Statistical Mechanics of a Fluid in an External Potential

Abstract

Classical statistical mechanics is used to obtain expressions for the density $n(\mathrm{r})$, the distribution functions, and other properties of a fluid in equilibrium in a potential $V(\mathrm{r})$. They are expressed in terms of their values in a uniform system, plus a series of terms involving the derivatives of $V(\mathrm{r})$. The result for the density is exemplified by applying it to a one-dimensional fluid of hard rods, in a potential. The expression for the two-particle distribution function is used to evaluate the stress tensor and moment stress tensor from formulas derived previously. It is found that the stress tensor is no longer a scalar, and the moment stress tensor does not vanish, when the terms involving the derivatives of $V(\mathrm{r})$ are taken into account. The method of analysis is that of Lebowitz and Percus. They showed how to express the distribution functions in terms of $n(\mathrm{r})$ and its derivatives. However, they did not obtain an expression for $n(\mathrm{r})$ itself. Their result is used here to determine the stress tensor in a surface layer. From it an expression for the surface tension is obtained.