Relativistic Corrections in the Statistical Mechanics of Interacting Particles

In previous work on statistical mechanics, the consequences of the theory of relativity have only been investigated for the ideal gas. This study is concerned with relativistic effects for systems of interacting particles. The noninstantaneous nature of the forces leads to dynamical equations which cannot be treated by known mathematical methods; however, the interaction terms can be expanded to obtain a description of the system in terms of the positions and their derivatives of all orders at a single time. If one stops at the ${\left(\frac{v}{c}\right)}^2$ approximation, a specification in terms of positions and velocities is obtained; in electrodynamics this corresponds to the Darwin Hamiltonian. A system described by this Hamiltonian is investigated with the methods of equilibrium statistical mechanics. The cluster expansion with subsequent summation of diagrams as employed by Mayer for the Coulomb case is used; the modifications necessary due to the presence of momentum-dependent terms in the interaction are developed. In evaluating the lowest order nonvanishing (ring) approximation, mathematical difficulties peculiar to the relativistic interactions force restriction to calculation of the relativistic short-range correlation effects in the charged system. A modified Debye-Hückel law is obtained, including a relativistic correction term which is small compared to the static one. At the high temperatures required for appearance of relativistic effects, the static term is itself negligible except at very high densities. Thus the relativistic contribution can in effect be neglected in this approximation. The difficulty of extending our method to mesonic or gravitational interactions is discussed.