Multifractal spectrum of phase space related to generalized thermostatistics

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 nonadditivity parameter is equal to τ̄(q) ≡ 1/τ(q) > 1, and the multifractal function τ(q) = qdq − f(dq) is the specific heat determined with multifractal parameter q ∈ [1,∞). 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 τ(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→ 1.