Covariant Statistical Mechanics and the Stress-Energy Tensor

After recapitulating the covariant formalism of equilibrium statistical mechanics in special relativity and extending it to the case of a nonvanishing spin tensor, we show that the relativistic stress-energy tensor at thermodynamical equilibrium can be obtained from a functional derivative of the partition function with respect to the inverse temperature four-vector beta. For usual thermodynamical equilibrium, the stress-energy tensor turns out to be the derivative of the relativistic thermodynamic potential current with respect to the four-vector beta, i.e., T-mu nu = -partial derivative Phi(mu)/partial derivative beta(nu). This formula establishes a relation between the stress-energy tensor and the entropy current at equilibrium, possibly extendable to nonequilibrium hydrodynamics.