Article (Scientific journals)
Nonconvexity of the relative entropy for Markov dynamics: A Fisher information approach
Polettini, Matteo; Esposito, Massimiliano
2013In Physical Review. E., 88, p. 012112
Peer reviewed
 

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Abstract :
[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.
Disciplines :
Physics
Author, co-author :
Polettini, Matteo ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Physics and Materials Science Research Unit
Esposito, Massimiliano  ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Physics and Materials Science Research Unit
Language :
English
Title :
Nonconvexity of the relative entropy for Markov dynamics: A Fisher information approach
Publication date :
11 July 2013
Journal title :
Physical Review. E.
Publisher :
American Physical Society
Volume :
88
Pages :
012112
Peer reviewed :
Peer reviewed
Available on ORBilu :
since 11 November 2013

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