Range of validity of transport equations

Abstract

Transport equations can be derived from quantum field theory assuming a loss of information about the details of the initial state and a gradient expansion. While the latter can be systematically improved, the assumption about a memory loss is in general not controlled by a small expansion parameter. We determine the range of validity of transport equations for the example of a scalar g 2 Φ 4 theory. We solve the nonequilibrium time evolution using the threeloop 2PI effective action. The approximation includes off-shell and memory effects and assumes no gradient expansion. This is compared to transport equations to lowest order (LO) and beyond (NLO). We find that the earliest time for the validity of transport equations is set by the characteristic relaxation time scale t damp = −2ω/Σ (eq) ̺ , where −Σ (eq) ̺ /2 denotes the on-shell imaginary-part of the self-energy. This time scale agrees with the characteristic time for partial memory loss, but is much shorter than thermal equilibration times. For times larger than about t damp the gradient expansion to NLO is found to describe the "full" results rather well for g 2 1. *

2PI nonequilibrium versus transport equations for an ultracold Bose gas

“…See also Refs. [25,26,27,28,29,30,31,32] in the context of relativistic physics. A comparison of dynamic equations with their kinetic approximation for relativistic dynamics has been given in Refs.…”

Attractive versus repulsive interactions in the Bose-Einstein condensation dynamics of relativistic field theories

“…From (B9), the time evolution of the particle momentum distribution can be obtained. The latter can be defined according to [53] fðt; pÞ ¼ Z ∞ 0 dp 0 2π 2p 0ρ ðpÞfðt; pÞ; ðB11Þ where Fðt; pÞ ¼ ðfðt; pÞ þ 1 2 ÞρðpÞ. 20 From (B9), the time evolution of the distribution function defined in that way is given by ∂fðt; pÞ ∂t ¼ Z ∞ 0 dp 0 2π ½Σ ρ ðt; pÞFðt; pÞ − Σ F ðt; pÞρðpÞ :…”

“…Remarkably, kinetic theory can still be extended to the highly occupied regime, with the help of a systematic resummation of scattering processes by use of the 1=N expansion to NLO that was discussed above. This has been done for the φ 4 theory [36,37,53] by deriving the corresponding truncated 2PI equations of motion and recasting them into a form similar to the Boltzmann equation, under additional assumptions such as a comparably smooth time evolution, an effective memory loss, and on-shell quasiparticles. In this section, we generalize that derivation to our case.…”

“…Transport equations are obtained from the 2PI equations of motion under several assumptions [37,53]. One of them concerns the effective memory loss of the system about the initial conditions, which allows one to take the limit t 0 → −∞.…”

Progress in nonequilibrium quantum field theory III

“…This has led to successful descriptions of far-from-equilibrium dynamics and subsequent thermalization in scalar and fermionic quantum field theories in various dimensions [ 11,15,16,9,5,10,12,17]. They have also been used to compute transport coefficients [ 18] and to determine the range of validity of transport or semi-classical approaches [ 19,20,21].…”