Thermodynamics of rotating fermions
We consider the thermodynamic properties of a rotating gas of free fermions. We begin by constructing the thermodynamic potential $Φ$ and its associated current $ϕ^μ$ within the grand canonical ensemble of a macroscopic rigidly rotating body, where the ensemble parameters are the temperature $T_0$ and chemical potential $μ_0$ on the rotation axis, as well as the rotation angular velocity $Ω_0$. We then consider the problem of local thermodynamics, where the thermodynamic state is defined by the local temperature $T$ and chemical potential $μ$, as well as the local spin potential tensor, $Ω_{μν}$. We find the thermodynamic pressure $P$, given as the sum of the usual classical (non-quantum) pressure and other corrections due to the spin potential and the kinematic state of the fluid, based on the local expectation value of the energy-momentum tensor. We compute the associated entropy, charge and spin densities, and show they are consistent with the Euler relation.