Hyperstatistics

Abstract

We propose a general approach, named by us hyperstatistics, to treat complex systems, in which Boltzmann-Gibbs statistics breaks down at the domain level of the system. Hyperstatistics preserves the concavity of the nonadditive $q$-entropy. We obtain analytical closed-form expressions for the here proposed ($q, n$)-generalized Boltzmann factor $B_q^{n}$ considering uniform, $γ$, Log-normal, F, and the $q$-$γ$ probability distribution functions. Remarkably, for all investigated distribution functions, $B_q^{n}$ reduces to a $q$-exponential-type function. To demonstrate the applicability of hyperstatistics, we use a table top experiment of the discharge of a capacitor considering $γ$-distributed relaxation times, the pressure decay over time associated with the pumping of $^4$He lines of a closed cycle cryostat, midrapidity data for $\textit{p}$-Pb collisions at the LHC, as well as data set for acceleration distribution in turbulent systems. Furthermore, we deduce the power-law-like $\textit{universal}$ dielectric response using a $q$-generalized $γ$-distribution function proposed in this work. Hyperstatistics is applicable to systems with inherent non-Boltzmann-Gibbsian statistics at the domain level. We demonstrate that at the domain level hyperstatistics and superstatistics are equivalent, while at the macroscopic level hyperstatistics recovers superstatistics for $q \to 1$ because the Boltzmann-Gibbs character of the domains is restored, as well as Boltzmann-Gibbs statistics for $n \to \infty$.