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Journal of Physics Conference Series

Comparing Boltzmann vs. Kadanoff-Baym

Markus Michael Müller

Published Apr 1, 2006• 5 citations
Publisher

Abstract

(Classical) Boltzmann equations suffer from several shortcomings as compared to their quantum mechanical generalizations, the so-called Kadanoff-Baym equations. Nevertheless, in practical calculations approximations to Boltzmann equations are widely used. A prominent example is the computation of the baryon asymmetry of the universe. In this work, we present a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar phi 4 quantum field theory. We solve the Kadanoff-Baym and the Boltzmann equations numerically in 3 + 1 dimensions and compare their predictions on the evolution of systems out of thermal equilibrium for various initial conditions. The obtained numerical solutions reveal significant discrepancies between both types of equations. Most notably, the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations.

DOI
10.1088/1742-6596/35/1/036

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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.

•Mar 24, 2010•0 citations
(Classical) Boltzmann equations suffer from several shortcomings as compared to their quantum mechanical generalizations, the so-called Kadanoff-Baym equations. Nevertheless, in practical calculations approximations to Boltzmann equations are widely used. A prominent example is the computation of the baryon asymmetry of the universe. In this work, we present a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar phi 4 quantum field theory. We solve the Kadanoff-Baym and the Boltzmann equations numerically in 3 + 1 dimensions and compare their predictions on the evolution of systems out of thermal equilibrium for various initial conditions. The obtained numerical solutions reveal significant discrepancies between both types of equations. Most notably, the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations.
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after relativistic heavy ion collisions. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this paper a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory in 3+13+13+1 space-time dimensions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by the Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for the Boltzmann equation.
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this thesis a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of relativistic quantum field theories in 3+1 space-time dimensions. In a first step, for simplicity we consider a real scalar Phi^4 quantum field theory and in a second step we generalize our results to a chirally invariant Yukawa-type quantum field theory including fermions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for Boltzmann equations.
Boltzmann equations are often used to describe the nonequilibrium time-evolution of many-body systems in particle physics. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process, which is described by so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the complete Kadanoff-Baym equations. Therefore, we present in this article a detailed comparison of the Boltzmann and the Kadanoff-Baym equations in the framework of a chirally invariant Yukawa-type quantum field theory including fermions and scalars. The obtained numerical results reveal significant differences between both types of equations. Apart from quantitative differences, on a qualitative level the late-time universality respected by the Kadanoff-Baym equations is severely restricted in the case of the Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. In contrast to this standard Boltzmann equations cannot describe the process of quantum-chemical equilibration, and consequently also cannot feature the above separation of time scales.
We solve the Kadanoff-Baym equations for nonequilibrium initial configurations of the φ 4-theory in 2+1 dimensions and compare to explicit solutions of generalized transport equations for the same theory. The latter transport equations are derived from the Kadanoff-Baym equation in a first order gradient expansion in phase space and explicitly retain the off-shell dynamics as inherent in the timedependent spectral functions. The solutions of these equations compare very well with the exact solutions of the full Kadanoff-Baym equations with respect to the occupation numbers of the individual modes, the spectral evolution as well as the chemical equilibration process. Furthermore, the proper equilibrium off-shell distribution is reached for large times contrary to the quasiparticle Boltzmann limit. We additionally present a direct comparison of the solution of the generalized transport equations in the Kadanoff-Baym and Botermans-Malfliet form; both solutions are found to agree very well with each other. PACS: 06.60.+w; 05.70.Ln; 11.10.Wx; 24.10.-i; 25.75.-q Keywords: Many-body theory; finite temperature field theory; out-of-equilibrium quantum field theory; nonequilibrium statistical physics; relativistic heavy-ion collisions Part of the PhD thesis of S. Juchem
The nonequilibrium dynamics of quantum fields is an initial-value problem, which can be described by Kadanoff-Baym equations. Typically, and, in particular, when numerical solutions are demanded, these Kadanoff-Baym equations are restricted to Gaussian initial states. However, physical initial states are non-Gaussian correlated initial states. In particular, renormalizability requires the initial state to feature nnn-point correlations that asymptotically agree with the vacuum correlations at short distances. In order to identify physical nonequilibrium initial states, it is therefore a precondition to describe the vacuum correlations of the interacting theory within the nonequilibrium framework. In this paper, Kadanoff-Baym equations for non-Gaussian correlated initial states describing vacuum and thermal equilibrium are derived from the 2PI effective action. A diagrammatic method for the explicit construction of vacuum and thermal initial correlations from the 2PI effective action is provided. We present numerical solutions of Kadanoff-Baym equations for a real scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory, which take the thermal initial four-point correlation as the leading non-Gaussian correction into account. We find that this minimal non-Gaussian initial condition yields an approximation to the complete equilibrium initial state that is quantitatively and qualitatively significantly improved as compared to Gaussian initial states.
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.