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 f ₀ = sin(πx)/π and f _± = sin(πx)/π ± sin(2πx)/2π, with f_ chosen to be particularly flat (locally cubic) at the equilibrium point x = 0, and f ₊ particularly flat at the unstable fixed point x = 1. We numerically solve the Fokker-Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ, with μ in the range [0, 1]. The strength D of the stochastic noise is in the range 10 ^-4-10 ^-6. 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 x = 0, the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1, 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 L _∞, 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 L _∞ as a function of initial position m is qualitatively similar to the force, including the differences between f ₀ = sin(πx)/π and f _± = sin(πx)/π ± sin(2πx)/2π, illustrating the value of information length as a useful diagnostic of the underlying force in the system.