The effect of spacetime curvature on statistical distributions

The Boltzmann distribution of an ideal gas is determined by the Hamiltonian function generating single particle dynamics. Systems with higher complexity often exhibit topological constraints, which are independent of the Hamiltonian and may affect the shape of the distribution function as well. Here, we study a further source of heterogeneity, the curvature of spacetime arising from the general theory of relativity. The present construction relies on three assumptions: first, the statistical ensemble is made of particles obeying geodesic equations, which define the phase space of the system. Next, the metric coefficients are time-symmetric, implying that, if thermodynamic equilibrium is achieved, all physical observables are independent of coordinate time. Finally, ergodicity is enforced with respect to proper time, so that ambiguity in the choice of a time variable for the statistical ensemble is removed. Under these hypothesis, we derive the distribution function of thermodynamic equilibrium, and verify that it reduces to the Boltzmann distribution in the non-relativistic limit. We further show that spacetime curvature affects physical observables, even far from the source of the metric. Two examples are analyzed: an ideal gas in Schwarzschild spacetime and a charged gas in Kerr–Newman spacetime. In the Schwarzschild case, conservation of macroscopic constraints, such as angular momentum, combined with relativistic distortion of the distribution function can produce configurations with decreasing density and growing azimuthal rotation velocity far from the event horizon of the central mass. In the Kerr–Newman case, it is found that kinetic energy associated with azimuthal rotations is an increasing function of the radial coordinate, and it eventually approaches a constant value corresponding to non-relativistic equipartition, even though spatial particle density decreases.