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arXiv:hep-ph/0303221

Kinetic Equations from the Two-Particle-Irreducible 1/N-Expansion

Muller, Markus Michael

Published Mar 26, 2003• 0 citations
arXivHTMLPDF

Abstract

We present kinetic equations that describe the evolution of O(N)-symmetric real scalar quantum fields out of thermal equilibrium in a systematic nonperturbative approximation scheme. This description starts from the 1/N-expansion of the 2PI effective action to next-to-leading order, which includes scattering and memory effects. From this starting point one is lead to evolution equations for the propagator, which are nonlocal in time. Numerical solutions showed that the propagator depends only very slightly on the center coordinates already after moderate times, and that correlations between earlier and later times are suppressed exponentially, which causes an effective memory loss. Exploiting these two observations, we combine a first order gradient expansion with a Wigner transformation to derive our kinetic equations, which are local in time, from the nonlocal evolution equations. In contrast to standard descriptions based on loop expansions, our kinetic equations remain valid even for nonperturbatively large fluctuations. Additionally, employing a quasi-particle approximation, we eventually arrive at a generalized Boltzmann equation.

DOI
10.48550/arxiv.hep-ph/0303221

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We present kinetic equations that describe the evolution of O(N)-symmetric real scalar quantum fields out of thermal equilibrium in a systematic nonperturbative approximation scheme. This description starts from the 1/N-expansion of the 2PI effective action to next-to-leading order, which includes scattering and memory effects. From this starting point one is lead to evolution equations for the propagator, which are nonlocal in time. Numerical solutions showed that the propagator depends only very slightly on the center coordinates already after moderate times, and that correlations between earlier and later times are suppressed exponentially, which causes an effective memory loss. Exploiting these two observations, we combine a first order gradient expansion with a Wigner transformation to derive our kinetic equations, which are local in time, from the nonlocal evolution equations. In contrast to standard descriptions based on loop expansions, our kinetic equations remain valid even for nonperturbatively large fluctuations. Additionally, employing a quasi-particle approximation, we eventually arrive at a generalized Boltzmann equation.
We consider the nonequilibrium evolution of an O(N)-symmetric scalar quantum field theory using a systematic two-particle irreducible 1/N-expansion to next-to-leading order, which includes scattering and memory effects. The corresponding ``full Kadanoff-Baym equations'' are solved numerically without further approximations. This allows one to obtain a controlled nonperturbative description of far-from-equilibrium dynamics and the late-time approach to quantum thermal equilibrium. Employing in addition a first-order gradient expansion for the Wigner transformed correlators we derive kinetic equations. In contrast to standard descriptions based on loop expansions, our equations remain valid for nonperturbatively large fluctuations. As an application, we discuss the fluctuation dominated regime following parametric resonance in quantum field theory.
We derive the nonequilibrium real-time evolution of an O(N)O(N)O(N)-invariant scalar quantum field theory in the presence of a nonvanishing expectation value of the quantum field. Using a systematic 1/N1/N1/N expansion of the 2PI effective action to next-to-leading order, we obtain nonperturbative evolution equations which include scattering and memory effects. The equivalence of the direct method, which requires the resummation of an infinite number of skeleton diagrams, with the auxiliary-field formalism, which involves only one diagram at next-to-leading order, is shown.
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We present a theoretical study of thermalization of quantum fields with Kadanoff-Baym (KB) equation. First we introduce a field theoretical technique known as 2 Particle Irreducible (2PI) effective action and review derivation of equation of motion for mean fields and fluctuations, then derive Kadanoff-Baym equation in scalar and gauge theories. In order to analyze thermalization processes we introduce relativistic entropy current based on the first order gradient expansion of Kadanoff-Baym equation. Then we show that in taking into account next-to-leading order (NLO) self-energy of the coupling expansion in φ4 theory (in symmetric phase <φ^> = 0), it is possible to prove the H-theorem within the 1st order gradient expansion. We also show that it is possible to prove the H-theorem for KB equation with NLO self-energy of 1/N expansion in O(N) theory (in symmetric phase). Furthermore we suggest a possibility for H-theorem to be satisfied in KB equation with leading-order (LO) self-energy of the coupling expansion in non-Abelian gauge theory. In d+1 dimensions (d＞2) LO particle number changing processes, such as 0-to-3, 1-to-2 and 2-to-1, which are prohibited in the normal Boltzmann approach with on-shell particles, may contribute to entropy production in KB approach although we must deal with gauge dependence and infrared divergence to complete the proof. Next we perform numerical simulations in spatially homogeneous configurations to investigate thermalization properties of the system in 1+1 and 2+1 dimensions for φ4 theory with NLO self-energy of the coupling expansion and 1+1 dimensions for O(N) theory with NLO self-energy of 1/N expansion. We also estimate the time evolution of the kinetic entropy in the case of scalar φ4 and O(N) theory. Then we find that at later times X0≫1/m, where m represents mass of particles, the kinetic entropy increases monotonically and approaches the equilibrium value, although the limited time interval at the earlier times X0～1/m invalidates the use of it. No thermalization occurs in the simulations with the normal Boltzmann equation with 2-to-2 collision term in 1+1 dimensions, while it occurs in the simulations with KB equation in 1+1 dimensions. It is due to two types of offshell effects: memory effects and spectral functions with the decay width.
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