Correlated Brownian Motions and the Depletion Phenomenon

Abstract

We derive correlated Brownian motions from an infinite system of deterministic coupled oscillators for finitely many large and infinitely many small particles. The only source of randomness are the initial positions and velocities of the small particles and the interaction between small and large particles is a mean-field interaction (simplifying an underlying collision dynamics). Then we analyze the correlation functional for the limiting Brownian motions (for the large particles). We obtain a forth order even polynomial in the distance between the positions of two Brownian particles and the correlation length parameter.  This polynomial has two positive roots. Between 0 and the first root the behavior of the two particles is attractive, between the two roots it is repulsive and "far away" the two Brownian particles are essentially independent Brownian motions. The attractive zone confirms experiments conducted in the empirical sciences (s. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954) -...- B. Goetzelmann, R. Evans and S. Dietrich, "Depletion forces in fluids, Physical Review E. Vol. 57, Number 6, (1998) and the references therein). These experiments together with theoretical considerations lead to the conclusion that the fluid between two heavy particles gets "depleted" if the two heavy particles are very close to one another.