Unidirectional random growth with resetting

Abstract

We review and classify stochastic processes without detailed balance condition. We obtain stationary distributions and investigate their stability in terms of generalized entropic distances beyond the Kullback-Leibler formula. A simple stochastic model with local growth rates and direct resetting to the ground state is investigated and applied to various networks, scientific citations and Facebook popularity, hadronic yields in high energy particle reactions, income and wealth distributions, biodiversity and settlement size distributions.

Entropic Divergence and Entropy Related to Nonlinear Master Equations

“…By the generalized entropy formula, important properties, like non-negativity, expansibility, convexity are still to be satisfied [25]. It is, however, hard to convince about one or the other "next to simplest" generalization without investigating physical phenomena which realize or-if we are lucky-even suggest preferring a given formula against others [26,27]. First, in a deductive way someone suggests a formula and then checks its fundamental properties before applying to a number of physical observations.…”

“…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.…”

“…Finally we would like to review a few applications of this simplified LGGR treatment. As it has been published in [26], stationary PDF-s can be reproduced in several socio-economic or physical models. Socio-economic examples include popularity ranking from citation numbers [52][53][54], income [55][56][57] and settlement sizes [58].…”

Telegraphic processes with stochastic resetting

“…The random dynamics of the process and the resetting mechanism are taken to be independent of each other, while resettings to a given position occur at Poissonian times. Besides few antecedents in physics [1] and in the mathematics literature (see [2] for more information), the topic has been revived and further developed by the works of Evans, Majumdar, and collaborators [3][4][5][6][7][8][9] as well as an increasing number of different investigators of which we cite a very small sample [2,[10][11][12][13][14][15].…”

Gintropy: Gini Index Based Generalization of Entropy

“…Our next example is the natural distribution, mimicking the Boltzmann–Gibbs exponential energy distribution, known from statistical physics. This is not necessarily an equilibrium distribution, it may also be the stationary limit of “growth and resetting” type processes with quantity-independent rates [ 23 ]. The PDF is a scaling one: .…”