NDLI: Coarse-grained entropy in dynamical systems

Content Provider	Springer Nature Link
Author	Piftankin, G. Treschev, D.
Copyright Year	2010
Abstract	Let M be the phase space of a physical system. Consider the dynamics, determined by the invertible map T: M → M, preserving the measure µ on M. Let ν be another measure on M, dν = ρdµ. Gibbs introduced the quantity s(ρ) = −∝ρ log ρdµ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy.First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information.Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms ν in the following way: ν → ν $_{n}$, dν $_{n}$ = ρ ○ T $^{−n }$ dµ. Hence, we obtain the sequence of densities ρ $_{n}$ = ρ ○ T $^{−n }$ and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map T. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.
Starting Page	575
Ending Page	597
Page Count	23
File Format	PDF
ISSN	15603547
Journal	Regular and Chaotic Dynamics
Volume Number	15
Issue Number	4-5
e-ISSN	14684845
Language	English
Publisher	SP MAIK Nauka/Interperiodica
Publisher Date	2010-08-06
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Gibbs entropy nonequilibrium thermodynamics Lyapunov exponents Gibbs ensemble Dynamical Systems and Ergodic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistical and Nonlinear Physics Mathematical Physics Mechanical Engineering Modeling and Simulation

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