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arXiv:hep-ph/0306124 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Approach to thermalization in the classical φ4 theory in 1+1 dimensions: Energy cascades and universal scaling

D. Boyanovsky, C. Destri, Hector J. de Vega

Published Feb 9, 2004• 56 citations
arXivHTMLPDF

Abstract

We study the dynamics of thermalization and the approach to equilibrium in the classical ${\ensuremath{\varphi}}^{4}$ theory in $1+1$ spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. In this context, comparing to the exact results we find that the Hartree approximation is remarkably accurate in equilibrium. The nonequilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities. The long time evolution is described by several distinct stages, all characterized by a cascade of energy towards the ultraviolet. After an initial transient stage, the spatiotemporal gradient terms become larger than the nonlinear term, and there emerges a stage of universal cascade. This cascade starts at a time scale ${t}_{0}$ independent of the initial conditions (except for very low energy density). During this stage the power spectra feature universal scaling behavior and the front of the cascade $\overline{k}(t)$ moves to the ultraviolet as a power law $\overline{k}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\lesssim}0.25$ an exponent weakly dependent on the energy density alone. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperature ${T}_{\mathrm{eff}}(t),$ which slowly decreases with time as $\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\alpha}}.$ Two well separated time scales emerge: while ${T}_{\mathrm{eff}}(t)$ varies slowly, the wave vectors in the wake with $k<\overline{k}(t)$ attain LTE on much shorter time scales. This universal scaling stage ends when the front of the ultraviolet cascade reaches the cutoff at a time scale ${t}_{1}\ensuremath{\sim}{a}^{\ensuremath{-}1/\ensuremath{\alpha}}.$ Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.

DOI
10.1103/physrevd.69.045003

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We study the dynamics of thermalization and the approach to equilibrium in the classical phi^4 theory in 1+1 spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. We find that the Hartree approximation is remarkably accurate in equilibrium. The non-equilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities.The time evolution is described by several stages with a cascade of energy towards the ultraviolet. After a transient stage, the spatio-temporal gradient terms become larger than the nonlinear term and a stage of universal cascade emerges.This cascade starts at a time scale t_0 independent of the initial conditions (except for very low energy density). Here the power spectra feature universal scaling behavior and the front of the cascade k(t) grows as a power law k(t) sim t^alpha with alpha lesssim 0.25. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperatureTeff(t) which slowly decreases as sim t^{-alpha}.Two well separated time scales emerge while Teff(t) varies slowly, the wavectors in the wake with k < k(t) attain LTE on much shorter time scales.This universal scaling stage ends when the front of the cascade reaches the cutoff at a time t_1 sim a^{-1/alpha}. Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.

