Comments on thermodynamic integration methods for the determination of nonequilibrium entropy

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Abstract
Over many years there have been suggestions that one ought to be able to determine entropy differences between nonequilibrium steady states by a generalization of the thermodynamic integration methods used for determining entropy differences between equilibrium states. We discuss a number of proposals for how this might be done and rule out some of the possibilities. We suggest that the entropy might be computed by integrating the excess rate at which heat is removed from a system that moves infinitely slowly and isoenergetically, between different nonequilibrium steady states. We argue that this excess heat should be measured relative to the average heat rate of the corresponding steady state. Further, we argue that the memory effects due to linear viscoelasticity can make no contribution to the entropy differences. Using our entropy definition we observe that the entropy of shearing isoenergetic, isochoric systems is a monotonic decreasing function of the strain rate. It is also apparent that the thermodynamic temperature computed using this entropy is always less than the corresponding kinetic temperature of nonequilibrium steady states.

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