Reference : Nonconvexity of the relative entropy for Markov dynamics: A Fisher information approach
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Physics
http://hdl.handle.net/10993/10453
Nonconvexity of the relative entropy for Markov dynamics: A Fisher information approach
English
Polettini, Matteo mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Physics and Materials Science Research Unit >]
Esposito, Massimiliano mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Physics and Materials Science Research Unit >]
11-Jul-2013
Physics Review E
American Physical Society
88
012112
Yes
International
[en] We show via counterexamples that relative entropy between the solution of a Markovian master equation and the steady state is not a convex function of time. We thus disprove the hypotheses that a general evolution principle of thermodynamics based on the decrease of the nonadiabatic entropy production could hold. However, we argue that a large separation of typical decay times is necessary for nonconvex solutions to occur, making concave transients extremely short lived with respect to the main relaxation modes. We describe a general method based on the Fisher information matrix to discriminate between generators that admit nonconvex solutions and those that do not. While initial conditions leading to concave transients are shown to be extremely fine-tuned, by our method we are able to select nonconvex initial conditions that are arbitrarily close to the steady state. Convexity does occur when the system is close to satisfying detailed balance or, more generally, when certain normality conditions of the decay modes are satisfied. Our results circumscribe the range of validity of a conjecture by Maes et al. [ Phys. Rev. Lett. 107 010601 (2011)] regarding monotonicity of the large deviation rate functional for the occupation probability, showing that while the conjecture might hold in the long-time limit, the conditions for Lyapunov's second criterion for stability are not met.
Researchers
http://hdl.handle.net/10993/10453
10.1103/PhysRevE.88.012112
http://pre.aps.org/abstract/PRE/v88/i1/e012112

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