A statistical approximation for solving some ordinary linear differential equations

Abstract

We propose a physical analogy between finding the solution of somelinear ordinary differential equations (ODEs) and doing the same for anN-particle problem in statistical mechanics. It makes use of the fact that finding the solutionof an ODE is equivalent to obtaining the minimum of a functional. Then, welink these two notions, proposing this functional to be the interaction potentialenergy or thermodynamic potential of an equivalent particle problem. Therefore,solving this statistical mechanics problem amounts to solving the ODE. If onlyone solution exists, our method provides the unique solution of the ODE. If wetreat an eigenvalue equation for which infinitely many solutions exist, we obtainthe absolute minimum of the corresponding functional or fundamental mode.As a result, it is possible to establish a general relationship between statisticalmechanics and ODEs which allows us not only to solve the related problemsfrom a physical perspective, but also to obtain all relevant thermodynamicalequilibrium variables of the equivalent particle system related to the differentialequation.