  We study the nonequilibrium steady state realized in a general stochastic
system attached to multiple heat baths and/or driven by an external force.
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
${\Phi}(\eta)=-\sum_k\beta_k\mathcal{E}_k(\eta)$ is the excess entropy change.
Here $\mathcal{E}_k(\eta)$ is the difference between two kinds of conditioned
path ensemble averages of excess heat transfer from the $k$-th heat bath whose
inverse temperature is $\beta_k$. Our expression may be verified experimentally
in nonequilibrium states realized, for example, in mesoscopic systems.
