Abstract
We present a general method to determine the entropy current of relativistic matter at local thermodynamic equilibrium in quantum statistical mechanics. Provided that the local equilibrium operator is bounded from below and its lowest lying eigenvector is non-degenerate, it is proved that, in general, the logarithm of the partition function is extensive, meaning that it can be expressed as the integral over a 3D space-like hypersurface of a vector current, and that an entropy current exists. We work out a specific calculation for a non-trivial case of global thermodynamic equilibrium, namely a system with constant comoving acceleration, whose limiting temperature is the Unruh temperature. We show that the integral of the entropy current in the right Rindler wedge is the entanglement entropy.
Highlights
In recent years there has been a considerable interest in the foundations of relativistic hydrodynamics
We have presented the condition of existence of an entropy current and a general method to calculate it
An entropy current can be obtained if the spectrum of the local equilibrium operator—which boils down to the Hamiltonian multiplied by 1=T in the simplest case of global homogeneous equilibrium—is bounded from below
Summary
In recent years there has been a considerable interest in the foundations of relativistic hydrodynamics. There have been attempts [10] to formulate relativistic hydrodynamics without an entropy current In most of these studies, the structure of the entropy current is postulated based on some classical form of the thermodynamics laws supplemented by more elaborate methods to include dissipative corrections [11,12], but, strictly speaking, it is not derived. Obtained by taking functional derivatives with respect to some external source).1 Another reason for this indeterminacy is the fact that, while in quantum statistical mechanics the total entropy has a precise definition in terms of the density operator (von Neumann formula) S 1⁄4 −trðρlog ρÞ, the entropy current, which should be a more fundamental quantity than the total entropy in a general-relativistic framework, does not. In some region which, once integrated, gives rise to its total entanglement entropy
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