Generalization of the Beck-Cohen superstatistics

Universal Non-Extensive Statistical Physics Temporal Pattern of Major Subduction Zone Aftershock Sequences

Large subduction-zone earthquakes generate long-lasting and wide-spread aftershock sequences. The physical and statistical patterns of these aftershock sequences are of considerable importance for better understanding earthquake dynamics and for seismic hazard assessments and earthquake risk mitigation. In this work, we analyzed the statistical properties of 42 aftershock sequences in terms of their temporal evolution. These aftershock sequences followed recent large subduction-zone earthquakes of M ≥ 7.0 with focal depths less than 70 km that have occurred worldwide since 1976. Their temporal properties were analyzed by investigating the probability distribution of the interevent times between successive aftershocks in terms of non-extensive statistical physics (NESP). We demonstrate the presence of a crossover behavior from power-law (q ≠ 1) to exponential (q = 1) scaling for greater interevent times. The estimated entropic q-values characterizing the observed distributions range from 1.67 to 1.83. The q-exponential behavior, along with the crossover behavior observed for greater interevent times, are further discussed in terms of superstatistics and in view of a stochastic mechanism with memory effects, which could generate the observed scaling patterns of the interevent time evolution in earthquake aftershock sequences.

Maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems

A formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem-representing the region where the measurements are made-in contact with a set of "nested heat reservoirs" corresponding to the hierarchical structure of the system, where the temperatures of the reservoirs are allowed to fluctuate owing to the complex interactions between degrees of freedom at different scales. The probability distribution function (pdf) of the temperature of the reservoir at a given scale, conditioned on the temperature of the reservoir at the next largest scale in the hierarchy, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox H functions. The distribution of states of the small subsystem is then computed by averaging the quasiequilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of H functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Macêdo et al. [Phys. Rev. E 95, 032315 (2017)10.1103/PhysRevE.95.032315] from a stochastic dynamical approach to the problem.

First-passage time for superstatistical Fokker-Planck models

The first-passage-time (FPT) problem is studied for superstatistical models assuming that the mesoscopic system dynamics is described by a Fokker-Planck equation. We show that all moments of the random intensive parameter associated to the superstatistical approach can be put in one-to-one correspondence with the moments of the FPT. For systems subjected to an additional uncorrelated external force, the same statistical information is obtained from the dependence of the FPT moments on the external force. These results provide an alternative technique for checking the validity of superstatistical models. As an example, we characterize the mean FPT for a forced Brownian particle.

Quantum entanglement and temperature fluctuations

In this paper, we consider entanglement in a system out of equilibrium, adopting the viewpoint given by the formalism of superstatistics. Such an approach yields a good effective description for a system in a slowly fluctuating environment within a weak interaction between the system and the environment. For this purpose, we introduce an alternative version of the formalism within a quantum mechanical picture and use it to study entanglement in the Heisenberg XY model, subject to temperature fluctuations. We consider both isotropic and anisotropic cases and explore the effect of different temperature fluctuations (χ^{2}, log-normal, and F distributions). Our results suggest that particular fluctuations may enhance entanglement and prevent it from vanishing at higher temperatures than those predicted for the same system at thermal equilibrium.

Nonthermal and suprathermal distributions as a consequence of superstatistics

We propose to put the well-known nonthermal and suprathermal empirical distributions, used in plasma physics, onto a more rigorous foundation. Their use is frequently criticized because of a lack of formal derivation and physical explanation. A connection between these non-Maxwellian distributions and the Beck-Cohen superstatistics is suggested. They are perceived as a consequence of typical temperature fluctuations. We show that the suprathermal distribution is generated by the Γ distribution of the inverse temperature, in the same way as the Tsallis q statistics. The nonthermal distribution also follows from the χ(2) distribution, with a small variance. Our contribution provides a possible physical meaning for these ad hoc distributions.

Bose-Einstein condensation of light: general theory

A theory of Bose-Einstein condensation of light in a dye-filled optical microcavity is presented. The theory is based on the hierarchical maximum entropy principle and allows one to investigate the fluctuating behavior of the photon gas in the microcavity for all numbers of photons, dye molecules, and excitations at all temperatures, including the whole critical region. The master equation describing the interaction between photons and dye molecules in the microcavity is derived and the equivalence between the hierarchical maximum entropy principle and the master equation approach is shown. The cases of a fixed mean total photon number and a fixed total excitation number are considered, and a much sharper, nonparabolic onset of a macroscopic Bose-Einstein condensation of light in the latter case is demonstrated. The theory does not use the grand canonical approximation, takes into account the photon polarization degeneracy, and exactly describes the microscopic, mesoscopic, and macroscopic Bose-Einstein condensation of light. Under certain conditions, it predicts sub-Poissonian statistics of the photon condensate and the polarized photon condensate, and a universal relation takes place between the degrees of second-order coherence for these condensates. In the macroscopic case, there appear a sharp jump in the degrees of second-order coherence, a sharp jump and kink in the reduced standard deviations of the fluctuating numbers of photons in the polarized and whole condensates, and a sharp peak, a cusp, of the Mandel parameter for the whole condensate in the critical region. The possibility of nonclassical light generation in the microcavity with the photon Bose-Einstein condensate is predicted.

Multicanonical distribution: statistical equilibrium of multiscale systems

A multicanonical formalism is introduced to describe the statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.

Hierarchical maximum entropy principle for generalized superstatistical systems and Bose-Einstein condensation of light

A principle of hierarchical entropy maximization is proposed for generalized superstatistical systems, which are characterized by the existence of three levels of dynamics. If a generalized superstatistical system comprises a set of superstatistical subsystems, each made up of a set of cells, then the Boltzmann-Gibbs-Shannon entropy should be maximized first for each cell, second for each subsystem, and finally for the whole system. Hierarchical entropy maximization naturally reflects the sufficient time-scale separation between different dynamical levels and allows one to find the distribution of both the intensive parameter and the control parameter for the corresponding superstatistics. The hierarchical maximum entropy principle is applied to fluctuations of the photon Bose-Einstein condensate in a dye microcavity. This principle provides an alternative to the master equation approach recently applied to this problem. The possibility of constructing generalized superstatistics based on a statistics different from the Boltzmann-Gibbs statistics is pointed out.