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

Maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems

A formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem-representing the region where the measurements are made-in contact with a set of "nested heat reservoirs" corresponding to the hierarchical structure of the system, where the temperatures of the reservoirs are allowed to fluctuate owing to the complex interactions between degrees of freedom at different scales. The probability distribution function (pdf) of the temperature of the reservoir at a given scale, conditioned on the temperature of the reservoir at the next largest scale in the hierarchy, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox H functions. The distribution of states of the small subsystem is then computed by averaging the quasiequilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of H functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Macêdo et al. [Phys. Rev. E 95, 032315 (2017)10.1103/PhysRevE.95.032315] from a stochastic dynamical approach to the problem.

Stochastic Model for Phonemes Uncovers an Author-Dependency of Their Usage

We study rank-frequency relations for phonemes, the minimal units that still relate to linguistic meaning. We show that these relations can be described by the Dirichlet distribution, a direct analogue of the ideal-gas model in statistical mechanics. This description allows us to demonstrate that the rank-frequency relations for phonemes of a text do depend on its author. The author-dependency effect is not caused by the author's vocabulary (common words used in different texts), and is confirmed by several alternative means. This suggests that it can be directly related to phonemes. These features contrast to rank-frequency relations for words, which are both author and text independent and are governed by the Zipf's law.

Explaining Zipf's law via a mental lexicon

Zipf's law is the major regularity of statistical linguistics that has served as a prototype for rank-frequency relations and scaling laws in natural sciences. Here we show that Zipf's law-together with its applicability for a single text and its generalizations to high and low frequencies including hapax legomena-can be derived from assuming that the words are drawn into the text with random probabilities. Their a priori density relates, via the Bayesian statistics, to the mental lexicon of the author who produced the text.

Multicanonical distribution: statistical equilibrium of multiscale systems

A multicanonical formalism is introduced to describe the statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.

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

Based on a combination of the distortive lattice gas model and the maximum information entropy approach, the thermodynamics of insertion into disordered hosts is analyzed. It is found that the isotherm specificities can be explained as a cooperative interplay of the host volume expansion and the internal distortions, which tend to optimize the host structure inducing a local lowering of the insertion energetic cost. Behavior of amorphous LixWO3 films of different thicknesses is discussed in this context.

Constrained equilibrium as a tool for characterization of deformable porous media

A new method for characterizing the deformable porous materials with noncritical adsorption probes is proposed. The mechanism is based on driving the adsorbate through a sequence of constrained equilibrium states with the insertion isotherms forming a pseudocritical point or a van der Waals-type loop. In the framework of a perturbation theory and Monte Carlo simulations we have found a link between the loop parameters and the host morphology. This allows one to characterize porous matrices through analyzing a shift of the pseudocritical point and a shape of the pseudospinodals.