matrix, which describes the scattering processes taking place in the thermodynamical system of interest. Such a formulation is necessary for studying the systems whose microscopic constituents behave according to the laws of relativistic quantum mechanics. Our result is a simple prescription for calculating the grand canonical potential of any gaseous system given the free-particle energies and

S

-matrix elements. When applied to a nonrelativistic gas, it gives a simple prescription for calculating all virial coefficients. Simplified relativistic gas models are considered as examples of application. A general form of the Levinson's Theorem for any number of particles follows immediately from our formalism. Its applications in statistical mechanics are briefly discussed.

DOI: 10.1103/physreva.6.851.2

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We have formulated the statistical mechanics in terms of the

S

S

We have formulated the statistical mechanics in terms of the

S

S

S

matrices consistent with unitarity that correspond to purely resonance-dominated dynamics. These are then used in the

S

-matrix formulation of statistical mechanics to obtain the grand partition function, which is seen to reveal the above result. In the following paper, operational criteria are presented for determining when in general narrow resonances and loosely bound states may be treated as "elementary" particles.

k

space, $n(k, \ensuremath{\beta})$. Various schemes for calculating these distribution functions directly are discussed. An approximate integral equation for $n(k, \ensuremath{\beta})$ is found, which can be solved by a converging iteration process. Some remarks are included on the application of the virial theorem.