# Derivation of the Langevin equation from the principle of detailed balance

## TL;DR

This paper derives the Langevin equation using detailed balance, applicable to anisotropic phase-spaces and curvilinear coordinates, with an application to superconductivity and a discussion on stochastic calculus interpretations.

## Contribution

It introduces a simple method to derive Langevin equations from detailed balance, accommodating complex coordinate systems and anisotropic phase-spaces.

## Key findings

- Method enables derivation of Langevin equations in complex coordinate systems.
- Application to Kramer--Watts-Tobin equation demonstrates practical utility.
- Discussion clarifies the choice between Itô and Stratonovich interpretations.

## Abstract

For a system at given temperature, with energy known as a function of a set of variables, we obtain the thermal fluctuation of the evolution of the variables by replacing the phase-space with a lattice and invoking the principle of detailed balance. Besides its simplicity, the asset of this method is that it enables us to obtain the Langevin equation when the phase-space is anisotropic and when the system is described by means of curvilinear coordinates. As an illustration, we apply our results to the Kramer--Watts-Tobin equation in superconductivity. The choice between the It\^{o} and the Stratonovich procedures is discussed.