Entropy of a nonequilibrium system

Three definitions of entropy for a nonequilibrium system of particles, driven homogeneously by external forces and thermostatted homogeneously by a feedback mechanism, are discussed. The first is proposed to be S(t)=-k〈ln$f_{\xi }$〉, i.e., the nonequilibrium ensemble average of the logarithm of the thermostatted equilibrium distribution function $f_{\xi }$. We show here, for a specific example, namely, the Nosé-Hoover thermostat, that the entropy so defined reduces properly to the equilibrium result when the external forces are turned off, that this entropy behaves correctly when the thermostat is turned off, and that the thermostatted steady state is achievable. A reasonable alternative definition from information theory, namely replacing $f_{\xi }$ by the nonequilibrium distribution function f, is shown to give incorrect results. If, however, the distribution function f is coarse grained in time to give f¯, then the resulting coarse-grained information-theory entropy, like the first definition, satisfies the requirements of the nonequilibrium entropy, with the added advantage of being easier to interpret in terms of the number of accessible states. Additional implications are discussed.