How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions

TL;DR

This paper demonstrates how the statistics of driven non-equilibrium systems are determined by the nature of their driving rates, showing that different state-dependent driving functions produce various well-known distributions.

Contribution

It introduces a framework linking driving rate functions to stationary distributions in non-equilibrium systems, extending the applicability of sample space reducing processes.

Findings

Constant driving rates produce power-law distributions.

State-dependent driving functions yield diverse distributions like exponential, Gamma, and normal.

Logarithmic and power-law dependencies lead to log-normal and stretched exponential distributions.

Abstract

Sample space reducing (SSR) processes offer a simple analytical way to understand of the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to cascading pro- cesses. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function gives the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process.