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

1 Nonequilibrium Quantum Fields and the Classical Field Theory Limit

Jürgen Berges

Published Jan 1, 2002• 4 citations
OpenAlex

Abstract

We calculate the far-from-equilibrium dynamics and thermalization both for the quantum and the classical O(N)–model. The early and late-time behavior can be described from the 2PI–loop expansion for weak couplings or the nonperturbative 2PI–1/N expansion of the effective action beyond leading order. A comparison with exact simulations in 1+1 dimensions in the classical limit shows that the 2PI–1/N expansion at next-to-leading order gives quantitatively precise results already for moderate values of N. We derive a criterion for the validity of the classical approximation and verify it by comparing far-fromequilibrium quantum and classical dynamics. At late times one observes the expected deviations due to the difference between classical and quantum thermal equilibrium. In recent years we have witnessed an enormous increase of interest in the dynamics of quantum fields out of equilibrium. Strong motivation comes from a wide range of applications including current and upcoming relativistic heavy-ion collision experiments, phase transitions in the early universe or the dynamics of Bose-Einstein condensation. Directly simulating quantum fields in real time, such as solving the Schrödinger equation for the wave functional is prohibitively difficult and one has to find suitable approximations. The

Related papers

QUANTUM FIELDS FAR FROM EQUILIBRIUM AND THERMALIZATION

Jürgen Berges

I review the use of the 2PI effective action in nonequilibrium quantum field theory. The approach enables one to find approximation schemes which circumvent long-standing problems of non-thermal or secular (unbounded) late-time evolutions encountered in standard loop or 1/N expansions of the 1PI effective action. It is shown that late-time thermalization can be described from a numerical solution of the three-loop 2PI effective action for a scalar $ϕ^4$--theory in 1+1 dimensions (with Jurgen Cox, hep-ph/0006160). Quantitative results far from equilibrium beyond the weak coupling expansion can be obtained from the 1/N expansion of the 2PI effective action at next-to-leading order (NLO), calculated for a scalar O(N) symmetric quantum field theory (hep-ph/0105311). Extending recent calculations in classical field theory by Aarts et al. (hep-ph/0007357) and by Blagoev et al. (hep-ph/0106195) to $N>1$ we show that the NLO approximation converges to exact (MC) results already for moderate values of $N$ (with Gert Aarts, hep-ph/0107129). I comment on characteristic time scales in scalar quantum field theory and the applicability of classical field theory for sufficiently high initial occupation numbers.

arXiv:hep-ph/0109170•Workshop on Non-Perturbative Quantum Chromodynamics•Apr 1, 2002•0 citations

Nonequilibrium dynamics in the O(N) model to next-to-next-to-leading order in the 1/N expansion

Gert Aarts A, Anders Tranberg B

Nonequilibrium dynamics in quantum field theory has been studied extensively using truncations of the 2PI effective action. Both $1/N$ and loop expansions beyond leading order show remarkable improvement when compared to mean-field approximations. However, in truncations used so far, only the leading-order parts of the self-energy responsible for memory loss, damping and equilibration are included, which makes it difficult to discuss convergence systematically. For that reason we derive the real and causal evolution equations for an $O(N)$ model to next-to-next-to-leading order in the 2PI-$1/N$ expansion. Because of the appearance of internal vertices the resulting equations appear intractable for a full-fledged $3+1$ dimensional field theory. Instead, we solve the closely related three-loop approximation in the auxiliary-field formalism numerically in $0+1$ dimensions (quantum mechanics) and compare to previous approximations and the exact numerical solution of the Schr\"odinger equation.

arXiv:hep-th/0604156•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 6, 2006•29 citations

Introduction to Nonequilibrium Quantum Field Theory

Jürgen Berges

There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.

•Sep 20, 2004•257 citations

Classical Aspects of Quantum Fields Far from Equilibrium

Gert Aarts A, Jürgen Berges

We consider the time evolution of nonequilibrium quantum scalar fields in the O(N) model, using the next-to-leading order 1/N expansion of the two-particle irreducible effective action. A comparison with exact numerical simulations in 1+1 dimensions in the classical limit shows that the 1/N expansion gives quantitatively precise results already for moderate values of N. For sufficiently high initial occupation numbers the time evolution of quantum fields is shown to be accurately described by classical physics. Eventually the correspondence breaks down due to the difference between classical and quantum thermal equilibrium.

arXiv:hep-ph/0107129•Physical Review Letters•Jan 15, 2002•181 citations

Introduction to Nonequilibrium Quantum Field Theory

Jürgen Berges

There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.

arXiv:hep-ph/0409233•AIP conference proceedings•Jan 1, 2004•462 citations

Nonequilibrium quantum fields from first principles

Jürgen Berges, Szabolcs Borsányi

Calculations of nonequilibrium processes become increasingly feasable in quantum field theory from first principles. There has been important progress in our analytical understanding based on 2PI generating functionals. In addition, for the first time direct lattice simulations based on stochastic quantization techniques have been achieved. The quantitative descriptions of characteristic far-from-equilibrium time scales and thermal equilibration in quantum field theory point out new phenomena such as prethermalization. They determine the range of validity of standard transport or semi-classical approaches, on which most of our ideas about nonequilibrium dynamics were based so far. These are crucial ingredients to understand important phenomena in high-energy physics related to collision experiments of heavy nuclei, early universe cosmology and complex many-body systems.

