Generalized thermodynamic potential for Markoff systems in detailed balance and far from thermal equilibrium

We analyze the question why there exist systems far from thermal equilibrium, e.g. lasers, whose stationary state may be described by a potential function, which has all the properties of a thermodynamic potential. It is argued, that the physical property, common to thermal equilibrium states and to these more general stationary states is detailed balance.The proof of this argument is given explicitly for systems whose macroscopic, collective variables can be described as a continuous Markoffian random process.We show that in order to preserve detailed balance the coefficients of the governing Fokker-Planck equation have to fulfill a set of restrictive conditions, special cases of which are well known in the mathematical theory of Markoff processes under the name of “potential conditions”. These restrictive conditions are derived as a consequence of the fact that the condition of detailed balance has to be compatible with the simultaneous validity of the Fokker-Planck equation (forward equation) and its adjoint equation (backward equation).By means of the potential conditions it is always possible to calculate the generalized thermodynamic potential explicitly. We further show that the validity of the potential conditions in turn implies that detailed balance holds.As the validity of the potential conditions has already been demonstrated for important cases in laser theory and in nonlinear optics the existence of detailed balance can now be inferred for those systems. When we specialize our present treatment to closed systems, we obtain the general Fokker-Planck equation derived for the first time by M. S. Green. In that paper detailed balance was implicitly assumed.This shows that detailed balance can provide a basis for a unified description of both the thermal equilibrium systems and some more general stationary states far from thermal equilibrium. Our potential function appears as a natural generalization of thermodynamic potentials to more general stationary states with detailed balance.