Nonthermal and suprathermal distributions as a consequence of superstatistics

Reaction rates in quasiequilibrium states

Non-Maxwellian distributions are commonly observed across a wide range of systems and scales. While direct observations provide the strongest evidence for these distributions, they also manifest indirectly through their influence on processes and quantities that strongly depend on the energy distribution, such as reaction rates. In this paper, we investigate reaction rates in the general context of quasiequilibrium systems, which exhibit only local equilibrium. The hierarchical structure of these systems allows their statistical properties to be represented as a superposition of statistics, i.e., superstatistics. Focusing on the three universality classes of superstatistics-χ^{2}, inverse-χ^{2}, and log-normal-we examine how these nonequilibrium distributions influence reaction rates. We analyze, both analytically and numerically, reaction rates for processes involving tunneling phenomena, such as fusion, and identify conditions under which quasiequilibrium distributions outperform Maxwellian distributions in enhancing fusion reactivities. To provide a more detailed quantitative analysis, we further employ semiempirical cross sections to evaluate the effect of these nonequilibrium distributions on ionization and recombination rates in a plasma.

Superstatistics Applied to Cucurbitaceae DNA Sequences

The short and long statistical correlations are essential in the genomic sequence. Such correlations are long-range for introns, whereas, for exons, these are short. In this study, we employed superstatistics to investigate correlations and fluctuations in the distribution of nucleotide sequence lengths of the Cucurbitaceae family. We established a time series for exon sizes to probe these correlations and fluctuations. We used data from the National Center for Biotechnology Information (NCBI) gene database to extract the temporal evolution of exon sizes, measured in terms of the number of base pairs (bp). To assess the model's viability, we utilized a timescale extraction method to determine the statistical properties of our time series, including the local distribution and fluctuations, which provide the exon size distributions based on the q-Gamma and inverse q-Gamma distributions. From the Bayesian statistics standpoint, both distributions are excellent for capturing the correlations and fluctuations from the data.

Superstatistics from a dynamical perspective: Entropy and relaxation

Distributions that deviate from equilibrium predictions are commonly observed across a broad spectrum of systems, ranging from laboratory experiments to astronomical phenomena. These distributions are generally regarded as a manifestation of a quasiequilibrium state and can very often be represented as a superposition of statistics, i.e., superstatistics. The underlying idea in this methodology is that the nonequilibrium system consists of a collection of smaller subsystems that remain infinitely close to equilibrium. This procedure has been effectively implemented in a kinetic setting, but thus far, only in the collisionless regime, limiting its scope of application. In this paper, we address the effect of collisions on the relaxation process and time evolution of superstatistical systems. After confronting the superstatistical distributions with experimental and simulation data, relevant to our analysis, we first study the effect of superstatistics on entropy. We explicitly show that, in the absence of long-range interactions, the extensivity of entropy is preserved, albeit influenced by the specific class of temperature fluctuations. Then, we examine how collisions drive the system towards a global equilibrium state, characterized by a maximum entropy, by employing the relaxation time approximation. This allows us to define a dynamical version of superstatistics, characterized by a singular time-varying parameter q(t), which undergoes a continuous evolution towards equilibrium. We show how this approach enables the determination of the evolution of the underlying temperature distribution under the influence of collisions, which act as stochastic forces, gradually narrowing the temperature distribution over time.

Kappa distribution from particle correlations in nonequilibrium, steady-state plasmas

Kappa-distributed velocities in plasmas are common in a wide variety of settings, from low-density to high-density plasmas. To date, they have been found mainly in space plasmas, but are recently being considered also in the modeling of laboratory plasmas. Despite being routinely employed, the origin of the kappa distribution remains, to this day, unclear. For instance, deviations from the Maxwell-Boltzmann distribution are sometimes regarded as a signature of the nonadditivity of the thermodynamic entropy, although there are alternative frameworks such as superstatistics where such an assumption is not needed. In this work we recover the kappa distribution for particle velocities from the formalism of nonequilibrium steady-states, assuming only a single requirement on the dependence between the kinetic energy of a test particle and that of its immediate environment. Our results go beyond the standard derivation based on superstatistics, as we do not require any assumption about the existence of temperature or its statistical distribution, instead obtaining them from the requirement on kinetic energies. All of this suggests that this family of distributions may be more common than usually assumed, widening its domain of application in particular to the description of plasmas from fusion experiments. Furthermore, we show that a description of kappa-distributed plasma is simpler in terms of features of the superstatistical inverse temperature distribution rather than the traditional parameters κ and the thermal velocity v_{th}.

