Investigating Information Geometry in Classical and Quantum Systems through Information Length

• Published in: Entropy (Basel, Switzerland) Vol. 20; no. 8; p. 574
• Main Author: Kim, Eun-Jin
• Format: Journal Article
• Language: English
• Published: Switzerland MDPI 03-08-2018; MDPI AG
• Subjects: attractor; chaos; Fisher information; Fokker–Planck equation; information length; Langevin equation; probability density function; relaxation; stochastic processes; attractor; chaos; Fokker–Planck equation; Langevin equation; relaxation; stochastic processes; Fisher information; information length; probability density function
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e20080574

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.