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

Dissipative processes abound in most areas of sciences and can often be abstractly written as ${\mathrm{\partial }}_{t}z=K(z)\delta S(z)\mathrm{/}\delta z$, 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.