dQ=TdSfar from equilibrium

It has been pointed out that heat flow in the slowly modulated dissipative steady state in some circuits far from equilibrium obeys a relationship $dQ-d{Q}_0=T_N\mathrm{dS}$. $d{Q}_0$ is the heat flow calculated in a macroscopic way by multiplying the ensemble averages of currents and voltages for the dissipative elements. $T_N$ is the temperature characterizing the fluctuations in the degree of freedom under consideration. $S$ is the entropy defined in the usual statistical-mechanical fashion. This relationship is illustrated through a particularly simple example: The divergent heat flow into a circuit as a second-order transition is approached from the high-symmetry side. The general case is more complex and discussed subsequently, with special attention to the complications caused by asymmetric distribution functions, and the fact that the dissipation in slowly shifted steady state differs from that in the exact steady state. The relationship, expressed in a form which involves entropy changes, is valid only if the distribution functions in the dissipative case mimic equilibrium distribution functions very closely. Other forms have a much broader applicability.