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

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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 = dL dt 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.

Original languageEnglish
Article number1393
Number of pages24
Issue number11
Early online date24 Oct 2021
Publication statusPublished - Nov 2021

Bibliographical note

This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution (CC BY) license (

Funding Information:
Acknowledgments: Eun-jin Kim acknowledges the Leverhulme Trust Research Fellowship (RF-2018-142-9) and thanks the collaborators, especially, James Heseltine who contributed to the works in this paper.

Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.




  • Entropy
  • Fluctuations
  • Fokker-planck equation
  • Information geometry
  • Information length
  • Information rate
  • Langevin equations
  • Self-organization
  • Time-dependent probability density functions

ASJC Scopus subject areas

  • Information Systems
  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)
  • Electrical and Electronic Engineering


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