We formulate a dynamical fluctuation theory for stationary non-equilibrium
states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics.
Within this theory we derive the following results: the modification of the
Onsager–Machlup theory in the SNS; a general Hamilton–Jacobi equation for
the macroscopic entropy; a non-equilibrium, nonlinear fluctuation dissipation
relation valid for a wide class of systems; an H theorem for the entropy. We
discuss in detail two models of stochastic boundary driven lattice gases: the zero
range and the simple exclusion processes. In the first model the invariant
measure is explicitly known and we verify the predictions of the general theory.
For the one dimensional simple exclusion process, as recently shown by Derrida,
Lebowitz, and Speer, it is possible to express the macroscopic entropy in terms
of the solution of a nonlinear ordinary differential equation; by using the
Hamilton–Jacobi equation, we obtain a logically independent derivation of this
result.