Coarse-grained entropy in dynamical systems
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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.
Key wordsGibbs entropy nonequilibrium thermodynamics Lyapunov exponents Gibbs ensemble
MSC2000 numbers37A05 37A60
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