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Shannon versus Kullback-Leibler entropies in nonequilibrium random motion. (English) Zbl 1171.82318
Summary: We analyze dynamical properties of the Shannon information entropy of a continuous probability distribution, which is driven by a standard diffusion process. This entropy choice is confronted with another option, employing the conditional Kullback-Leibler entropy. Both entropies discriminate among various probability distributions, either statically or in the time domain. An asymptotic approach towards equilibrium is typically monotonic in terms of the Kullback entropy. The Shannon entropy time rate needs not to be positive and is a sensitive indicator of the power transfer processes (removal/supply) due to an active environment. In the case of Smoluchowski diffusions, the Kullback entropy time rate coincides with the Shannon entropy “production” rate.

82C03 Foundations of time-dependent statistical mechanics
94A17 Measures of information, entropy
Full Text: DOI
[1] Risken, H., The fokker – planck equation, (1989), Springer-Verlag Berlin · Zbl 0665.60084
[2] Hasegawa, H., Prog. theor. phys., 57, 1523, (1977)
[3] Vilar, J.M.G.; Rubi, J.M., Proc. natl. acad. sci. (N.Y.), 98, 11081, (2001)
[4] Shannon, C.E.; Shannon, C.E., Bell syst. technol. J., Bell syst. technol. J., 27, 623, (1948)
[5] Cover, T.M.; Thomas, J.A., Elements of information theory, (1991), Wiley New York · Zbl 0762.94001
[6] Sobczyk, K., Mech. systems signal proc., 15, 475, (2001)
[7] Lasota, A.; Mackey, M.C., Chaos, fractals and noise, (1994), Springer-Verlag Berlin · Zbl 0784.58005
[8] Jiang, D.-Q.; Qian, M.; Qian, M.-P., Mathematical theory of nonequilibrium steady states, Lecture notes in mathematics, vol. 1833, (2004), Springer-Verlag Berlin
[9] Mackey, M.C.; Tyran-Kamińska, M.
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