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Information Geometry of Spatially Periodic Stochastic Systems

Rainer Hollerbach,Eun-jin Kim

Published 2019 in Entropy

ABSTRACT

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f0=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 μ is qualitatively similar to the force, including the differences between f0=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.

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