2018/05/15 by Li, Xiaoguai, Dirr, Nicolas, Embacher, Peter +2
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1805.05788
Dissipative processes abound in most areas of sciences and can often be abstractly written as ∂t z = K(z) δS(z)/δz, 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.