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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sum_{i,j} q_i \cdot {\nabla} U(q_i-q_j)$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $u_t = {1\over2} \Delta P(u)$\end{document} under a certain hypothetical postulate concerning concentration of particles.