Analytical computation of frequency distributions of pathdependent processes by means of a nonmultinomial maximum entropy approach
Abstract
Pathdependent stochastic processes are often nonergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of pathdependent processes. Their statistics is typically nonmultinomial in the sense that the multiplicities of the occurrence of states is not a multinomial factor. The maximum entropy principle is tightly related to multinomial processes, noninteracting systems, and to the ensemble picture; It loses its meaning for pathdependent processes. Here we show that an equivalent to the ensemble picture exists for pathdependent processes, such that the nonmultinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of a relative entropy. We demonstrate this for selfreinforcing Pólya urn processes, which explicitly generalise multinomial statistics. We demonstrate the adequacy of this constructive approach towards nonmultinomial pendants of entropy by computing frequency and rank distributions of Pólya urn processes. We show how microscopic update rules of a pathdependent process allow us to explicitly construct a nonmultinomial entropy functional, that, when maximized, predicts the timedependent distribution function.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.00414
 Bibcode:
 2015arXiv151100414H
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Computer Science  Information Theory
 EPrint:
 13 pages, 3 figures