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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2010, Volume 15, Issue 4-5, Pages 575–597 (Mi rcd517)

Coarse-grained entropy in dynamical systems

G. Piftankin, D. Treschev

V.A. Steklov Mathematical Institute, RAS, Gubkina str. 8, Moscow 119991, Russia

Abstract: Let $M$ be the phase space of a physical system. Consider the dynamics, determined by the invertible map $T:M\to M$, preserving the measure $\mu$ on $M$. Let $\nu$ be another measure on $M$, $d\nu=\rho d\mu$. Gibbs introduced the quantity $s(\rho)=-\int \rho\log\rho d\mu$ 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 $\nu$ in the following way: $\nu\mapsto\nu_n$, $d\nu_n=\rho\circ T^{-n} d\mu$. Hence, we obtain the sequence of densities $\rho_n=\rho\circ T^{-n}$ 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.

Keywords: Gibbs entropy, nonequilibrium thermodynamics, Lyapunov exponents, Gibbs ensemble

DOI: https://doi.org/10.1134/S156035471004012X

Bibliographic databases:

MSC: 37A05, 37A60
Accepted:24.12.2009
Language:

Citation: G. Piftankin, D. Treschev, “Coarse-grained entropy in dynamical systems”, Regul. Chaotic Dyn., 15:4-5 (2010), 575–597

Citation in format AMSBIB
\Bibitem{PifTre10} \by G. Piftankin, D. Treschev \paper Coarse-grained entropy in dynamical systems \jour Regul. Chaotic Dyn. \yr 2010 \vol 15 \issue 4-5 \pages 575--597 \mathnet{http://mi.mathnet.ru/rcd517} \crossref{https://doi.org/10.1134/S156035471004012X} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2679766} \zmath{https://zbmath.org/?q=an:1203.37008}