# Isotropic finite-difference discretization of stochastic conservation laws preserving detailed balance

## TL;DR

This paper introduces an isotropic finite-difference discretization method for stochastic conservation laws that preserves detailed balance, ensuring accurate and faithful simulation of thermally fluctuating systems.

## Contribution

The authors develop a finite-difference scheme that maintains detailed balance and isotropy, with improved spatial accuracy and stable temporal integration for stochastic conservation laws.

## Key findings

- Method accurately reproduces analytical results for model B dynamics.
- Discretization preserves detailed balance and isotropy on the lattice.
- Benchmark tests show excellent agreement with theoretical predictions.

## Abstract

The dynamics of thermally fluctuating conserved order parameters are described by stochastic conservation laws. Thermal equilibrium in such systems requires the dissipative and stochastic components of the flux to be related by detailed balance. Preserving this relation in spatial and temporal discretization is necessary to obtain solutions that have fidelity to the continuum. Here, we propose a finite-difference discretization that preserves detailed balance on the lattice, has spatial error that is isotropic to leading order in lattice spacing, and can be integrated accurately in time using a delayed difference method. We benchmark the method for model B dynamics with a $\phi^{4}$ Landau free energy and obtain excellent agreement with analytical results.