Statistical Mechanics of a Fluid in an External Potential

Classical statistical mechanics is used to obtain expressions for the density $n\left(\mathrm{r}\right)$, the distribution functions, and other properties of a fluid in equilibrium in a potential $V\left(\mathrm{r}\right)$. They are expressed in terms of their values in a uniform system, plus a series of terms involving the derivatives of $V\left(\mathrm{r}\right)$. 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 $V\left(\mathrm{r}\right)$ 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 $n\left(\mathrm{r}\right)$ and its derivatives. However, they did not obtain an expression for $n\left(\mathrm{r}\right)$ 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.