Multifractal spectrum of phase space related to generalized thermostatistics

We consider a self-similar phase space with specific fractal dimension dd being distributed with spectrum function f(d)f(d). Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter equals to τ̄(q)≡1/τ(q)>1, and the multifractal function τ(q)=qdq−f(dq)τ(q)=qdq−f(dq) is the specific heat determined with multifractal parameter q∈[1,∞]q∈[1,∞]. At that, the equipartition law is shown to take place. Optimization of the multifractal spectrum function f(d)f(d) arrives at the relation between the statistical weight and the system complexity. It is shown that the statistical weight exponent τ(q)τ(q) can be modeled by hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponential functions to describe arbitrary multifractal phase space explicitly. The spectrum function f(d)f(d) is proved to increase monotonically from minimum value f=−1f=−1 at d=0d=0 to maximum one f=1f=1 at d=1d=1. At the same time, the number of monofractals increases with the growth of the phase-space volume at small dimensions dd and falls down in the limit d→1d→1.