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SCALAR FIELD DYNAMICS: CLASSICAL, QUANTUM AND IN BETWEEN ∗

M. Sallé, Jan Smit A, Jeroen C. Vink, C.P. Korthals Altes

Published Jan 1, 2000• 1 citation
OpenAlex

Abstract

Using a Hartree ensemble approximation, we investigate the dynamics of the \\phi^4 model in 1+1 dimensions. We find that the fields initially thermalize with a Bose-Einstein distribution for the fields. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved.

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On Thermalization in Classical Scalar Field Theory

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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.

•Jan 1, 2000•34 citations
Using a Hartree ensemble approximation, we investigate the dynamics of the \phi^4 model in 1+1 dimensions. We find that the fields initially thermalize with a Bose-Einstein distribution for the fields. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved.
We describe a Hartree ensemble method to approximately solve the Heisenberg equations for the ϕ 4 model in 1 + 1 dimensions. We compute the energies and number densities of the quantum particles described by the ϕ field and find that the particles initially thermalize with a Bose-Einstein distribution for the particle density. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved. 1. Hartree ensemble approximation Non-perturbative equilibrium properties of quantum field theories are studied with great success using numerical lattice techniques. Similarly one would like to investigate quantum field dynamics non-perturbatively. This has obvious applications to e.g. heavy ion collisions and early universe physics. Furthermore, it could be used to study equilibrium properties in cases where Monte Carlo methods do not work, such as in QCD at finite density or chiral gauge theories. Using Monte Carlo techniques to compute the Minkowski time path integral is, of course, not possible. Neither can one directly solve the Heisenberg equations for the quantum field operators. Hence one has to resort to approximations, such as classical dynamics [2] (for recent work see [3]), large n or Hartree (see e.g. [4]). Here we discuss a different type of Hartree or gaussian approximation than used previously, in which we use an ensemble of gaussian wavefunctions to compute Green functions. Ref. [1] also contains a discussion of this method and a detailed presentation will appear elsewhere. We use the lattice ϕ4 model in 1+1 dimensions as a test model, which has the following Heisenberg operator equations, ˙ˆϕ = ˆπ, ˙ ˆπ = ( ∆ − µ
We describe a Hartree ensemble method to approximately solve the Heisenberg equations for the ϕ 4 model in 1 + 1 dimensions. We compute the energies and number densities of the quantum particles described by the ϕ field and find that the particles initially thermalize with a Bose-Einstein distribution for the particle density. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved. 1. Hartree ensemble approximation Non-perturbative equilibrium properties of quantum field theories are studied with great success using numerical lattice techniques. Similarly one would like to investigate quantum field dynamics non-perturbatively. This has obvious applications to e.g. heavy ion collisions and early universe physics. Furthermore, it could be used to study equilibrium properties in cases where Monte Carlo methods do not work, such as in QCD at finite density or chiral gauge theories. Using Monte Carlo techniques to compute the Minkowski time path integral is, of course, not possible. Neither can one directly solve the Heisenberg equations for the quantum field operators. Hence one has to resort to approximations, such as classical dynamics [2] (for recent work see [3]), large n or Hartree (see e.g. [4]). Here we discuss a different type of Hartree or gaussian approximation than used previously, in which we use an ensemble of gaussian wavefunctions to compute Green functions. Ref. [1] also contains a discussion of this method and a detailed presentation will appear elsewhere. We use the lattice ϕ4 model in 1+1 dimensions as a test model, which has the following Heisenberg operator equations, ˙ˆϕ = ˆπ, ˙ ˆπ = ( ∆ − µ
For homogeneous initial conditions, Hartree (Gaussian) dynamical approximations are known to have problems with thermalization because of insufficient scattering. We attempt to improve on this by writing an arbitrary density matrix as a superposition of Gaussian pure states and applying the Hartree approximation to each member of such an ensemble. Particles can then scatter via their back reaction on the typically inhomogeneous mean fields. Starting from initial states that are far from equilibrium we numerically compute the time evolution of particle distribution functions and observe that they indeed display approximate thermalization on intermediate time scales by approaching a Bose-Einstein form. However, for very large times the distributions drift towards classical-like equipartition.
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.