# Drift-diffusion processes from elimination of fast variables under inhomogeneous conditions

## TL;DR

This paper develops a unified method to eliminate fast variables in inhomogeneous environments, deriving effective drift-diffusion processes for particles, applicable to both passive and active motion, with detailed generator calculations.

## Contribution

It introduces a straightforward approach to derive effective drift-diffusion models from complex systems with position-dependent velocities and fast variables, enhancing understanding of environmental effects on particle motion.

## Key findings

- Unified framework for eliminating fast variables
- Explicit drift-diffusion generators for various models
- Insights into inhomogeneity-induced drift in passive and active particles

## Abstract

The problem of eliminating fast-relaxing variables to obtain an effective drift-diffusion process in position is solved in a uniform and straightforward way for models with velocity a function jointly of position and fast variables. A more unified view is thereby obtained of the effect of environmental inhomgeneity on the motion of a diffusing particle, in particular, whether a drift is induced, covering both passive and active particles. Infinitesimal generators (equivalently, drift-diffusion fields) for the contracted processes are worked out in detail for several models.