Continuum versus periodic lattice Monte Carlo approach to classical field theory

TL;DR

This paper compares momentum space and periodic lattice Monte Carlo methods in classical ^4 field theory, showing they converge to the same result in the continuum limit despite initial mismatches.

Contribution

It provides a detailed comparison between momentum space and lattice Monte Carlo approaches, highlighting their convergence in the continuum limit.

Findings

Both methods converge to the same thermalized value in the continuum limit.

Initial mismatches in ^2_{ ext{cl}}(t) cause shifts at large times.

The convergence validates the equivalence of the two approaches in the continuum limit.

Abstract

We compare the momentum space with the standard periodic lattice approach to Monte Carlo calculations in classical $\phi^4$ field theory. We show that the mismatch in the initial value of $\phi^2_{\text{cl}}(t)$, results in a shift in the ``thermalized'' value, at large times. The two approaches converge to the same result in the continuum limit.