Diffusion of particles moving with constant speed

TL;DR

This paper models photon propagation in scattering media as a constant-speed Brownian motion, deriving a Fokker-Planck equation, analyzing moments, and confirming the velocity randomization time aligns with experimental observations.

Contribution

It introduces a novel phase space model for constant-speed particles, deriving analytic solutions, and analyzing velocity randomization and persistence in scattering media.

Findings

Velocity distribution randomizes after about 8 mean free times.

Derived analytic expressions for displacement moments and probability distribution.

Calculated a persistence exponent of approximately 0.435 in two dimensions.

Abstract

The propagation of light in a scattering medium is described as the motion of a special kind of a Brownian particle on which the fluctuating forces act only perpendicular to its velocity. This enforces strictly and dynamically the constraint of constant speed of the photon in the medium. A Fokker-Planck equation is derived for the probability distribution in the phase space assuming the transverse fluctuating force to be a white noise. Analytic expressions for the moments of the displacement $<x^{n}>$ along with an approximate expression for the marginal probability distribution function $P(x,t)$ are obtained. Exact numerical solutions for the phase space probability distribution for various geometries are presented. The results show that the velocity distribution randomizes in a time of about eight times the mean free time ($8t^*$) only after which the diffusion approximation becomes valid. This factor of eight is a well known experimental fact. A persistence exponent of $0.435 \pm 0.005$ is calculated for this process in two dimensions by studying the survival probability of the particle in a semi-infinite medium. The case of a stochastic amplifying medium is also discussed.