•Jan 1, 2006•4 citations
We calculate the far-from-equilibrium dynamics and thermalization both for the quantum and the classical O(N)–model. The early and late-time behavior can be described from the 2PI–loop expansion for weak couplings or the nonperturbative 2PI–1/N expansion of the effective action beyond leading order. A comparison with exact simulations in 1+1 dimensions in the classical limit shows that the 2PI–1/N expansion at next-to-leading order gives quantitatively precise results already for moderate values of N. We derive a criterion for the validity of the classical approximation and verify it by comparing far-fromequilibrium quantum and classical dynamics. At late times one observes the expected deviations due to the difference between classical and quantum thermal equilibrium. In recent years we have witnessed an enormous increase of interest in the dynamics of quantum fields out of equilibrium. Strong motivation comes from a wide range of applications including current and upcoming relativistic heavy-ion collision experiments, phase transitions in the early universe or the dynamics of Bose-Einstein condensation. Directly simulating quantum fields in real time, such as solving the Schrödinger equation for the wave functional is prohibitively difficult and one has to find suitable approximations. The
I review the use of the 2PI effective action in nonequilibrium quantum field theory. The approach enables one to find approximation schemes which circumvent long-standing problems of non-thermal or secular (unbounded) late-time evolutions encountered in standard loop or 1/N expansions of the 1PI effective action. It is shown that late-time thermalization can be described from a numerical solution of the three-loop 2PI effective action for a scalar ϕ4ϕ^4ϕ4--theory in 1+1 dimensions (with Jurgen Cox, hep-ph/0006160). Quantitative results far from equilibrium beyond the weak coupling expansion can be obtained from the 1/N expansion of the 2PI effective action at next-to-leading order (NLO), calculated for a scalar O(N) symmetric quantum field theory (hep-ph/0105311). Extending recent calculations in classical field theory by Aarts et al. (hep-ph/0007357) and by Blagoev et al. (hep-ph/0106195) to N>1N>1N>1 we show that the NLO approximation converges to exact (MC) results already for moderate values of NNN (with Gert Aarts, hep-ph/0107129). I comment on characteristic time scales in scalar quantum field theory and the applicability of classical field theory for sufficiently high initial occupation numbers.
Nonequilibrium dynamics in quantum field theory has been studied extensively using truncations of the 2PI effective action. Both 1/N1/N1/N and loop expansions beyond leading order show remarkable improvement when compared to mean-field approximations. However, in truncations used so far, only the leading-order parts of the self-energy responsible for memory loss, damping and equilibration are included, which makes it difficult to discuss convergence systematically. For that reason we derive the real and causal evolution equations for an O(N)O(N)O(N) model to next-to-next-to-leading order in the 2PI-1/N1/N1/N expansion. Because of the appearance of internal vertices the resulting equations appear intractable for a full-fledged 3+13+13+1 dimensional field theory. Instead, we solve the closely related three-loop approximation in the auxiliary-field formalism numerically in 0+10+10+1 dimensions (quantum mechanics) and compare to previous approximations and the exact numerical solution of the Schr\"odinger equation.
There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.
We consider the time evolution of nonequilibrium quantum scalar fields in the O(N) model, using the next-to-leading order 1/N expansion of the two-particle irreducible effective action. A comparison with exact numerical simulations in 1+1 dimensions in the classical limit shows that the 1/N expansion gives quantitatively precise results already for moderate values of N. For sufficiently high initial occupation numbers the time evolution of quantum fields is shown to be accurately described by classical physics. Eventually the correspondence breaks down due to the difference between classical and quantum thermal equilibrium.
There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.
Calculations of nonequilibrium processes become increasingly feasable in quantum field theory from first principles. There has been important progress in our analytical understanding based on 2PI generating functionals. In addition, for the first time direct lattice simulations based on stochastic quantization techniques have been achieved. The quantitative descriptions of characteristic far-from-equilibrium time scales and thermal equilibration in quantum field theory point out new phenomena such as prethermalization. They determine the range of validity of standard transport or semi-classical approaches, on which most of our ideas about nonequilibrium dynamics were based so far. These are crucial ingredients to understand important phenomena in high-energy physics related to collision experiments of heavy nuclei, early universe cosmology and complex many-body systems.