Microscopic Derivation of Causal Diffusion Equation using Projection Operator Method

TL;DR

This paper derives a causal diffusion equation from microscopic principles using the projection operator method, highlighting the importance of memory effects and challenging traditional definitions of diffusion constants.

Contribution

It provides a microscopic derivation of a causal diffusion equation that incorporates memory effects and aligns with physical constraints, unlike traditional acausal models.

Findings

Derived a causal integrodifferential diffusion equation

Confirmed the equation's consistency with causality and conservation laws

Showed that current-current correlations may not define diffusion constants accurately

Abstract

We derive a coarse-grained equation of motion of a number density by applying the projection operator method to a non-relativistic model. The derived equation is an integrodifferential equation and contains the memory effect. The equation is consistent with causality and the sum rule associated with the number conservation in the low momentum limit, in contrast to usual acausal diffusion equations given by using the Fick's law. After employing the Markov approximation, we find that the equation has the similar form to the causal diffusion equation. Our result suggests that current-current correlations are not necessarily adequate as the definition of diffusion constants.