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Temporal Behavior of the Conditional and Gibbs’ Entropies
Authors:Michael C Mackey  Marta Tyran-Kamińska
Institution:(1) Departments of Physiology, Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University, 3655 Promenade Sir William Osler, Montreal, QC, Canada, H3G 1Y6;(2) Institute of Mathematics, Silesian University, ul. Bankowa 14, 40-007 Katowice, Poland
Abstract:We study the temporal approach to equilibrium of the Gibbs’ and conditional entropies for stochastic systems in the presence of white noise. The conditional entropy will either remain constant or monotonically increase to its maximum of zero. However, the Gibbs’ entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach. We have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically.
Keywords:conditional entropy  Gibbs’  entropy  asymptotic stability  Ornstein-Uhlenbeck process  noisy harmonic oscillator
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