The theory of superstatistics is a generalization of Boltzmann-Gibbs statistical mechanics that allows for temperature fluctuations and builds steady-state ensembles from the distribution of these fluctuations. Although it has been widely used for non-equilibrium steady states in complex systems, recent work has shown that superstatistics is not always applicable because certain conditions must be satisfied by the so-called fundamental inverse temperature function /3F. In this work, we present a complete set of sufficient conditions under which a steadystate model can be represented using superstatistics. We show that /3F alone, along with its derivatives, fully determines the existence and form of the underlying temperature distribution. Moreover, we provide explicit expressions for the moments and cumulants of the conditional distribution of /3 given the energy E, in terms of /3F, and demonstrate that superstatistical models different from the canonical require /3F to be infinitely differentiable, which excludes all polynomial cases. These results strengthen the theoretical foundations of superstatistics and offer a practical way to assess its relevance in real-world applications, such as turbulence, finance, and plasma physics.

The electronic structure of material surfaces has significant impact for the design of catalysts, semiconductors, and potentially quantum materials. However, given the complexity of the electronic structure and its accurate modeling, understanding the isolated role of various design controls (for example, chemistry, orientation and surface layer thickness) leads to significant challenges in the design and understanding of surface behaviors. To address these challenges, we apply a multi-level informatics analysis on the density of states (DOS) of CuNbS3. From the analysis, we identify that the electronic behavior associated with the Cu-S bond is the most impacted by changing the surface character, while the isolated behavior near the Fermi energy is most related with Nb-S. These various interpretations, achieved through integration of manifold mappings and regression models, provide different levels of information contributing to a holistic design framework. While this work provides an approach for developing design guidelines from small data using a limited test case, the developed analytical framework is applicable to a broader range of ceramic surface design applications.

Kappa distributions are widely used in space plasma physics to model velocity distribution functions with heavy tails. Parameter estimation in these distributions is, however, complicated by the fact that the kappa distribution does not belong to the exponential family, so it admits no sufficient statistics and direct maximum likelihood requires numerical optimization without analytically closed-form update equations. Working within the Beck-Cohen superstatistics framework, where a gamma-distributed inverse temperature β generates the kappa distribution upon marginalization, we treat β as a latent variable. This hierarchical description restores the exponential family structure that the marginal kappa distribution lacks, and yields an analytically tractable implementation of the expectation-maximization (EM) algorithm whose E-step and M-step admit closed-form expressions in terms of sufficient statistics. Applied to synthetic data drawn from the model, the algorithm converges monotonically to a stationary point of the marginal kappa log-likelihood and recovers the generating parameters consistently across the explored range of κ. EM thus offers a tractable and transparent route to inference in superstatistical systems with local temperature fluctuations.

Recently (Physica A 660 130370, 2025), emergence of superstatistical behavior in driven classical systems has been shown for systems with constant microcanonical heat capacity. As an application of this result, in this work we show that a system of N particles with inverse gamma distribution of total kinetic energies must have kappa-distributed single-particle velocities in the limit N →∞. Our results provide insight into the nature of kappa distributions outside the theory of nonextensive statistical mechanics, while also bringing forward a practical method for the generation of kappa velocities via Monte Carlo Metropolis simulation in the inverse gamma ensemble.

In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based on expectation-calculus identities associated with the Fluctuation-Dissipation Theorem and the Conjugate Variables Theorem, allowing the derivation of evolution equations directly for arbitrary observables and fluctuations without explicitly solving the full probability-density equation. The resulting relations provide a classical Ehrenfest-type formulation for observable dynamics and fluctuations under linear probability-density evolution. While the resulting equations are not closed in general, since they typically involve higher-order moments, correlations, or derivatives, the formalism offers a unified operational framework for studying observable dynamics under suitable approximations or closure assumptions. We illustrate the procedure with examples involving Fokker-Planck and Liouville dynamics and discuss the scope, limitations, and possible applications of the framework in nonequilibrium statistical mechanics. In particular, we emphasize that the method is intended as a systematic observable-based formulation for systems governed by linear evolution equations, rather than as a universal closure scheme for arbitrary nonequilibrium dynamics.

