Maximum entropy approach to power-law distributions in coupled dynamic-stochastic systems

Abstract

Statistical properties of coupled dynamic-stochastic systems are studied within a combination of the maximum information principle and the superstatistical approach. The conditions at which the Shannon entropy functional leads to power-law statistics are investigated. It is demonstrated that, from a quite general point of view, the power-law dependencies may appear as a consequence of "global" constraints restricting both the dynamic phase space and the stochastic fluctuations. As a result, at sufficiently long observation times the dynamic counterpart is driven into a nonequilibrium steady state whose deviation from the usual exponential statistics is given by the distance from the conventional equilibrium.

Interplay of host volume variations and internal distortions in the course of intercalation into disordered matrices

“…Therefore, we infer the distribution using the maximum information entropy approach. 9,18,19 The procedure consists in maximizing the Shannon entropy measure…”

“…Despite a considerable progress 5 in the field, the consequences of the disordered host morphology are still poorly understood. In particular, recent theoretical studies 8,9 of adsorption into disordered porous media suggest that, because of the host disorder, the adsorption thermodynamics might depend on a driving path ͑e.g., controlled injections of prescribed portions or an equilibrium with a bulk reservoir͒. In the context of intercalation processes this means that different electrochemical methods ͑e.g., chronopotentiometry and voltammetry͒ could give different results.…”

Entropy minimization within the maximum entropy approach

“…For instance, as we have demonstrated [10,11], if θ(µ|x) is of logarithmic form, then (9) transforms into a Γ-distribution. Physically this corresponds to a non-equilibrium stationary state whose thermodynamic entropy is held at a given distance from the maximum that corresponds to the equilibrium.…”

“…This relation does not depend on the form of the entropic functional (the form (1) is quite general) and thus could be helpful for drawing conclusions of general validity. A brief discussion of (15) for the case of the Shannon form can be found in [10,11]. Anyway it is clear that the behavior of H(µ) depends on what is known/measured θ(µ) and on our conditional estimation θ(µ|x).…”

“…In this paper we explore such a possibility in application to coupled dynamic-stochastic systems [9,10,11], where the dynamic counterpart may serve as a probe for determining the behavior f (x|µ) of the stochastic subsystem. In this way the maximum entropy approach allows one to find a functional form of the distribution f (x|µ) while the subsequent minimization of the maximized entropy permits to reduce the uncertainty on the relevant parameters.…”