Extending the definition of entropy to nonequilibrium steady states

We study the nonequilibrium statistical mechanics of a finite classical system subjected to nongradient forces $\xi$ 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 $\rho_\xi$. When $\xi$ is replaced by $\xi+\delta\xi$ one can compute the change $\delta\rho$ of $\rho_\xi$ (linear response) and define an entropy change $\delta S$ based on energy considerations. When $\xi$ 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 $\xi$ is a gradient) we show that the curvature is zero, and that the entropy $S(\xi+\delta\xi)$ near equilibrium is well defined to second order in $\delta\xi$.