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