Rate of Mutual Information Between Coarse-Grained Non-Markovian Variables

The problem of calculating the rate of mutual information between two coarse-grained variables that together specify a continuous time Markov process is addressed. As a main obstacle, the coarse-grained variables are in general non-Markovian, therefore, an expression for their Shannon entropy rates in terms of the stationary probability distribution is not known. A numerical method to estimate the Shannon entropy rate of continuous time hidden-Markov processes from a single time series is developed. With this method the rate of mutual information can be determined numerically. Moreover, an analytical upper bound on the rate of mutual information is calculated for a class of Markov processes for which the transition rates have a bipartite character. Our general results are illustrated with explicit calculations for four-state networks.

[1]  Udo Seifert,et al.  Efficiency of a Brownian information machine , 2012, 1203.0184.

[2]  Jordan M. Horowitz,et al.  Thermodynamic reversibility in feedback processes , 2011, 1104.0332.

[3]  V. Lecomte,et al.  Thermodynamic Formalism for Systems with Markov Dynamics , 2007 .

[4]  Masahito Ueda,et al.  Generalized Jarzynski equality under nonequilibrium feedback control. , 2009, Physical review letters.

[5]  Christopher Jarzynski,et al.  Work and information processing in a solvable model of Maxwell’s demon , 2012, Proceedings of the National Academy of Sciences.

[6]  Gernot Schaller,et al.  Stochastic thermodynamics for “Maxwell demon” feedbacks , 2012, 1204.5671.

[7]  Neri Merhav,et al.  Hidden Markov processes , 2002, IEEE Trans. Inf. Theory.

[8]  M. Esposito.  Stochastic thermodynamics under coarse graining. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Philippe Jacquet,et al.  On the entropy of a hidden Markov process , 2008, Theor. Comput. Sci..

[10]  Touchette,et al.  Information-theoretic limits of control , 1999, Physical review letters.

[11]  Andrea J. Goldsmith,et al.  Capacity of Finite State Channels Based on Lyapunov Exponents of Random Matrices , 2006, IEEE Transactions on Information Theory.

[12]  Sosuke Ito,et al.  Effects of error on fluctuations under feedback control. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Masahito Ueda,et al.  Nonequilibrium thermodynamics of feedback control. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Yuhong Yang.  Elements of Information Theory (2nd ed.). Thomas M. Cover and Joy A. Thomas , 2008 .

[15]  A. Crisanti,et al.  Products of random matrices in statistical physics , 1993 .

[16]  J. Parrondo,et al.  Dissipation: the phase-space perspective. , 2007, Physical review letters.

[17]  A. Crisanti,et al.  Nonequilibrium and information: the role of cross correlations. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Angelo Vulpiani,et al.  Products of Random Matrices , 1993 .

[19]  Yuhai Tu,et al.  The energy-speed-accuracy tradeoff in sensory adaptation , 2012, Nature Physics.

[20]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[21]  M. Feito,et al.  Thermodynamics of feedback controlled systems. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  J. Schnakenberg.  Network theory of microscopic and macroscopic behavior of master equation systems , 1976 .

[23]  A. C. Barato,et al.  Information-theoretic vs. thermodynamic entropy production in autonomous sensory networks , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  U. Seifert,et al.  Extracting work from a single heat bath through feedback , 2011, 1102.3826.

[25]  Masahito Ueda,et al.  Sagawa and Ueda Reply , 2010 .

[26]  U. Seifert.  Stochastic thermodynamics, fluctuation theorems and molecular machines , 2012, Reports on progress in physics. Physical Society.

[27]  Masahito Ueda,et al.  Fluctuation theorem with information exchange: role of correlations in stochastic thermodynamics. , 2012, Physical review letters.

[28]  Udo Seifert,et al.  An autonomous and reversible Maxwell's demon , 2013, 1302.3089.

[29]  Christopher Jarzynski,et al.  Fluctuation relations and coarse-graining , 2007, 0708.2437.

[30]  Wentian Li.  Mutual information functions versus correlation functions , 1990 .

[31]  Suriyanarayanan Vaikuntanathan,et al.  Nonequilibrium detailed fluctuation theorem for repeated discrete feedback. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Juan M R Parrondo,et al.  Entropy production and Kullback-Leibler divergence between stationary trajectories of discrete systems. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Holger Kantz,et al.  Thermodynamic cost of measurements. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Gernot Schaller,et al.  Thermodynamics of a physical model implementing a Maxwell demon. , 2012, Physical review letters.

[35]  David J Schwab,et al.  Energetic costs of cellular computation , 2012, Proceedings of the National Academy of Sciences.

[36]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[37]  Udo Seifert,et al.  Thermodynamics of genuine nonequilibrium states under feedback control. , 2011, Physical review letters.