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 + deltaxi, one can compute the change deltarho of rho(xi) (linear response) and define an entropy change deltaS 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. However, at equilibrium (i.e., if xi is a gradient) we show that the curvature is zero, and that the entropy S(xi + deltaxi) near equilibrium is well defined to second order in deltaxi.