Extending the definition of entropy to nonequilibrium steady states

Abstract

We study the nonequilibrium statistical mechanics of a finite classical system subjected to nongradient forces $ξ$ and maintained at fixed kinetic energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that there is a natural nonequilibrium steady state $ρ_ξ$. When $ξ$ is replaced by $ξ+δξ$ one can compute the change $δρ$ of $ρ_ξ$ (linear response) and define an entropy change $δS$ based on energy considerations. When $ξ$ is varied around a loop, the total change of $S$ need not vanish: outside of equilibrium the entropy has curvature. But at equilibrium (i.e. if $ξ$ is a gradient) we show that the curvature is zero, and that the entropy $S(ξ+δξ)$ near equilibrium is well defined to second order in $δξ$.