Expression for the stationary distribution in nonequilibrium steady states

We study the nonequilibrium steady state realized in a general stochastic system attached to multiple heat baths. Starting from the detailed fluctuation theorem, we derive concise and suggestive expressions for the corresponding stationary distribution which are correct up to the second order in thermodynamic forces. The probability of a microstate eta is proportional to exp[Phi(eta)] where Phi(eta)=-Sigma(k)beta(k)E(k)(eta) is the excess entropy change. Here, E(k)(eta) is the difference between two kinds of conditioned path ensemble averages of excess heat transfer from the kth heat bath whose inverse temperature is beta(k). This result can be easily extended to steady states maintained with other sources, e.g., particle current driven by an external force. Our expression may be verified experimentally in nonequilibrium states realized, for example, in mesoscopic systems.