Statistical thermodynamics of a relativistic gas

Abstract

Classical statistical thermodynamics is one of the oldest, most well‐established physical theories and its basis and results have not been challenged within its domain since the time of Boltzmann. Special relativity, however, introduces some constraints as well as ambiguities into such a theory. For example, the cornerstone of classical statistical mechanics, the Maxwell‐Boltzmann (MB) distribution does not respect the maximal velocity of light, the cornerstone of special relativity. Additionally, the Lorentz transformation of temperature, i.e. how a moving body’s temperature compares to its rest frame value, has long caused controversies. Special relativity also introduces a new concept of proper time, which could potentially affect fundamental concepts of ergodicity and time‐averaging in thermodynamics. In this work, we propose a model of a relativistic hard‐sphere gas, and via molecular dynamics simulations, investigate all the above issues. In particular we show that the so‐called Jüttner distribution is the correct relativistic generalization of the MB distribution. Introducing proper time averaging simply rescales such distribution by similar energy factor γ. We also show that temperature is best understood as an invariant quantity, i.e. temperature does not change under the motion of inertial frames, and is not affected by time reparametrization. Additionally, we have studied this model under a temperature gradient and have shown that the model satisfies the minimal ingredients to study nonequilibrium transport properties, i.e. the existence of a non‐equilibrium steady state and local thermal equilibrium. This will allow us to study generalizations of transport properties to relativistic regimes.