Soft bounds on diffusion produce skewed distributions and Gompertz growth

TL;DR

This paper investigates how soft bounds on diffusion processes lead to skewed distributions and Gompertz growth, revealing steady states with entropic flux and a saturating maximum, with implications for understanding constrained systems.

Contribution

It provides a theoretical analysis of soft bounds in diffusion, showing their effects on steady-state distributions, flux, and growth laws, and proposes a universal scaling mechanism.

Findings

Steady state exhibits a skewed distribution with a shoulder.

The maximum point follows Gompertz growth law.

Soft bounds induce a net probability flux with entropic origin.

Abstract

Constraints can affect dramatically the behavior of diffusion processes. Recently, we analyzed a natural and a technological system and reported that they perform diffusion-like discrete steps displaying a peculiar constraint, whereby the increments of the diffusing variable are subject to configuration-dependent bounds. This work explores theoretically some of the revealing landmarks of such phenomenology, termed "soft bound". At long times, the system reaches a steady state irreversibly (i.e., violating detailed balance), characterized by a skewed "shoulder" in the density distribution, and by a net local probability flux, which has entropic origin. The largest point in the support of the distribution follows a saturating dynamics, expressed by the Gompertz law, in line with empirical observations. Finally, we propose a generic allometric scaling for the origin of soft bounds. These findings shed light on the impact on a system of such "scaling" constraint and on its possible generating mechanisms.