Information Geometry, Fluctuations, Non-Equilibrium Thermodynamics, and Geodesics in Complex Systems

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 Γ=dLdt 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, L 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.

Exact Time-Dependent Solutions and Information Geometry of a Rocking Ratchet

“…Equation (18) shows explicitly that the non-zero entropy production S T > 0 or heat flow Q reduces the efficiency.…”

“…As a measure of time-irreversibility, entropy production, fluctuation theorems (e.g., see [13][14][15] and references therein), etc., are investigated. Furthermore, since the proposal of Feymann's ratchet, there has been growing interest in information theory [13,[16][17][18] to understand or else optimise a Brownian motor [19][20][21][22][23].…”

“…In this paper, we propose to investigate the dynamics of a Brownian motor from the point of geometry and the distance (metric tensor) by utilising the information geometric theory-the application of the differential geometry to probability and statistics [18,[24][25][26]. It is a powerful tool for elucidating the disparity between different probabilities as well as for linking complexity and geometry (e.g., see [18,24] and references therein).…”

“…In this paper, we propose to investigate the dynamics of a Brownian motor from the point of geometry and the distance (metric tensor) by utilising the information geometric theory-the application of the differential geometry to probability and statistics [18,[24][25][26]. It is a powerful tool for elucidating the disparity between different probabilities as well as for linking complexity and geometry (e.g., see [18,24] and references therein). To capture a temporal variation, our focus will be on the path-dependent information geometric concept (information length and rate) that quantifies the time evolution of a system in terms of a dimensionless distance in a statistical space [18,[27][28][29][30][31][32][33][34][35][36][37][38] or the change in information.…”

“…It is a powerful tool for elucidating the disparity between different probabilities as well as for linking complexity and geometry (e.g., see [18,24] and references therein). To capture a temporal variation, our focus will be on the path-dependent information geometric concept (information length and rate) that quantifies the time evolution of a system in terms of a dimensionless distance in a statistical space [18,[27][28][29][30][31][32][33][34][35][36][37][38] or the change in information. Its key properties are summarised in Section 2.…”