Brownian sampling in an unbounded space

Abstract

As a particle undergoes translational Brownian motion in an unbounded space, the particle samples the space. Traditionally the sampling in N dimensions is quantified in terms of average squared distance traversed (〈 x · x 〉) . However, another quantitative measure of the sampled space is the total number ( n) of equispaced regions (of size L N ) sampled after a particle moves with a diffusion coefficient ( D) for a time ( t). Calculations show that the average 〈 n〉= a( Dt/ L 2) b . Results are given for a and b for 1, 2, 3, and 4 dimensions.