scholarly journals Unified framework for the entropy production and the stochastic interaction based on information geometry

2020 ◽  
Vol 2 (3) ◽  
Author(s):  
Sosuke Ito ◽  
Masafumi Oizumi ◽  
Shun-ichi Amari
Keyword(s):  
Entropy Production ◽  
Information Geometry ◽  
Unified Framework

scholarly journals Information geometry, trade-off relations, and generalized Glansdorff-Prigogine criterion for stability

2021 ◽  
Author(s):  
Sosuke Ito
Keyword(s):  
Master Equation ◽  
Entropy Production ◽  
Production Rate ◽  
Excess Entropy ◽  
Information Geometry ◽  
Entropy Production Rate ◽  
Trade Off ◽  
Geometric Criterion ◽  
Excess Entropy Production ◽  
Nonlinear Master Equation

Abstract We discuss a relationship between information geometry and the Glansdorff-Prigogine criterion for stability. For the linear master equation, we found a relation between the line element and the excess entropy production rate. This relation leads to a new perspective of stability in a nonequilibrium steady-state. We also generalize the Glansdorff-Prigogine criterion for stability based on information geometry. Our information-geometric criterion for stability works well for the nonlinear master equation, where the Glansdorff-Prigogine criterion for stability does not work well. We derive a trade-off relation among the fluctuation of the observable, the mean change of the observable, and the intrinsic speed. We also derive a novel thermodynamic trade-off relation between the excess entropy production rate and the intrinsic speed. These trade-off relations provide a physical interpretation of our information-geometric criterion for stability. We illustrate our information-geometric criterion for stability by an autocatalytic reaction model, where dynamics are driven by a nonlinear master equation.

scholarly journals Unified framework for information integration based on information geometry

2016 ◽  
Vol 113 (51) ◽  
pp. 14817-14822 ◽  
Author(s):  
Masafumi Oizumi ◽  
Naotsugu Tsuchiya ◽  
Shun-ichi Amari
Keyword(s):  
Probability Distribution ◽  
Information Integration ◽  
Information Transfer ◽  
Holistic Approach ◽  
Information Geometry ◽  
Unified Framework ◽  
Information Theoretic ◽  
Research Fields ◽  
Information Theoretic Measures ◽  
Integrated Information

Assessment of causal influences is a ubiquitous and important subject across diverse research fields. Drawn from consciousness studies, integrated information is a measure that defines integration as the degree of causal influences among elements. Whereas pairwise causal influences between elements can be quantified with existing methods, quantifying multiple influences among many elements poses two major mathematical difficulties. First, overestimation occurs due to interdependence among influences if each influence is separately quantified in a part-based manner and then simply summed over. Second, it is difficult to isolate causal influences while avoiding noncausal confounding influences. To resolve these difficulties, we propose a theoretical framework based on information geometry for the quantification of multiple causal influences with a holistic approach. We derive a measure of integrated information, which is geometrically interpreted as the divergence between the actual probability distribution of a system and an approximated probability distribution where causal influences among elements are statistically disconnected. This framework provides intuitive geometric interpretations harmonizing various information theoretic measures in a unified manner, including mutual information, transfer entropy, stochastic interaction, and integrated information, each of which is characterized by how causal influences are disconnected. In addition to the mathematical assessment of consciousness, our framework should help to analyze causal relationships in complex systems in a complete and hierarchical manner.

FROM NEW FUNDAMENTAL EQUATION OF STATISTICAL PHYSICS TO THE FORMULA FOR ENTROPY PRODUCTION RATE

2004 ◽  
Vol 18 (17n19) ◽  
pp. 2432-2440 ◽  
Author(s):  
XIU-SAN XING
Keyword(s):  
Entropy Production ◽  
Production Rate ◽  
Statistical Physics ◽  
Classical Mechanics ◽  
Fundamental Equation ◽  
Entropy Production Rate ◽  
Unified Framework ◽  
Nonequilibrium States ◽  
The Difference ◽  
Thermodynamic Processes

Why the classical mechanics and quantum mechanics are reversible while the macroscopic thermodynamic processes are irreversible? Is it because both of the two are different fundamental laws? If yes, what is the difference between them? What is the fundamental equation of nonequilibrium statistical physics? Can it provide a unified framework of statistical physics including nonequilibrium states and equilibrium states? How can we derive rigorously the hydrodynamic equations from microscopic kinetics? Does nonequilibrium entropy obey any evolution equation? What is the form of this equation if it exist? What is the microscopic physical basis of entropy production rate namely the law of entropy increase? Can it be described by a quantitative concise formula? What mechanism is responsible for the processes of approach to equlibrium? How to quantitatively describe it? In this paper we try to solve all these problems from a new fundamental equation of statistical physics in a unified fasion.

scholarly journals Stochastic Reasoning, Free Energy, and Information Geometry

2004 ◽  
Vol 16 (9) ◽  
pp. 1779-1810 ◽  
Author(s):  
Shiro Ikeda ◽  
Toshiyuki Tanaka ◽  
Shun-ichi Amari
Keyword(s):  
Free Energy ◽  
Stochastic Models ◽  
Statistical Physics ◽  
Belief Propagation ◽  
Information Geometry ◽  
Universal Method ◽  
Free Energy Function ◽  
Unified Framework ◽  
Exact Inference ◽  
New Algorithms

Belief propagation (BP) is a universal method of stochastic reasoning. It gives exact inference for stochastic models with tree interactions and works surprisingly well even if the models have loopy interactions. Its performance has been analyzed separately in many fields, such as AI, statistical physics, information theory, and information geometry. This article gives a unified framework for understanding BP and related methods and summarizes the results obtained in many fields. In particular, BP and its variants, including tree reparameterization and concave-convex procedure, are reformulated with information-geometrical terms, and their relations to the free energy function are elucidated from an information-geometrical viewpoint. We then propose a family of new algorithms. The stabilities of the algorithms are analyzed, and methods to accelerate them are investigated.

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