguish
Figure: E0 = 30 Ec and energy densities arenormalized to E 2
0 /8π.
three kinds of pairs energy
ρrest± = (n− + n+) me c2
ρkin± = ρrest
±
√ 〈p‖〉2±m2
ec2+ 1− 1
ρin± = ρ± − ρrest
± − ρkin±
The rest energy of pairssaturates to a small fractionof the total initial energy.The initial energy is mainlyconverted into internal energy of pairs while the kinetic one decreasesprogressively. The energy stored in the electric field becomes muchsmaller than the pairs internal energy
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
We define the maximum achievable pairs number density
nmax =E 2
0
8πme' 9.4 · 1031
(E0
Ec
)2
cm−3
We expect the final equilibrated thermal electron-positron-photon plasmato be characterized by the temperature
Teq = 4
√ρ0
a' 1.7
√E0
EcMeV
The temperature of the system can be estimated looking the spreading ofthe distribution function in the momentum space
〈p2i 〉ν =
1
nν
∫d3p fν (pi − 〈pi 〉ν)2 , i =‖,⊥
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
There is initial anisotropy between parallel and orthogonal spreading ofthe distribution functions. In order to achieve kinetic equilibrium,√〈p2⊥〉± and
√〈p2‖〉± have to converge such as particles acquire the
common temperature Teq, but nonzero chemical potential. If also3-particle interactions would be accounted for, the thermal equilibriumcould be reached meaning that the photon chemical potential is zero[Aksenov et al. (2009)].
E/Ec Teq
√〈p2⊥〉±
√〈p2‖〉± 〈p‖〉1 n1/nmax ns/nmax
1 1.7 0.4 75 160 0.006 0.018
3 2.9 0.8 37 82 0.018 0.037
10 5.4 1.3 35 77 0.013 0.041
30 9.3 2.0 87 192 0.005 0.016
100 17 3.5 127 284 0.003 0.011
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
After a time much longer that the average oscillation period, the energydensity of photons equals and then overcomes the pairs energy density.
At later times the rates of electron-positron annihilation into photons andits inverse process are perfectly balanced.
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Figure: Didascalia comune alle due figureAlberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma
Introduction Framework Results Conclusions
Conclusions
For the first time we solved the relativistic Boltzmann-Vlasovequations for electrons, positrons and photons starting from aninitial overcritical electric field up to reaching kinetic equilibrium forcreated pair plasma.
At early times, the time dependence of pairs number density, bulkparallel momentum and electric field are very similar to thoseobtained in the literature. However, after a short period, we obtainsubstantially different results.
The number density of pairs always saturates to a small fraction ofthe maximum achievable one. The initial energy stored in theelectric field is mainly converted into internal and kinetic pairenergies, but the former becomes predominant as time advances.
For higher initial fields interactions appear to be efficient sooner.
A perfect symmetry between pair annihilation and creation rates isachieved and the system evolves toward the kinetic equilibrium.
Alberto Benedetti, G.V. Vereshchagin, R. Ruffini
On energy conversion from overcritical electric fields toward thermalized pair-photon plasma