Einstein's random walk and thermal diffusion

TL;DR

This paper explains Ludwig's thermal diffusion through an extended Einstein's random walk model that incorporates spatial heterogeneity, providing a clearer mathematical foundation for understanding thermal diffusion.

Contribution

It introduces a novel spatially heterogeneous random walk model to explain thermal diffusion, extending Einstein's classical homogeneous walk framework.

Findings

The model successfully captures the temperature gradient effects.

It provides a rigorous mathematical basis for thermal diffusion.

The approach bridges classical and modern understanding of the phenomenon.

Abstract

Thermal diffusion has been studied for over 150 years. Despite of the long history and the increasing importance of the phenomenon, the physics of thermal diffusion remains poorly understood. In this paper Ludwig's thermal diffusion is explained using Einstein's random walk. The only new structure added is the spatial heterogeneity of the random walk to reflect the temperature gradient of thermal diffusion. Hence, the walk length and the walk speed are location dependent functions in this paper. Then, a mathematical understanding of such a random walk gives the foundation of the thermal diffusion as clearly as the original homogeneous case of Einstein.