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 $d\to 1$.

16 pages, 7 figures

References in corpus (1)

• Nonconcave entropies in multifractals and the thermodynamic formalism

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