Energy and Momentum Conservation for Diffusion - A Stochastic Mechanics Approximation - Part I

TL;DR

This paper models classical diffusion using stochastic mechanics, showing energy conservation and deriving relationships between particle distances, collision times, and velocity correlations, with implications for elastic collisions.

Contribution

It introduces a stochastic differential equation approach to model diffusion, demonstrating energy conservation and linking collision dynamics to Lorentz transformations.

Findings

Energy and momentum are conserved in the model.

The particle's position distribution aligns with Schrödinger's equation.

Collision dynamics relate to Minkowski invariants and Lorentz transformations.

Abstract

This paper models the classical diffusion of a main particle through a heatbath by means of a pre-limit microscopic representation of its drifted momentum and energy transfers at collision times. The collision point linear interpolated path can be approximated by the solution to the "inscribed" continuous stochastic differential equation using the same drift function. Employing results from stochastic mechanics it is then shown that the combined main particle/heatbath system does not exchange or radiate energy if the probability distribution for the position of the main particle is derived from Schroedinger's equation. Furthermore it is shown that the main particle distance traveled between collisions and the mean inter-collision time must satisfy a type of Minkowski invariant. Hence if there is a correlation between the pre- and post-collision velocities of the main particle through a collision point then the mean distance traveled can be related to the mean inter-particle collision times via a Lorentz transformation. The last Section shows that this approach can be applied to all elastic main particle/heatbath particle collisions either via direct calculation involving modeling the collision scattering or by altering the properties of the heatbath.