Entropy Changes in the Steady State far from Equilibrium

A relationship $dS=\frac{\mathrm{dQ}}{T}$ is derived for slow changes in a steady-state system far from equilibrium. Systems consisting of one conservative degree of freedom, coupled to a dissipative nonequilibrium system without energy storage are considered. The entropy $S$ is defined in the conventional statistical mechanical sense by an integral $-k\int \rho \ln \rho \mathrm{dq}$ over the conservative degree of freedom. The fluctuations are assumed to be narrow and the temperature $T$ is the noise temperature, i.e., the temperature characterizing the intensity of the fluctuations in stored energy. $-dQ$ is the energy given up to the reservoir to the extent that it exceeds that predicted for steady-state losses on the basis of the macroscopic equations. Generalizations to the case of two or more degrees of freedom are successful only in special cases. These include the case where one degree of freedom adjusts rapidly compared to the other. It also includes cases where the different degrees of freedom see the same noise temperature.