Probabilistic Derivation of Spatiotemporal Correlation Functions in the Hydrodynamic Limit

In this paper we use probability theory to prove, in suitable conditions, the equivalence of time correlation functions of microscopic density fields in the equilibrium ensemble with the time correlation functions of local macroscopic density fields evolved by hydrodynamics in (approximate) phenomenological continuum theories of matter. We further discuss a useful and rigorous numerical algorithm which can be derived from the same principles and permits to compute macroscopic space- and time-dependent be- haviors (such as, in suitable conditions, the hydrodynamical) from molecular dynamics simulations.