A statistical approximation to solve ordinary differential equations

TL;DR

This paper introduces a novel approach to solving ordinary differential equations by mapping them onto a statistical mechanics problem, enabling solutions through physical analogies and thermodynamic concepts.

Contribution

It establishes a new link between solving ODEs and statistical mechanics, providing a physical perspective and a method to find solutions via energy minimization.

Findings

Method successfully finds unique solutions for single-solution ODEs.

For eigenvalue problems, it identifies the fundamental mode as the minimum of the functional.

Establishes a general relationship connecting ODE solutions with thermodynamic variables.

Abstract

We propose a physical analogy between finding the solution of an ordinary differential equation (ODE) and a $N$ particle problem in statistical mechanics. It uses the fact that the solution of an ODE is equivalent to obtain the minimum of a functional. Then, we link these two notions, proposing this functional to be the interaction potential energy or thermodynamic potential of an equivalent particle problem. Therefore, solving this statistical mechanics problem amounts to solve the ODE. If only one solution exists, our method provides the unique solution of the ODE. In case we treat an eigenvalue equation, where infinite solutions exist, we obtain the absolute minimum of the corresponding functional or fundamental mode. As a result, it is possible to establish a general relationship between statistical mechanics and ODEs which allows not only to solve them from a physical perspective but also to obtain all relevant thermodynamical equilibrium variables of that particle system related to the differential equation.