The Classical Gas in the Canonical Formalism

  • Ragnar Ekholm
  • Leonard D. Kohn
  • Seymour H. Wollman
Part of the Graduate Texts in Contemporary Physics book series (GTCP)

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)

Keywords

Partition Function Canonical Formalism Thermodynamic Limit Hard Core Pair Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Ragnar Ekholm
    • 1
  • Leonard D. Kohn
    • 2
  • Seymour H. Wollman
    • 2
  1. 1.University of GöteborgGöteborgSweden
  2. 2.National Institutes of HealthBethesdaUSA

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