Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics

TL;DR

This paper unifies different formulations of inhomogeneous transport equations using statistical physics and develops methods to estimate spatially varying diffusion coefficients from simple measurements, validated through multiple case studies.

Contribution

It demonstrates the physical equivalence of Smoluchowski and Fokker-Planck equations and introduces an efficient estimation technique for inhomogeneous diffusion coefficients based on first-passage statistics.

Findings

The two transport equations are physically equivalent.

Estimated diffusion coefficients match simulation data.

Water mobility decreases near surfactant monolayers.

Abstract

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g. the diffusion coefficient or the heat conductivity which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated, because two alternative formulations of the inhomogeneous transport equations exist in the literature -- the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Buettikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.