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arXiv:hep-ph/0010239 · AIP conference proceedings

Thermalization of inhomogeneous quantum scalar fields in 1+1D

M. Sallé

Published Jan 1, 2001• 0 citations
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Abstract

Using an improved version of the Hartree approximation, allowing for ensembles of inhomogeneous configurations, we show in a λφ 4 theory, that initially the system thermalises with a Bose-Einstein distribution.For later times and larger couplings we see deviations.
DOI
10.1063/1.1363556

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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.
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
We study the dynamics of a spatially inhomogeneous quantum $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in 1+11+11+1 dimensions in the Hartree approximation. In particular, we investigate the long-time behavior of this approximation in a variety of controlled situations, both at zero and finite temperature. The observed behavior is much richer than that in the spatially homogeneous case and, unlike the latter, shows some universality at long times. Nevertheless, we show that, within a family of Gaussian spatial profiles for the mean field, the evolution fails to thermalize in a quantum canonical sense, as expected from analogous results in closely related (mean field) transport theory. We argue that this dynamical approximation is best suited as a means to study the short-time decay of spatially inhomogeneous fields and in the dynamics of coherent quasiclassical inhomogeneous configurations (e.g. topological defects) in a background of dynamical self-consistent quantum fluctuations.