Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Kadanoff-Baym Theory for Thermalization of Quantum Fields

Akihiro Nishiyama

Published Mar 24, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.15083/00002503

Related papers

Comparison of Boltzmann Kinetics with Quantum Dynamics for Relativistic Quantum Fields

Markus Michael Müller

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.

•Jan 1, 2006•2 citations

Quantum dynamics and thermalization for out-of-equilibrium φ4 theory

S. Juchem, W. Cassing, Carsten Greiner

The quantum time evolution of ${\ensuremath{\varphi}}^{4}$-field theory for a spatially homogeneous system in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym equations including the tadpole and sunset self-energies. Whereas the tadpole self-energy yields a dynamical mass, the sunset self-energy is responsible for dissipation and an equilibration of the system. In particular we address the dynamics of the spectral (``off-shell'') distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. The investigation explicitly demonstrates that the only translation invariant solutions representing the stationary fixed points of the coupled equation of motions are those of full thermal equilibrium. They agree with those extracted from the time integration of the Kadanoff-Baym equations for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ Furthermore, a detailed comparison of the full quantum dynamics to more approximate and simple schemes such as that of a standard kinetic (on-shell) Boltzmann equation is performed. Our analysis shows that the consistent inclusion of the dynamical spectral function has a significant impact on relaxation phenomena. The different time scales that are involved in the dynamical quantum evolution towards a complete thermalized state are discussed in detail. We find that far off-shell $1\ensuremath{\leftrightarrow}3$ processes are responsible for chemical equilibration, which is missed in the Boltzmann limit. Finally, we briefly address the case of (bare) massless fields. For sufficiently large couplings $\ensuremath{\lambda}$ we observe the onset of Bose condensation, where our scheme within symmetric ${\ensuremath{\varphi}}^{4}$ theory breaks down.

arXiv:hep-ph/0307353•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 30, 2004•139 citations

Nonequilibrium Quantum-Field Dynamics and Off-Shell Transport

S. Juchem

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

•Jun 12, 2015•2 citations

Nonequilibrium quantum fields: Closed-time-path effective action, Wigner function, and Boltzmann equation

Esteban Calzetta, B. L. Hu

This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 15, 1988•539 citations

Comparison of Boltzmann equations with quantum dynamics for scalar fields

M. Lindner, Markus Michael Müller

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 $3+1$ 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.

arXiv:hep-ph/0512147•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jun 5, 2006•50 citations

Quantum kinetic field theory

Esteban Calzetta, B. L. Hu

The Kadanoff–Baym equations Quantum kinetic field theory is the theme of this chapter. In this section we get right to the heart of it by showing a derivation of the celebrated Kadanoff-Baym (KB) equations [KadBay62]. The basic idea is that close to equilibrium, propagators are nearly translation invariant. It is possible to define a partial Fourier transform with respect to the relative position of the arguments. The Kadanoff – Baym equations then determine how the partial Fourier transform depends on the average (or “center of mass”) of the arguments in the original propagators. Besides the presentation in Kadanoff and Baym's textbook, there are several derivations of these equations in the literature [Dub67, Dan84a, MroDan90, MroHei94, ZhuHei98]. We shall follow [CalHu88, CaHuRa00]. See also [Hen95, IvKnVo00, KnIvVo01, Nie02, Koi02]. References [BoVeWa00, WBVS00] follow a different path towards quantum kinetic theory, based on the so-called dynamical renormalization group. The model . To better appreciate the main points in this derivation, we shall consider a simple model, namely, the KB equations for the theory of a single real self-interacting λΦ 4 scalar field, in the absence of background fields. Actually, the key ideas are not sensitive to the particular models, but for concreteness it will be helpful to have a model in mind. The classical action is given by equation (6.106). A translation-invariant propagator G ab depends on its arguments x and x ′ only through the so-called “relative” variable u = x − x ′.

Cambridge University Press eBooks•Jul 24, 2008•0 citations
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.
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.
The quantum time evolution of ${\ensuremath{\varphi}}^{4}$-field theory for a spatially homogeneous system in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym equations including the tadpole and sunset self-energies. Whereas the tadpole self-energy yields a dynamical mass, the sunset self-energy is responsible for dissipation and an equilibration of the system. In particular we address the dynamics of the spectral (``off-shell'') distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. The investigation explicitly demonstrates that the only translation invariant solutions representing the stationary fixed points of the coupled equation of motions are those of full thermal equilibrium. They agree with those extracted from the time integration of the Kadanoff-Baym equations for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ Furthermore, a detailed comparison of the full quantum dynamics to more approximate and simple schemes such as that of a standard kinetic (on-shell) Boltzmann equation is performed. Our analysis shows that the consistent inclusion of the dynamical spectral function has a significant impact on relaxation phenomena. The different time scales that are involved in the dynamical quantum evolution towards a complete thermalized state are discussed in detail. We find that far off-shell $1\ensuremath{\leftrightarrow}3$ processes are responsible for chemical equilibration, which is missed in the Boltzmann limit. Finally, we briefly address the case of (bare) massless fields. For sufficiently large couplings $\ensuremath{\lambda}$ we observe the onset of Bose condensation, where our scheme within symmetric ${\ensuremath{\varphi}}^{4}$ theory breaks down.
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
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
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.
The Kadanoff–Baym equations Quantum kinetic field theory is the theme of this chapter. In this section we get right to the heart of it by showing a derivation of the celebrated Kadanoff-Baym (KB) equations [KadBay62]. The basic idea is that close to equilibrium, propagators are nearly translation invariant. It is possible to define a partial Fourier transform with respect to the relative position of the arguments. The Kadanoff – Baym equations then determine how the partial Fourier transform depends on the average (or “center of mass”) of the arguments in the original propagators. Besides the presentation in Kadanoff and Baym's textbook, there are several derivations of these equations in the literature [Dub67, Dan84a, MroDan90, MroHei94, ZhuHei98]. We shall follow [CalHu88, CaHuRa00]. See also [Hen95, IvKnVo00, KnIvVo01, Nie02, Koi02]. References [BoVeWa00, WBVS00] follow a different path towards quantum kinetic theory, based on the so-called dynamical renormalization group. The model . To better appreciate the main points in this derivation, we shall consider a simple model, namely, the KB equations for the theory of a single real self-interacting λΦ 4 scalar field, in the absence of background fields. Actually, the key ideas are not sensitive to the particular models, but for concreteness it will be helpful to have a model in mind. The classical action is given by equation (6.106). A translation-invariant propagator G ab depends on its arguments x and x ′ only through the so-called “relative” variable u = x − x ′.