Coarse-grained Entropy in Dynamical Systems

Let M be the phase space of a physical system. Consider the dynamics, determined by the invertible map T : M. M, preserving the measure mu on M. Let. be another measure on M, d nu = rho d mu. Gibbs introduced the quantity s(rho) = - integral 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 bar right arrow nu(n), d nu(n) = rho o T(-n) d mu. Hence, we obtain the sequence of densities rho(n) = rho o 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.