Continuum Derrida Approach to Drift and Diffusivity in Random Media

TL;DR

This paper applies Derrida's finite-size sampling approach to analyze effective drift and diffusivity in random media, establishing identities and relationships that enhance understanding of particle transport in complex systems.

Contribution

It introduces a Derrida-based method for computing asymptotic transport properties in random media, revealing identities and constraints not previously verified beyond perturbation theory.

Findings

Proves the equality of effective drift and diffusivity tensors under certain conditions.

Derives a relationship between diffusivity in fluctuating media and gradient drift problems.

Establishes a Ward identity for effective transport tensors in specific random media.

Abstract

By means of rather general arguments, based on an approach due to Derrida that makes use of samples of finite size, we analyse the effective diffusivity and drift tensors in certain types of random medium in which the motion of the particles is controlled by molecular diffusion and a local flow field with known statistical properties. The power of the Derrida method is that it uses the equilibrium probability distribution, that exists for each {\em finite} sample, to compute asymptotic behaviour at large times in the {\em infinite} medium. In certain cases, where this equilibrium situation is associated with a vanishing microcurrent, our results demonstrate the equality of the renormalization processes for the effective drift and diffusivity tensors. This establishes, for those cases, a Ward identity previously verified only to two-loop order in perturbation theory in certain models. The technique can be applied also to media in which the diffusivity exhibits spatial fluctuations. We derive a simple relationship between the effective diffusivity in this case and that for an associated gradient drift problem that provides an interesting constraint on previously conjectured results.