
Coarsegrained 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
(finegrained) entropy the so called coarsegrained entropy.
First we obtain a formula for the difference between the coarsegrained
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 coarsegrained 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 coarsegrained entropy.
We show that while the Gibbs entropy remains constant, the
coarsegrained 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 coarsegrained 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 Received: 17.12.2009 Accepted:24.12.2009
Language:
Citation:
G. Piftankin, D. Treschev, “Coarsegrained entropy in dynamical systems”, Regul. Chaotic Dyn., 15:45 (2010), 575–597
Citation in format AMSBIB
\Bibitem{PifTre10}
\by G. Piftankin, D. Treschev
\paper Coarsegrained entropy in dynamical systems
\jour Regul. Chaotic Dyn.
\yr 2010
\vol 15
\issue 45
\pages 575597
\mathnet{http://mi.mathnet.ru/rcd517}
\crossref{https://doi.org/10.1134/S156035471004012X}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2679766}
\zmath{https://zbmath.org/?q=an:1203.37008}
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