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.
Highlights
The spontaneous emergence of order out of disorder is one of the most fascinating phenomena in nature and laboratory experiments, attracting ever-increasing interest
We investigate the information geometry of non-equilibrium processes involved in disorder-order transitions between the critical and subcritical states in a bistable system
We investigated information geometry associated with order-to-disorder and disorder-to-order transitions in a 0D Ginzburg–Landau model where the formation of an ordered state is modelled by the transition from a unimodal to bimodal Probability Density Functions (PDFs) of a stochastic variable x
Summary
The spontaneous emergence of order out of disorder is one of the most fascinating phenomena in nature and laboratory experiments, attracting ever-increasing interest. The aim of this paper is to understand order-disorder transition from the perspective of information change associated with transition and uncover a geometrical structure in a statistical space, which can be utilised to understand ever-increasing experimental/observational data To this end, we investigate a bistable stochastic system that is often invoked as a canonical model of self-regulating systems, e.g., in electric circuits [17], in various cellular processes such as cycles, differentiation and apoptosis, regulation of heart, brain, etc. We aim to elucidate what might be an optimal initial “resting” state that minimizes the information change, sustaining a robust geodesic solution To this end, we provide detailed comparison of on-quenching processes in this paper with off-quenching processes, where the control parameter changes between subcritical and supercritical, reported in [40]. Appendices contain the derivation of equations used in the main text
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