Abstract
Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance by quantifying how information unfolds over time as a probability density function (PDF) evolves in time. Here, we discuss some recent developments in information geometric theory focusing on time-dependent dynamic aspects of non-equilibrium processes (e.g., time-varying mean value, time-varying variance, or temperature, etc.) and their thermodynamic and physical/biological implications. We compare different distances between two given PDFs and highlight the importance of a path-dependent distance for a time-dependent PDF. We then discuss the role of the information rate and relative entropy in non-equilibrium thermodynamic relations (entropy production rate, heat flux, dissipated work, non-equilibrium free energy, etc.), and various inequalities among them. Here, is the information length representing the total number of statistically distinguishable states a PDF evolves through over time. We explore the implications of a geodesic solution in information geometry for self-organization and control.
Highlights
Information geometry refers to the application of the techniques of differential geometry to probability and statistics
We show that the information length is useful in elucidating the correlation between two interacting species such as two competing components relaxing to the same equilibrium in the long time limit
To elucidate the utility of information geometric theory in understanding non-equilibrium thermodynamics, we review some of the important thermodynamic measures of irreversibility and dissipation [112] and relate them to information geometric measures Γ and K [29]
Summary
Information geometry refers to the application of the techniques of differential geometry to probability and statistics. It agrees with the expectation that a PDF gradient (the Fisher-information) increases with information [32] This concept has been generalized to non-equilibrium systems [36,37,38,39,40,41,42,43], including the utilization for controlling systems to minimize entropy production [38,40,42], the measurement of the statistical distance in experiments to validate theoretical predictions [41], etc.
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