Entropic Distance for Nonlinear Master Equation

Abstract

More and more works deal with statistical systems far from equilibrium, dominated by unidirectional stochastic processes, augmented by rare resets. We analyze the construction of the entropic distance measure appropriate for such dynamics. We demonstrate that a power-like nonlinearity in the state probability in the master equation naturally leads to the Tsallis (HavrdaCharvát, Aczél-Daróczy) q-entropy formula in the context of seeking for the maximal entropy state at stationarity. A few possible applications of a certain simple and linear master equation to phenomena studied in statistical physics are listed at the end.Keywords: q-entropy; entropic distance; Matthew principle Definition and Properties of Entropic DistanceDealing with the dynamics of classical probabilities, we would like to propose a general recipe for defining the corresponding formula for the entropic divergence between two probability distributions. Our goal is to handle complex systems with a stochastic dynamics, generalized to nonlinear dependence on the probabilities. For the study of quantum state probabilities and their distance measures we refer to a recent paper [1] and references therein.Entropic distance, more properly called "entropic divergence", is traditionally interpreted as a relative entropy, as a difference between entropies with a prior condition, and without [2]. It is also the Boltzmann-Shannon entropy of a distribution relative to another [3]. Looking at this construction, however, from the viewpoint of a generalized entropy [4], the simple difference or logarithm of a ratio cannot be held as a definition anymore.Instead, in this paper, we explore a reverse engineering concept: seeking an entropic divergence formula, which is subject to some wanted properties, we consider entropy as a derived quantity. More precisely, we seek entropic divergence formulas appropriate for given stochastic dynamics, shrinking during the approach to a stationary distribution, whenever it exists, and establish the entropy formula from this distance to the uniform distribution. By doing so we serve two goals: (i) having constructed a non-negative entropic distance we derive an entropy formula which is maximal for the uniform distribution; and (ii) we come as near as possible to the classical difference formula for the relative entropy.Starting from a given master equation, it is far from trivial that which is the most suitable entropy divergence formula for analyzing the stability of a stationary solution. In the present paper we provide a general procedure to obtain a general entropic divergence formula for atypical cases. Although we exemplify only the well-known cases of the logarithmic formula of the Kullback-Leibler and that of the Renyi divergence, our result readily generalizes to an infinite number of cases, distinguished by the dependence on the initial state probability at each transition term.

Scaling in income inequalities and its dynamical origin

“…We conclude that a simple master equation with state dependent growth and reset terms [32,33] performs beautifully in describing the income distribution on the whole range of income values. In agreement with the simple reasoning, the analyses of real-world data revealed that the growth in salaries is on average preferential, and the reset rate depends in a simple form as a function of income, being negative for the low income region and saturates at a constant positive value for high salaries.…”

“…In the above equation we denoted by δ(x) the Dirac functional. It has been proven, that under very general conditions the above dynamical evolution equation converges to a steady-state with a ρs(x) stationary probability density [33]. This stationary probability density is derived from the condition:…”

Entropic Divergence and Entropy Related to Nonlinear Master Equations

“…Although both ways are legitimate, in this paper we follow the second one. Given an evolution dynamics for the microstate probabilities, we first seek for an expedient formula for the entropic divergence [28]: a non-negative measure between two probability distributions (in a continuous model probability distribution functions (PDFs) which shrinks during the dynamical evolution. Possibly for two arbitrary initial distributions.…”

“…As part of this presentation, we repeat some formulas published already earlier [26,28]. Our purpose is to provide the reader with a self-contained chain of thoughts, followable without further reading.…”

Association Factor for Identifying Linear and Nonlinear Correlations in Noisy Conditions

“…We also compute Entropic Distance (ED) and compare it with AF. Entropic Distance, also called “relative entropy”, is the differences between entropies with and without a prior condition [ 20 ]. The conditional entropy of two variables X and Y taking values x and y , respectively, is defined by: where b is the logarithm base.…”