# Diffusion equations in inhomogeneous solid having arbitrary gradient concentration

## TL;DR

This paper derives a quantum kinetic equation for inhomogeneous solids with arbitrary concentration gradients, leading to a nonlinear diffusion equation that generalizes Fick's and Cahn's equations, enhancing understanding of atom migration.

## Contribution

It introduces a new quantum kinetic framework for inhomogeneous solids with arbitrary gradients, unifying and extending classical diffusion models.

## Key findings

- Derived a quantum kinetic equation for atom migration.
- Reduced the nonlinear diffusion equation to Fick's or Cahn's equations under specific conditions.
- Provided conditions for simplifying the complex diffusion equation.

## Abstract

A quantum kinetic equation is obtained for an inhomogeneous solid having arbitrary gradient concentration and chemical potential. We find, starting from nonequilibrium statistical operator, a new equation to describe atom migration in solid states. In continuous approximation, this equation turns into a non-linear diffusion equation. We derive conditions for which this equation can be reduced to Fick's or Cahn equation.