Controlling one-dimensional Langevin dynamics on the lattice

Abstract

Stochastic evolutions of classical field theories have recently become popular in the study of problems such as determination of the rates of topological transitions and the statistical mechanics of nonlinear coherent structures. To obtain high precision results from numerical calculations, a careful accounting of spacetime discreteness effects is essential, as well as the development of schemes to systematically improve convergence to the continuum. With a kink-bearing φ 4 field theory as the application arena, we present such an analysis for a 1+1-dimensional Langevin system. Analytical predictions and results from high resolution numerical solutions are found to be in excellent agreement.

Finite element discretization of non-linear diffusion equations with thermal fluctuations

“…In this case, the SPDE is the fundamental equation and one needs to modify the action (free energy) with counterterms that depend on the lattice spacing in order for the final theory to have a proper continuum limit [11][12][13][14][15] . On the other hand, when a "more fundamental" theory exists, as is the case in colloidal suspension where the fundamental theory are Hamilton's equations, the requirement of having a continuum limit is desirable but not essential.…”

“…(4) has a proper continuum limit in 1D but it is divergent in D > 1 due to the so called ultraviolet catastrophe. In this latter case, renormalization group techniques have been used in order to recover a continuum limit [11][12][13][14][15] . A rigorous mathematical analysis of the renormalization of SPDEs near the critical point has been conducted recently 16,17 .…”

Langevin dynamics of the pureSU(2)deconfining transition

“…In fact, in the presence of thermal noise, short and long wavelength modes are mixed during the dynamics, yielding an unphysical lattice-size sensitivity. The issue of obtaining robust results, as well as the correct ultraviolet behavior, in performing Langevin dynamics was discussed by several authors [19][20][21][22][23]. The problem, which is not a priori evident in the Langevin formulation, is related to the well-known Rayleigh-Jeans ultraviolet catastrophe in classical field theory.…”

Langevin simulation of scalar fields: Additive and multiplicative noises and lattice renormalization

“…For the standard GLL equation this was shown extensively in a series of previous papers [33,34,35,36], but we are not aware of the same demonstration for the case including multiplicative noise terms. From the results shown in Figs.…”

“…Such a lattice spacing sensitivity is also present in the numerical simulation of SGP equations [31,32]. The issue of obtaining robust results, as well as the correct ultraviolet behavior, in performing Langevin dynamics was discussed by several authors [33,34,35,36,37,38].…”