# Steady diffusion in a drift field: a comparison of large deviation techniques and multiple-scale analysis

## TL;DR

This paper compares large deviation techniques and multiple-scale analysis to derive effective advection-diffusion properties of particles with internal states in a force field, providing new methods and insights into their diffusion behavior.

## Contribution

It introduces two novel methods—large deviation and multiple-scale analysis—for deriving effective diffusion and drift in particles with internal states, applicable to both discrete and continuous cases.

## Key findings

- Effective drift velocity is proportional to the averaged mobility.
- Effective diffusion has an equilibrium term and a force-dependent quadratic term.
- Measuring diffusion components can determine kinetic rates from moments.

## Abstract

A particle with internal unobserved states diffusing in a force field will generally display effective advection-diffusion. The drift velocity is proportional to the mobility averaged over the internal states, or effective mobility, while the effective diffusion has two terms. One is of the equilibrium type and satisfies an Einstein relation with the effective mobility while the other is quadratic in the applied force. In this contribution we present two new methods to obtain these results, on the one hand using large deviation techniques, and on the other by a multiple-scale analysis, and compare the two. We consider both systems with discrete internal states and continuous internal states. We show that the auxiliary equations in the multiple-scale analysis can also be derived in second-order perturbation theory in a large deviation theory of a generating function (discrete internal states) or generating functional (continuous internal states). We discuss that measuring the two components of the effective diffusion give a way to determine kinetic rates from only first and second moments of the displacement in steady state.