The kappa distribution of velocities is frequently found instead of the Maxwellian distribution in collisionless plasmas present in Earth's magnetosphere, the solar wind among other contexts where particles do not reach thermal equilibrium. Although the origin of these distributions is sometimes explained by means of non-extensive statistics, they can also be recovered using alternative frameworks such as superstatistics, providing a closer connection with probability theory. In this work we take this approach and derive the multi-particle and single-particle kappa distributions from superstatistics while taking into account the scale invariance property of the superstatistical temperature distribution. The formalism presented here emphasizes the usefulness of superstatistics in the computation of expectation values under kappa distributions. Some consequences of a superstatistical interpretation of kappa distributions are also discussed, such as the connection between correlations and temperature uncertainty, the meaning of the superstatistical temperature and the entropy of kappa-distributed plasmas.

Among the statistical mechanical frameworks able to describe systems in non-equilibrium steady states such as collisionless plasmas, self-gravitating systems and other complex systems, superstatistics have gained recent attention. Superstatistics postulates a superposition of canonical systems with inverse temperatures /i described by a probability distribution depending on the external conditions. Unfortunately, the uncertainty about /i cannot be attributed to fluctuations of a phase space function, and this suggests that the distribution of /i is purely of statistical nature and must be inferred rather than measured. This lack of direct observability of the superstatistical temperature then becomes a conceptual issue in need of resolution. In this work we address this issue, showing that all the information relevant to determine the superstatistical /i is contained in the recently proposed fundamental temperature /iF, a model-dependent function of the energy. In this way, a mapping can be constructed from functions of /i to new functions of /iF such that their expectation values coincide. Our results provide new tools to access the superstatistical temperature from energy values, and we illustrate this by computing the superstatistical inverse temperature distribution of the q-canonical ensemble without the use of Laplace inversion.

Abstract Background FLASH radiotherapy, characterized by ultra-high dose rates (> 40 Gy/s), potentially spares normal tissues while maintaining antitumor efficacy (the FLASH effect). While electron and proton FLASH are explored, pulsed X-ray sources like plasma focus devices offer unique possibilities. Previous work has reported hyper-radiosensitivity in colorectal cancer cells exposed to ultra-high-dose-rate pulsed X-rays from a kilojoule plasma focus (PF) device, without significant effects on non-cancerous cells. This study further investigates the biological effects of ultra-high-dose-rate (~ 10⁷ Gy/min), low-total-dose pulsed X-rays generated by a PF-2 kJ device on colorectal cancer cell lines, focusing on DNA damage, cell cycle progression, and gene expression. Results Low-total-dose (~ 0.25 Gy), ultra-high-dose-rate pulsed X-rays (0.025 Gy/pulse, a total of 10 pulses, pulses temporally separated by 15–20 s) generated by a PF-2 kJ device induced a significant increase in the SubG1 population in HCT116 and DLD1 cells over 72 h, an effect indicative of apoptosis, which was not observed with conventional X-rays at similar total doses. In addition, pulsed X-rays induced apoptosis in radioresistant MCF-7 breast cancer cells. Whereas conventional X-rays did not cause a significant increase in double-strand breaks (DSBs), surrogate marker γ-H2AX and phosphor-P53(Ser15) signal were detected 30 min following pulsed X-ray exposure and persisted for up to 24 h, and no evidence of G2/M cell cycle arrest was detected in exposed cells. Gene expression analysis and preliminary transcriptomic data further suggest a DNA damage response leading to cell death and global change in general biological processes related to regulation of gene expression. Conclusion Low-total-dose (~ 0.25 Gy), ultra-high-dose-rate pulsed X-rays generated by a PF-2 kJ device induce significant and sustained DNA damage (DSBs) leading to increased apoptosis in colorectal (HCT-116, DLD-1) and breast (MCF-7) cancer cells, compared to conventional X-rays. These effects, coupled with distinct changes in gene expression, suggest that ultra-high-dose-rate pulsed X-rays may overcomeradio-resistancee without eliciting a conventional DNA damage repair or cell cycle checkpoint response. These findings support the potential of PF-generated pulsed X-rays as a novel sourcof e radiotherapy modality and warrant further investigation, particularly in in vivo models, to assess clinical applicability and safety.

