Analytical computation of frequency distributions of path-dependent processes by means of a non-multinomial maximum entropy approach

Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. Their statistics is typically non-multinomial 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, non-interacting systems, and to the ensemble picture; It loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, such that the non-multinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of an entropy. We demonstrate this for self-reinforcing Pólya urn processes that explicitly break multinomial structure. We demonstrate the new method by computing frequency and rank distributions of Polya urn processes. For the first time we are able to use detailed microscopic update rules of a path-dependent process to construct a non-multinomial entropy functional, that, when maximized, predicts the time-dependent distribution function.