Surprisal analysis of dissipative Fokker-Planck dynamics in phase-space

We show that in well-characterized special cases, a maximal entropy-motivated approach can provide an exact description of the dynamics in phase space of a classical dissipative system such as governed by the Fokker-Planck equation. This is achieved by identifying a set of constraints which, when held constant in time, make the system stationary. Even when such a set is not possible in general it can be identified for special limits. An example is a double well potential in the limit of a high barrier between the two wells. Technically we show that the surprisal of the distribution can be expressed as a linear combination of the constraints. The time-dependent expansion coefficients are identified as the Lagrange multipliers needed to impose a distribution of maximal entropy. For an exact solution, the Lagrange multipliers satisfy a closed set of equations of motion. Unlike the reversible case where these equations are linear, the dissipation introduces non-linear terms. Even so, for an exact solution, the equations are integrable. The constraints are a basis for time-dependent constants of the motion. When it is not possible to identify an exact solution the maximal entropy formalism delivers a tight numerical approximation for the distribution. For the Kramers barrier crossing problem for a high barrier, the surprisal analysis correctly determines the branching fraction between the two wells.