1 Damping and the Hartree Ensemble Approximation

M. Sallé, Jan Smit A, Jeroen C. Vink

Published Jan 1, 2000• 0 citations

Abstract

We study a Hartree ensemble approximation for real-time dynamics in the toy model of 1+1 dimensional scalar field theory. Damping behavior seen in numerical simulations is compared with analytical predictions based on perturbation theory in the original (non-Hartree-approximated) model. 1.

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1 Damping and the Hartree Ensemble Approximation

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Published Jan 1, 2000• 0 citations

OpenAlex

Abstract

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William Silvert

American Journal of PhysicsJan 1, 19720 citations

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

{N}_{f},

and

n\mathrm{PI}

-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)

-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.

We study the nonequilibrium dynamics of the

\mathrm{O}(N)

model in classical and quantum field theory in 1+1 dimensions, for

N>1.

We compare numerical results obtained using the Hartree approximation and two next to leading order approximations: the bare vertex approximation and the two-particle irreducible

1/N

expansion. The latter approximations differ through terms of order

{g}^{2},

where g is the scaled coupling constant, $g=\ensuremath{\lambda}/N.$ In this paper we investigate the statement regarding the convergence with respect to g. The differences between these two approximation schemes are computed and found to be already small for relatively small values of N, when $\ensuremath{\lambda}$ is fixed.