Diffusion of particles moving with constant speed

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 ac t only perpendicular to its velocity. This enforces strictly and dynamicall y the constraint of constant speed of the photon in the medium. A Fokker-Pl anck equation is derived for the probability distribution in the phase spac e assuming the transverse fluctuating force to be a white noise. Analytic e xpressions for the moments of the displacement (x(n))along with an approxim ate expression for the marginal probability distribution function P(x,t) ar e obtained. Exact numerical solutions for the phase space probability distr ibution for various. geometries are presented. The results show that the ve locity distribution randomizes in a time of about eight times the mean free time (8t(*)) only after which the diffusion approximation becomes valid. T his factor of 8 is a well-known experimental fact. A persistence exponent o f 0.435 +/- 0.005 is calculated for this process in two dimensions by study ing the survival probability of the particle in semi-infinite medium. The c ase of a stochastic amplifying medium is also discussed. [S1063-651X(99)038 08-8].