The Classical Gas in the Canonical Formalism

Abstract

A classical system of N monatomic particles of mass m may be given by the Hamiltonian

$$H = \sum\limits_{i = 1}^N {\frac{1}{{2m}}} \vec p_i^2 + \sum\limits_{i < j} {V(\left| {{{\vec r}_i} - {{\vec r}_j}} \right|)} ,$$

(6.1)

where $V(\left| {\vec r} \right|)$ is a pair potential, as sketched in figure 6.1. The potential of a realistic system displays a hard core, $V(r) \to \infty ,{\text{ for }}r = \left| {{{\vec r}_i} - {{\vec r}_j}} \right| \to 0$, and a suitably vanishing (for r → ∞) attractive part. The canonical partition function of this classical system, in contact with a heat reservoir at temperature T, in a container of volume V, is given by the integral in phase space,

$${Z_c} = \int { \cdots \int\limits_V {{d^3}{{\vec r}_1} \ldots {d^3}{{\vec r}_N}\int { \cdots \int {{d^3}{{\vec p}_1} \cdots {d^3}{{\vec p}_N}{\kern 1pt} \exp {\kern 1pt} ( - \beta H),} } } } $$

(6.2)

where the spatial coordinates are restricted to the region of volume V. The trivial integration over the momentum coordinates may be written as a product of 3N Gaussian integrals of the form

$$\int\limits_{ - \infty }^{ + \infty } {dp{\kern 1pt} \exp ( - \frac{{\beta {p^2}}}{{2m}})} = {(\frac{{2\pi m}}{\beta })^{1/2}}.$$

(6.3)