How to find the evolution operator of dissipative PDEs from particle fluctuations?

TL;DR

This paper introduces a numerical method to compute the evolution operator of dissipative PDEs directly from particle fluctuations, enabling efficient multiscale simulations without additional particle runs.

Contribution

The authors develop a fluctuation-based approach to determine the diffusion operator in dissipative PDEs from particle data, facilitating pre-computed macroscopic models.

Findings

Accurately computed the operator K from particle fluctuations.

Validated the method with a zero-range process example.

Achieved excellent agreement with analytical solutions.

Abstract

Dissipative processes abound in most areas of sciences and can often be abstractly written as $\partial_t z = K(z) \delta S(z)/\delta z$, which is a gradient flow of the entropy $S$. Although various techniques have been developed to compute the entropy, the calculation of the operator $K$ from underlying particle models is a major long-standing challenge. Here, we show that discretizations of diffusion operators $K$ can be numerically computed from particle fluctuations via an infinite-dimensional fluctuation-dissipation relation, provided the particles are in local equilibrium with Gaussian fluctuations. A salient feature of the method is that $K$ can be fully pre-computed, enabling macroscopic simulations of arbitrary admissible initial data, without any need of further particle simulations. We test this coarse-graining procedure for a zero-range process in one space dimension and obtain an excellent agreement with the analytical solution for the macroscopic density evolution. This example serves as a blueprint for a new multiscale paradigm, where full dissipative evolution equations --- and not only parameters --- can be numerically computed from particles.