Pressure in Classical Statistical Mechanics and Interacting Brownian Particles in Multi-dimensions

. Let P(u) denote the pressure at the density u defined in the Gibbs statistical mechanics determined by a 2 body potential U (_qi - _qj). The function U(x) is supposed rotationally invariant and of finite range but may be unbounded about the origin. We establish a representation of P(u) by means of the law of large numbers for the virial ?_i,j q_i ·? U(q_i-q_j)sum_{i,j} q_i cdot {nabla} U(q_i-q_j), whether or not there occur phase transitions. This result on P(u) is motivated by a study of the hydrodynamic behavior of a system of a large number of interacting Brownian particles moving on a d-dimensional torus (d = 1, 2, ...) in which the interaction is given by binary potential forces of potential U. Employing our representation of P(u), we also show that in the hydrodynamic limit of such a system there arises a non linear evolution equation of the form u_t = ¹/₂ DP(u)u_t = {1over2} Delta P(u) under a certain hypothetical postulate concerning concentration of particles.