Self-similarity degree of deformed statistical ensembles

TL;DR

This paper investigates the self-similarity degree of deformed statistical ensembles, revealing how phase space deformation affects the symmetry and self-similarity properties of probability distributions, with connections to the golden mean.

Contribution

It introduces a framework for analyzing self-similarity in deformed phase spaces, identifying how the self-similarity degree varies with deformation parameters and symmetries.

Findings

Self-similarity degree $q$ is fixed by invariance conditions.

$q$ equals the golden mean at a specific deformation.

Dilatation decreases the self-similarity degree $q$.

Abstract

We consider self-similar statistical ensembles with the phase space whose volume is invariant under the deformation that squeezes (expands) the coordinate and expands (squeezes) the momentum. Related probability distribution function is shown to possess a discrete symmetry with respect to manifold action of the Jackson derivative to be a homogeneous function with a self-similarity degree $q$ fixed by the condition of invariance under $(n+1)$-fold action of the dilatation operator related. In slightly deformed phase space, we find the homogeneous function is defined with the linear dependence at $n=0$, whereas the self-similarity degree equals the gold mean at $n=1$, and $q\to n$ in the limit $n\to\infty$. Dilatation of the homogeneous function is shown to decrease the self-similarity degree $q$ at $n>0$.