# Application of Onsager Machlup integral in solving dynamic equations in non-equilibrium systems

## TL;DR

This paper introduces an improved variational method based on the Onsager-Machlup integral for solving kinetic equations in non-equilibrium systems, overcoming previous limitations and enabling steady state determination.

## Contribution

The paper presents a novel approach using the Onsager-Machlup integral that allows for better approximate solutions and steady state analysis in non-equilibrium kinetic equations.

## Key findings

- Successfully applied to diffusion, capillary, and free boundary problems.
- Enables determination of steady states via variational calculus.
- Improves accuracy over previous small-time incremental methods.

## Abstract

In 1931, Onsager proposed a variational principle which has become the base of many kinetic equations for non-equilibrium systems. We have been showing that this principle is useful in obtaining approximate solutions for the kinetic equations, but our previous method has a weakness that it can be justified, strictly speaking, only for small incremental time. Here we propose an improved method which does not have this drawback. The new method utilizes the integral proposed by Onsager and Machlup in 1953, and can tell us which of the approximate solutions is the best solution without knowing the exact solution. The new method has an advantage that it allows us to determine the steady state in non-equilibrium system by a variational calculus. We demonstrate this using three examples, (a) simple diffusion problem, (b) capillary problem in a tube with corners, and (c) free boundary problem in liquid coating, for which the kinetic equations are written in second or fourth order partial differential equations.