Information Geometry of Spatially Periodic Stochastic Systems

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are [Formula: see text] and [Formula: see text] , with [Formula: see text] chosen to be particularly flat (locally cubic) at the equilibrium point [Formula: see text] , and [Formula: see text] particularly flat at the unstable fixed point [Formula: see text]. We numerically solve the Fokker–Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at [Formula: see text] , with [Formula: see text] in the range [Formula: see text]. The strength D of the stochastic noise is in the range [Formula: see text] – [Formula: see text]. We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point [Formula: see text] , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point [Formula: see text] , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length [Formula: see text] , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that [Formula: see text] as a function of initial position [Formula: see text] is qualitatively similar to the force, including the differences between [Formula: see text] and [Formula: see text] , illustrating the value of information length as a useful diagnostic of the underlying force in the system.