Relativistic Equilibrium Distribution by Relative Entropy Maximization

The equilibrium state of a relativistic gas has been calculated based on the maximum entropy principle. Though the relativistic equilibrium state was long believed to be the Jüttner distribution, a number of papers have been published in recent years proposing alternative equilibrium states. However, some of these papers do not pay enough attention to the covariance of distribution functions, resulting confusion in equilibrium states. Starting from a fully covariant expression to avoid this confusion, it has been shown in the present paper that the Jüttner distribution is the maximum entropy state if we assume the Lorentz symmetry. Introduction. – Little after the establishment of the theory of relativity, the equilibrium particle distribution of a relativistic gas was investigated. The distribution obtained, which is called Jüttner distribution [1, 2], has been long and widely believed. However, relatively recent years a number of papers have been published proposing equilibrium distribution functions other than the Jüttner distribution ( [3–7] and references therein). Dunkel and coworkers [7, 8] have examined the discrepancy in the equilibrium distributions as the maximum entropy state, and showed that the difference comes from the choice of the reference measure. The maximum entropy state cannot be uniquely determined when one naively defines the entropy such as S = − ∫ f (x, v) ln f (x, v) dxdv (symbols have conventional meaning in the present paper unless otherwise stated). For instance, the result would be different if we rewrite distribution function as a function of momentum p instead of velocity v. To overcome this difficulty, it was proposed in Ref [7] to maximize the following relative entropy