Open AccessArticle
Information Geometry of Spatially Periodic Stochastic Systems
^{1}
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
^{2}
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
^{*}
Author to whom correspondence should be addressed.
Received: 8 June 2019 / Revised: 4 July 2019 / Accepted: 10 July 2019 / Published: 12 July 2019

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Abstract
We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are
${\mathbf{f}}_{\mathbf{0}}=sin\left(\pi x\right)/\pi $ and
${\mathbf{f}}_{\pm}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}sin\left(\pi x\right)/\pi \pm sin\left(2\pi x\right)/2\pi $ , with
${\mathbf{f}}_{}$ chosen to be particularly flat (locally cubic) at the equilibrium point
$x=0$ , and
${\mathbf{f}}_{+}$ 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=\mu $ , with
$\mu $ 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
${\mathcal{L}}_{\infty}$ , 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
${\mathcal{L}}_{\infty}$ as a function of initial position
$\mu $ is qualitatively similar to the force, including the differences between
${\mathbf{f}}_{\mathbf{0}}=sin\left(\pi x\right)/\pi $ and
${\mathbf{f}}_{\pm}=sin\left(\pi x\right)/\pi \pm sin\left(2\pi x\right)/2\pi $ , illustrating the value of information length as a useful diagnostic of the underlying force in the system.
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