  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.