Evaluating the Adiabatic Invariants in Magnetized Plasmas Using a Classical Ehrenfest Theorem

In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov's. We introduce the Ehrenfest procedure as an alternative tool that promises to address these challenges more efficiently. Based on the conjugate variable theorem and the well-known fluctuation-dissipation theorem, this procedure offers a less expensive way of deriving time evolution Equations for macroscopic properties in systems far from equilibrium. We investigate the application of the Ehrenfest procedure for the study of adiabatic invariants in magnetized plasmas. We consider charged particles trapped in a dipole magnetic field and apply the procedure to the study of adiabatic invariants in magnetized plasmas and derive Equations for the magnetic moment, longitudinal invariant, and magnetic flux. We validate our theoretical predictions using a test particle simulation, showing good agreement between theory and numerical results for these observables. Although we observed small differences due to time scales and simulation limitations, our research supports the utility of the Ehrenfest procedure for understanding and modeling the behavior of particles in magnetized plasmas. We conclude that this procedure provides a powerful tool for the study of dynamical systems and statistical mechanics out of equilibrium, and opens perspectives for applications in other systems with probabilistic continuity.

Application of Cairns-Tsallis distribution to the dipole-type Hamiltonian mean-field model

We found that the rare distribution of velocities in quasisteady states of the dipole-type Hamiltonian mean-field model can be explained by the Cairns-Tsallis distribution, which has been used to describe nonthermal electron populations of some plasmas. This distribution gives us two interesting parameters which allow an adequate interpretation of the output data obtained through molecular dynamics simulations, namely, the characteristic parameter q of the so-called nonextensive systems and the α parameter, which can be seen as an indicator of the number of particles with nonequilibrium behavior in the distribution. Our analysis shows that fit parameters obtained for the dipole-type Hamiltonian mean-field simulated system are ad hoc with some nonthermality and nonextensivity constraints found by different authors for plasma systems described through the Cairns-Tsallis distribution.

Reconsideration of Temperature Determined by the Excited-State Population Distribution of Hydrogen Atoms Based on Tsallis Entropy and Its Statistics in Hydrogen Plasma in Non-Equilibrium State

In non-equilibrium plasmas, the temperature cannot be uniquely determined unless the energy-distribution function is approximated as a Maxwell-Boltzmann distribution. To overcome this problem, we applied Tsallis statistics to determine the temperature with respect to the excited-state populations in non-equilibrium state hydrogen plasma, which enables the description of its entropy that obeys q-exponential population distribution in the non-equilibrium state. However, it is quite difficult to apply the q-exponential distribution because it is a self-consistent function that cannot be solved analytically. In this study, a self-consistent iterative scheme was adopted to calculate q-exponential distribution using the similar algorithm of the Hartree-Fock method. Results show that the excited-state population distribution based on Tsallis statistics well captures the non-equilibrium characteristics in the high-energy region, which is far from the equilibrium-Boltzmann distribution. The temperature was calculated using the partial derivative of entropy with respect to the mean energy based on Tsallis statistics and using the coefficient of q-exponential distribution. An analytical expression was derived and compared with Boltzmann statistics, and the distribution was discussed from the viewpoint of statistical physics.

Superstatistical and DNA sequence coding of the human genome

In this work, by considering superstatistics we investigate the short-range correlations (SRCs) and the fluctuations in the distribution of lengths of strings of nucleotides. To this end, a stochastic model provides the distributions of the size of the exons based on the q-Gamma and inverse q-Gamma distributions. Specifically, we define a time series for exon sizes to investigate the SRC and the fluctuations through the superstatistics distributions. To test the model's viability, we use the Project Ensembl database of genes to extract the time evolution of exon sizes, calculated in terms of the number of base pairs (bp) in these biological databases. Our findings show that, depending on the chromosome, both distributions are suitable for describing the length distribution of human DNA for lengths greater than 10 bp. In addition, we used Bayesian statistics to perform a selection model approach, which revealed weak evidence for the inverse q-Gamma distribution for a considerable number of chromosomes.

