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Dissipative processes abound in most areas of sciences and can often beabstractly written as $\partial_t z = K(z) \delta S(z)/\delta z$, which is agradient flow of the entropy $S$. Although various techniques have beendeveloped to compute the entropy, the calculation of the operator $K$ fromunderlying particle models is a major long-standing challenge. Here, we showthat discretizations of diffusion operators $K$ can be numerically computedfrom particle fluctuations via an infinite-dimensional fluctuation-dissipationrelation, provided the particles are in local equilibrium with Gaussianfluctuations. A salient feature of the method is that $K$ can be fullypre-computed, enabling macroscopic simulations of arbitrary admissible initialdata, without any need of further particle simulations. We test thiscoarse-graining procedure for a zero-range process in one space dimension andobtain an excellent agreement with the analytical solution for the macroscopicdensity evolution. This example serves as a blueprint for a new multiscaleparadigm, where full dissipative evolution equations --- and not onlyparameters --- can be numerically computed from particles.

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