Statistical Mechanics of Kinks in (1+1)-Dimensions: Numerical Simulations and Double Gaussian Approximation

TL;DR

This paper combines numerical simulations and a double Gaussian approximation to analyze the thermal behavior of kinks in a 1+1 dimensional $^4$ field theory, identifying temperature regimes where different descriptions are valid.

Contribution

It introduces a double Gaussian variational method to analytically study kink properties and their breakdown temperature, aligning with simulation results.

Findings

Dilute gas approximation is accurate below a certain temperature.

Double Gaussian ansatz predicts kink number and correlation length accurately.

Specific heat peak is due to kink contributions near the breakdown temperature.

Abstract

We investigate the thermal equilibrium properties of kinks in a classical $\F^4$ field theory in $1+1$ dimensions. From large scale Langevin simulations we identify the temperature below which a dilute gas description of kinks is valid. The standard dilute gas/WKB description is shown to be remarkably accurate below this temperature. At higher, ``intermediate'' temperatures, where kinks still exist, this description breaks down. By introducing a double Gaussian variational ansatz for the eigenfunctions of the statistical transfer operator for the system, we are able to study this region analytically. In particular, our predictions for the number of kinks and the correlation length are in agreement with the simulations. The double Gaussian prediction for the characteristic temperature at which the kink description ultimately breaks down is also in accord with the simulations. We also analytically calculate the internal energy and demonstrate that the peak in the specific heat near the kink characteristic temperature is indeed due to kinks. In the neighborhood of this temperature there appears to be an intricate energy sharing mechanism operating between nonlinear phonons and kinks.