The Longitudinal Plasma Modes of κ-Deformed Kaniadakis Distributed Plasmas Carrying Orbital Angular Momentum

Based on plasma kinetic theory, the dispersion and Landau damping of Langmuir and ion-acoustic waves carrying finite orbital angular momentum (OAM) were investigated in the κ-deformed Kaniadakis distributed plasma system. The results showed that the peculiarities of the investigated subjects relied on the deformation parameter κ and OAM parameter η. For both Langmuir and ion-acoustic waves, dispersion was enhanced with increased κ, while the Landau damping was suppressed. Conversely, both the dispersion and Landau damping were depressed by OAM. Moreover, the results coincided with the straight propagating plane waves in a Maxwellian plasma system when κ=0 and η→∞. It was expected that the present results would give more insight into the trapping and transportation of plasma particles and energy.

Generalized statistical mechanics of stellar systems

The observed distributions of stellar parameters, in particular, rotational and radial velocities, often depart from the Maxwellian (Gaussian) distribution. In the absence of a consistent statistical framework, these distributions are, in general, accounted for phenomenologically by employing power-law distributions, such as Tsallis or Kaniadakis distributions. Here we argue that the observed distributions correspond to locally Gaussian distributions, whose characteristic width is regarded as a statistical variable, in accordance with common knowledge that this parameter is mass dependent. The distributions arising within this picture correspond to superstatistics-a formalism emerging naturally in the context of self-gravitating media. We discuss in detail the distributions arising within this formalism and confront them with observational data of open clusters. We compute their moments and show that the Chandrasekhar-Münch relation remains invariant in this scenario. We also address the effect of these distributions on the thermalization of a massive body, e.g., a supermassive black hole, immersed in a stellar gas. We further discuss how the superstatistical picture clarifies certain ambiguities while offering a whole family of distributions (of which asymptotic power laws represent a special case), opening possibilities for fitting observational data.

Fingerprints of nonequilibrium stationary distributions in dispersion relations

Distributions different from those predicted by equilibrium statistical mechanics are commonplace in a number of physical situations, such as plasmas and self-gravitating systems. The best strategy for probing these distributions and unavailing their origins consists in combining theoretical knowledge with experiments, involving both direct and indirect measurements, as those associated with dispersion relations. This paper addresses, in a quite general context, the signature of nonequilibrium distributions in dispersion relations. We consider the very general scenario of distributions corresponding to a superposition of equilibrium distributions, that are well-suited for systems exhibiting only local equilibrium, and discuss the general context of systems obeying the combination of the Schrödinger and Poisson equations, while allowing the Planck's constant to smoothly go to zero, yielding the classical kinetic regime. Examples of media where this approach is applicable are plasmas, gravitational systems, and optical molasses. We analyse in more depth the case of classical dispersion relations for a pair plasma. We also discuss a possible experimental setup, based on spectroscopic methods, to directly observe these classes of distributions.

Single-particle velocity distributions of collisionless, steady-state plasmas must follow superstatistics

The correct modeling of velocity distribution functions for particles in steady-state plasmas is a central element in the study of nuclear fusion and also in the description of space plasmas. In this paper, a statistical mechanical formalism for the description of collisionless plasmas in a steady state is presented, based solely on the application of the rules of probability and not relying on the concept of entropy. Beck and Cohen's superstatistical framework [Beck and Cohen, Physica A 322, 267 (2003)PHYADX0378-437110.1016/S0378-4371(03)00019-0] is recovered as a limiting case, and a "microscopic" definition of inverse temperature β is given. Nonextensivity is not invoked a priori but enters the picture only through the analysis of correlations between parts of the system.

Fractional superstatistics from a kinetic approach

Through a kinetic approach, in which temperature fluctuations are taken into account, we obtain generalized fractional statistics interpolating between Fermi-Dirac and Bose-Einstein statistics. The latter correspond to the superstatistical analogues of the Polychronakos and Haldane-Wu statistics. The virial coefficients corresponding to these statistics are worked out and compared to those of an ideal two-dimensional anyon gas. It is shown that the obtained statistics reproduce correctly the second and third virial coefficients of an anyon gas. On this basis, a link is established between the statistical parameter and the strength of fluctuations. A further generalization is suggested by allowing the statistical parameter to fluctuate. As a by-product, superstatistics of ewkons, introduced recently to deal with dark energy [Phys. Rev. E 94, 062115 (2016)2470-004510.1103/PhysRevE.94.062115], are also obtained within the same method.

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.