Investigating Information Geometry in Classical and Quantum Systems through Information Length

Stochastic processes are ubiquitous in nature and laboratories, and play a major role across traditional disciplinary boundaries. These stochastic processes are described by different variables and are thus very system-specific. In order to elucidate underlying principles governing different phenomena, it is extremely valuable to utilise a mathematical tool that is not specific to a particular system. We provide such a tool based on information geometry by quantifying the similarity and disparity between Probability Density Functions (PDFs) by a metric such that the distance between two PDFs increases with the disparity between them. Specifically, we invoke the information length L(t) to quantify information change associated with a time-dependent PDF that depends on time. L(t) is uniquely defined as a function of time for a given initial condition. We demonstrate the utility of L(t) in understanding information change and attractor structure in classical and quantum systems.

[1]  Mark Andrews.  Quantum mechanics with uniform forces , 2010 .

[2]  Rainer Hollerbach,et al.  Time-dependent probability density function in cubic stochastic processes. , 2016, Physical review. E.

[3]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[4]  Rainer Hollerbach,et al.  Signature of nonlinear damping in geometric structure of a nonequilibrium process. , 2017, Physical review. E.

[5]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[6]  A. Plastino,et al.  Fisher's information, Kullback's measure, and H-theorems , 1998 .

[7]  F. Schlögl,et al.  Thermodynamic metric and stochastic measures , 1985 .

[8]  Eun-Jin Kim,et al.  Structures in Sound: Analysis of Classical Music Using the Information Length , 2016, Entropy.

[9]  Anja Walter,et al.  Introduction To Stochastic Calculus With Applications , 2016 .

[10]  James Heseltine,et al.  Geometric structure and geodesic in a solvable model of nonequilibrium process. , 2016, Physical review. E.

[11]  W. Wootters.  Statistical distance and Hilbert space , 1981 .

[12]  Eun-Jin Kim,et al.  Information Geometry of Non-Equilibrium Processes in a Bistable System with a Cubic Damping , 2017, Entropy.

[13]  James Heseltine,et al.  Novel mapping in non-equilibrium stochastic processes , 2016 .

[14]  Massimiliano Esposito,et al.  Nonconvexity of the relative entropy for Markov dynamics: a Fisher information approach. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  B. Frieden.  Science from Fisher Information , 2004 .

[16]  Rainer Hollerbach,et al.  Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models , 2017 .

[17]  S. Nicholson,et al.  Investigation of the statistical distance to reach stationary distributions , 2015 .

[18]  Gavin E Crooks,et al.  Far-from-equilibrium measurements of thermodynamic length. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Bjarne Andresen,et al.  Quasistatic processes as step equilibrations , 1985 .

[20]  Jan Naudts,et al.  Quantum Statistical Manifolds , 2018, Entropy.

[21]  G. Ruppeiner,et al.  Thermodynamics: A Riemannian geometric model , 1979 .

[22]  Mw Hirsch,et al.  Chaos In Dynamical Systems , 2016 .

[23]  Eun Jin Kim,et al.  Information length in quantum systems , 2018 .

[24]  Eun-Jin Kim,et al.  Far-From-Equilibrium Time Evolution between Two Gamma Distributions , 2017, Entropy.

[25]  David A. Sivak,et al.  Thermodynamic metrics and optimal paths. , 2012, Physical review letters.