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Entropy 2017, 19(6), 268; doi:10.3390/e19060268

Information Geometry of Non-Equilibrium Processes in a Bistable System with a Cubic Damping

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.
Academic Editor: Geert Verdoolaege
Received: 29 April 2017 / Revised: 2 June 2017 / Accepted: 6 June 2017 / Published: 11 June 2017
(This article belongs to the Special Issue Information Geometry II)
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Abstract

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic systems and information geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the information length. By using this concept, we investigate the information geometry of non-equilibrium processes involved in disorder-order transitions between the critical and subcritical states in a bistable system. Specifically, we compute time-dependent PDFs, information length, the rate of change in information length, entropy change and Fisher information in disorder-to-order and order-to-disorder transitions and discuss similarities and disparities between the two transitions. In particular, we show that the total information length in order-to-disorder transition is much larger than that in disorder-to-order transition and elucidate the link to the drastically different evolution of entropy in both transitions. We also provide the comparison of the results with those in the case of the transition between the subcritical and supercritical states and discuss implications for fitness. View Full-Text
Keywords: stochastic processes; Fokker–Planck equation; fluctuations and noise; non-equilibrium statistical mechanics stochastic processes; Fokker–Planck equation; fluctuations and noise; non-equilibrium statistical mechanics
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Hollerbach, R.; Kim, E.-J. Information Geometry of Non-Equilibrium Processes in a Bistable System with a Cubic Damping. Entropy 2017, 19, 268.

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