Multifractal spectrum of phase space related to generalized thermostatistics

Abstract

We consider a self-similar phase space with specific fractal dimension $d$ being distributed with spectrum function $f(d)$. Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter is equal to $\bar{\tau }(q)\equiv 1/\tau (q)>1$, and the multifractal function $\tau (q)=q{d}_q-f(d_q)$ is the specific heat determined with multifractal parameter $q\in [1,\infty )$. In this way, the equipartition law is shown to take place. Optimization of the multifractal spectrum function $f(d)$ derives the relation between the statistical weight and the system complexity. It is shown the statistical weight exponent $\tau (q)$ can be modeled by hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponentials to describe arbitrary multifractal phase space explicitly. The spectrum function $f(d)$ is proved to increase monotonically from minimum value $f=-1$ at $d=0$ to maximum one $f=1$ at $d=1$. At the same time, the number of monofractals increases with growth of the phase space volume at small dimensions $d$ and falls down in the limit $d\to 1$.

arXiv comment16 pages, 7 figures

References in corpus (8)

• Non Linear Kinetics underlying Generalized Statistics
• Nonextensivity and multifractality in low-dimensional dissipative systems
• Nonextensive thermodynamic relations
• Tsallis' entropy maximization procedure revisited
• Generalized thermostatistics based on deformed exponential and logarithmic functions
• Microcanonical foundation of nonextensivity and generalized thermostatistics based on the fractality of the phase space
• Nonconcave entropies in multifractals and the thermodynamic formalism
• Effective temperature of self--similar time series: analytical and numerical developments

Cited by in corpus (11)

• Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions
• Maximum entropy principle and power-law tailed distributions
• Thermodynamics with fractal structure, Tsallis statistics and hadrons
• Relativistic Entropy and Related Boltzmann Kinetics
• Essential discreteness in generalized thermostatistics with non-logarithmic entropy
• Physical origin of the power-law tailed statistical distributions
• Relativistic Roots of $\kappa$-Entropy
• Relativistic kinetics and power-law tailed distributions
• Statistical field theories deformed within different calculi
• Power-Law tailed statistical distributions and Lorentz transformations
• Self-similarity degree of deformed statistical ensembles