Exact fixed points in discrete hydrodynamics

Abstract

Recently a discrete formulation of hydrodynamics was introduced, which was shown to be exactly renormalizable in a certain sense: a procedure was given for computing the equations of motion on a coarse space and time scale from those on a finer scale. In this paper we carry out this coarsening procedure explicitly, giving exact numerical results for a one-dimensional diffusive system. The coarsening transformation is found to have a one-parameter family of nontrivial fixed points, parameterized by a diffusion parameterD. This result gives a new way of understanding why so many systems obey Fick's lawj = − D'dn/dx. Any of an extremely broad class of microscopic equations of motion, when viewed on a coarse enough scale, obey the fixed-point equations (which are equivalent to Fick's law). The methods used here are equally applicable to higher-dimensionality systems such as fluids.