Self-similarity degree of deformed statistical ensembles

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 $n>0$.

17 pages, 1 figure

References in corpus (9)

• Statistical mechanics in the context of special relativity
• Discrete scale invariance and complex dimensions
• Thermostatistics of q-deformed boson gas
• Deformed logarithms and entropies
• Thermodynamic equilibrium and its stability for Microcanonical systems described by the Sharma-Taneja-Mittal entropy
• Generalized q-Analysis of Log-Periodicity: Applications to Critical Ruptures
• Multifractal spectrum of phase space related to generalized thermostatistics
• Numerical investigations of discrete scale invariance in fractals and multifractal measures
• Classical q-deformed dynamics

Cited by in corpus (2)

• Statistical field theories deformed within different calculi
• Statistical theory of self-similarly distributed fields