title: Soft bounds on diffusion produce skewed distributions and Gompertz growth.
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.
authors.affiliation: Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge Massachusetts, USA and Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy.
authors.forename: Salvatore
authors.initials: S
authors.lastname: Mandrà
authors.affiliation: Sorbonne Universités, UPMC Univ Paris 06, UMR 7238, Computational and Quantitative Biology, 15 rue de l'École de Médecine Paris, France and CNRS, UMR 7238, Paris, France.
authors.forename: Marco Cosentino
authors.initials: MC
authors.lastname: Lagomarsino
authors.affiliation: Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy.
authors.forename: Marco
authors.initials: M
authors.lastname: Gherardi
mesh_terms: D003198:Computer Simulation; D004058:Diffusion; D019277:Entropy; D008962:Models, Theoretical; D009010:Monte Carlo Method