Preprint Article Version 1 This version is not peer-reviewed

Information Geometry in Classical and Quantum Systems

Version 1 : Received: 19 July 2018 / Approved: 19 July 2018 / Online: 19 July 2018 (11:30:59 CEST)

A peer-reviewed article of this Preprint also exists.

Kim, E.-J. Investigating Information Geometry in Classical and Quantum Systems through Information Length. Entropy 2018, 20, 574. Kim, E.-J. Investigating Information Geometry in Classical and Quantum Systems through Information Length. Entropy 2018, 20, 574.

Journal reference: Entropy 2018, 20, 574
DOI: 10.3390/e20080574

Abstract

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic processes and geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the information length. By using this concept, we investigate classical and quantum systems and demonstrate the utility of the information length as a unique Lagrangian diagnostic to quantify the information change as a system continuously evolves in time and to map out attractor structure. We further elucidate quantum effects (uncertainty relation) and the dual role of the width of PDF in quantum systems.

Subject Areas

stochastic processes; Langevin equation; Fokker-Planck equation; information length; Fisher information; relaxation; chaos; attractor

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