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We study the dynamics of thermalization and the approach to equilibrium in the classical phi^4 theory in 1+1 spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. We find that the Hartree approximation is remarkably accurate in equilibrium. The non-equilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities.The time evolution is described by several stages with a cascade of energy towards the ultraviolet. After a transient stage, the spatio-temporal gradient terms become larger than the nonlinear term and a stage of universal cascade emerges.This cascade starts at a time scale t_0 independent of the initial conditions (except for very low energy density). Here the power spectra feature universal scaling behavior and the front of the cascade k(t) grows as a power law k(t) sim t^alpha with alpha lesssim 0.25. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperatureTeff(t) which slowly decreases as sim t^{-alpha}.Two well separated time scales emerge while Teff(t) varies slowly, the wavectors in the wake with k < k(t) attain LTE on much shorter time scales.This universal scaling stage ends when the front of the cascade reaches the cutoff at a time t_1 sim a^{-1/alpha}. Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.
We investigate the dynamics of thermalization and the approach to equilibrium in the classical ${\ensuremath{\phi}}^{4}$ theory in 3+13+13+1 spacetime dimensions. The nonequilibrium dynamics is studied by numerically solving the equations of motion in a light-cone-like discretization of the model for a broad range of initial conditions and energy densities. A smooth cascade of energy towards the ultraviolet is found to be the basic mechanism of thermalization. After an initial transient stage, at a time scale of several hundred inverse masses, the squared magnitude of the field spatial gradient becomes larger than the nonlinear term and there emerges a stage of universal cascade, independent of the details of the initial conditions. As the cascade progresses, the modes with higher wave numbers, but well behind the forefront of the cascade, exhibit weaker and weaker nonlinearities well described by the Hartree approximation, while the infrared modes retain strong self-interactions. As a consequence, two time scales for equilibration appear as characteristic of two time-dependent wave number regions. For ${k}^{2}\ensuremath{\gtrsim}\overline{{\ensuremath{\phi}}^{2}}(t)$, we observe an effective equilibration to a time-dependent powerlike spectrum with a time scale in the hundreds of inverse masses; cutoff effects are absent and the Hartree approximation holds for ${k}^{2}\ensuremath{\gg}\overline{{\ensuremath{\phi}}^{2}}(t)$. On the other hand, infrared modes with ${k}^{2}\ensuremath{\lesssim}\overline{{\ensuremath{\phi}}^{2}}(t)$ equilibrate only by time scales in the millions of inverse masses when the cutoff effects become dominant and complete thermalization is setting in. Accordingly, we observe in the field correlator a relatively large and long-lived deviation from the Hartree behavior of a nonperturbative character. There corresponds an effective mass governing the long distance behavior of the correlator which turns out to be significantly smaller than the Hartree mass which is exhibited by the modes with ${k}^{2}\ensuremath{\gtrsim}\overline{{\ensuremath{\phi}}^{2}}(t)$. Virialization and the equation of state start to set in much earlier than thermalization. The applicability of these results in quantum field theory for large occupation numbers and small coupling is analyzed.
Thermalization of classical fields is investigated in a \\phi^4 scalar field theory in 1+1 dimensions, discretized on a lattice. We numerically integrate the classical equations of motion using initial conditions sampled from various nonequilibrium probability distributions. Time-dependent expectation values of observables constructed from the canonical momentum are compared with thermal ones. It is found that a closed system, evolving from one initial condition, thermalizes to high precision in the thermodynamic limit, in a time-averaged sense. For ensembles consisting of many members with the same energy, we find that expectation values become stationary - and equal to the thermal values - in the limit of infinitely many members. Initial ensembles with a nonzero (noncanonical) spread in the energy density or other conserved quantities evolve to noncanonical stationary ensembles. In the case of a narrow spread, asymptotic values of primary observables are only mildly affected. In contrast, fluctuations and connected correlation functions will differ substantially from the canonical values. This raises doubts on the use of a straightforward expansion in terms of 1PI-vertex functions to study thermalization.
Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large Nf,{N}_{f},Nf​, and nPIn\mathrm{PI}nPI-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a (1+1)(1+1)(1+1)-dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.
Whether and how a system approaches equilibrium is a central issue in nonequilibrium statistical physics, underpinning our understanding of both thermalization and heat transport. Bogoliubov's three-stage (initial, kinetic, and hydrodynamic) evolution hypothesis provides a qualitative framework, but quantitative advances have mostly been restricted to near-integrable systems such as dilute gases. In this work, we investigate the relaxation dynamics of a one-dimensional diatomic hard-point gas and present a phase diagram that comprehensively characterizes relaxation behavior across the full parameter space, from near-integrable to far-from-integrable regimes. We first analyze thermalization (local energy relaxation in nonequilibrium states) and identify three universal dynamical regimes: (1) In the near-integrable regime, kinetic processes dominate (initial + kinetic phase), local energy relaxation decays exponentially, and the thermalization time τ scales with perturbation strength δ as τ ∝ δ − 2 . (2) In the far-from-integrable regime, hydrodynamic effects dominate (initial + hydrodynamic phase), local energy relaxation follows power-law decay, and thermalization time scales linearly with system size N . (3) In the intermediate regime, Bogoliubov phase emerges, marked by the transition from kinetic to hydrodynamic relaxation. The phase diagram further reveals that hydrodynamic behavior can emerge even in small systems when the system is sufficiently far from the integrable regime, challenging the conventional view that such effects are only significant in large systems. In the thermodynamic limit, the system's relaxation behavior depends on the order in which the limits ( N → ∞ or δ → 0 ) are taken. We then turn to heat transport (decay of heat-current fluctuations in equilibrium) and demonstrate that its behavior is fully consistent with the thermalization process, leading to the first unified theoretical description of thermalization and heat transport. Our results establish a comprehensive view of relaxation in classical many-body systems and suggest a route for studying similar dynamics in quantum systems, highlighting universal scaling laws and the interplay of kinetic and hydrodynamic processes across integrability regimes.
Recent progress in out-of-equilibrium closed quantum systems has significantly advanced the understanding of mechanisms behind their evolution toward thermalization. Notably, the concept of nonthermal fixed points (NTFPs)-responsible for the emergence of spatiotemporal universal scaling in far-from-equilibrium systems-has played a crucial role in both theoretical and experimental investigations. In this work, we introduce a differential equation that has the universal scaling associated with NTFPs as a solution. The advantage of working with a differential equation, rather than only with its solution, is that we can extract several insightful properties not necessarily present in the solution alone. How the differential equation is derived allows physical interpretation of the universal exponents in terms of the time dependence of the amplitude of the distributions and their momentum scaling. Employing two limiting cases of the equation, we determined the universal exponents related to the scaling using the distributions near just two momentum values. We established a solid agreement with previous investigations by validating this approach with three distinct physical systems. This consistency highlights the universal nature of scaling due to NTFPs and emphasizes the predictive capabilities of the proposed differential equation. Moreover, under specific conditions, the equation predicts a power-law related to the ratio of the two universal exponents, leading to implications concerning particle and energy transport. This suggests that the observed power-laws in far-from-equilibrium turbulent fluids could be related to the universal scaling due to NTFPs, potentially offering insights into the study of turbulence.