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Entropy 2014, 16(9), 4713-4748; doi:10.3390/e16094713

Information Anatomy of Stochastic Equilibria

1
Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA
2
Complexity Sciences Center, Department of Physics, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA
*
Authors to whom correspondence should be addressed.
Received: 17 March 2014 / Revised: 3 August 2014 / Accepted: 19 August 2014 / Published: 25 August 2014
(This article belongs to the Special Issue Information in Dynamical Systems and Complex Systems)
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Abstract

A stochastic nonlinear dynamical system generates information, as measured by its entropy rate. Some—the ephemeral information—is dissipated and some—the bound information—is actively stored and so affects future behavior. We derive analytic expressions for the ephemeral and bound information in the limit of infinitesimal time discretization for two classical systems that exhibit dynamical equilibria: first-order Langevin equations (i) where the drift is the gradient of an analytic potential function and the diffusion matrix is invertible and (ii) with a linear drift term (Ornstein–Uhlenbeck), but a noninvertible diffusion matrix. In both cases, the bound information is sensitive to the drift and diffusion, while the ephemeral information is sensitive only to the diffusion matrix and not to the drift. Notably, this information anatomy changes discontinuously as any of the diffusion coefficients vanishes, indicating that it is very sensitive to the noise structure. We then calculate the information anatomy of the stochastic cusp catastrophe and of particles diffusing in a heat bath in the overdamped limit, both examples of stochastic gradient descent on a potential landscape. Finally, we use our methods to calculate and compare approximations for the time-local predictive information for adaptive agents. View Full-Text
Keywords: Langevin equation; entropy rate; ephemeral information; bound information;time-local predictive information Langevin equation; entropy rate; ephemeral information; bound information;time-local predictive information
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This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Marzen, S.; Crutchfield, J.P. Information Anatomy of Stochastic Equilibria. Entropy 2014, 16, 4713-4748.

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