We derive an exact generalization of the well-known Lebowitz–Percus–Verlet (LPV) formula that relates the kinetic energy fluctuations of an isolated system to its specific heat. Our general formula, obtained by the application of expectation identities, is valid for finite system sizes within the broad class of steady–state ensembles whose microstate probability depends on the Hamiltonian only, P(Γ|λ)=ρ(H(Γ);λ), provided that the underlying system has constant microcanonical specific heat CkB, or equivalently, that the density of states is a power law Ω(E)∝EC. The usual microcanonical LPV formula can be readily recovered as a particular case where energy fluctuations vanish. We test the validity of the generalized formula by performing Monte Carlo simulations of a superstatistical system of harmonic oscillators, as well as by exact calculation of energy variances in a uniform–energy ensemble. Our results may prove useful in the study of non-equilibrium phase transitions in finite systems.

Superstatistics is an elegant framework for the description of steady-state thermodynamics, mostly used for systems with long-range interactions such as plasmas. In this work, we show that the potential energy distribution of a classical system under externally imposed energy fluctuations can also be described by superstatistics in the thermodynamic limit. As an example, we apply this formalism to the thermodynamics of a finite Lennard-Jones crystal with constant microcanonical heat capacity driven by sinusoidal energy oscillations. Our results show that molecular dynamics simulations of the Lennard-Jones crystal are in agreement with the provided theoretical predictions.

The formation of large social groups having uniform opinions influenced by mass media is currently an important topic in the social sciences. In this work, we explore and extend an off-lattice, two-dimensional Potts model (Eur. Phys. J. B 87, 78 [2014]) that describes the formation and dynamics of opinions in social groups according to individual consequence and agreement between neighbors. This model was originally obtained by the application of the maximum entropy principle, a general method in statistical inference, and using the same methodology we have now included the influence of mass media as a constant external field. By means of microcanonical Monte Carlo Metropolis simulations on a setup with two regions with opposing external influences, we have shown the presence of metastable states associated to the formation of clusters aligned with the locally imposed opinion. Our results suggest that, for some values of the total energy of the system, only a single cluster with a uniform opinion survives, thus the presence of two large, opposing groups is not a thermodynamically stable configuration.

An understanding of quantum theory in terms of new, underlying descriptions capable of explaining the existence of non-classical correlations, non-commutativity of measurements and other unique and counter-intuitive phenomena remains still a challenge at the foundations of our description of physical phenomena. Among some proposals, the idea that quantum states are essentially states of knowledge in a Bayesian framework is an intriguing possibility due to its explanatory power. In this work, the formalism of quantum theory is derived from the application of Bayesian probability theory to “fragile” systems, that is, systems that are perturbed by the measurement. Complex Hilbert spaces, non-commuting operators and the trace rule for expectations all arise naturally from the use of linear algebra to solve integral equations involving classical probabilities over hidden variables. The non-fragile limit of the theory, where all measurements are commutative and the theory becomes analogous to classical statistical theory is discussed as well.

The formalism of Quantum Mechanics is derived from the application of Bayesian probability theory to "fragile" systems, i.e. systems that are perturbed by the act of measurement. Complex Hilbert spaces, non-commuting operators and the trace rule for expectations all arise naturally from the use of linear algebra to solve integral equations involving classical probabilities over hidden variables. We comment on the case of non-local hidden variables, where violations of Bell's theorem can be produced, as well as the non-fragile limit of the theory, where all measurements are commutative and the theory becomes analogous to classical statistical mechanics.

Plasmas and other systems with long-range interactions are commonly found in non-equilibrium steady states that are outside traditional Boltzmann-Gibbs statistics, but can be described using generalized statistical mechanics frameworks such as superstatistics, where steady states are treated as superpositions of canonical ensembles under a temperature distribution. In this work we solve the problem of inferring the possible steady states of a composite system AB where subsystem A is described by superstatistics and E_AB = E_A + E_B. Our result establishes a closure property of superstatistics, namely that A is described by superstatistics if and only if AB and B are also superstatistical with the same temperature distribution. Some consequences of this result are discussed, such as the impossibility of local thermal equilibrium (LTE) for additive subsystems in non-canonical steady states.

In this work, using a microcanonical framework, we provide an explicit inversion formula that allows the calculation of the configurational density of states from the total density of states without resorting to the inversion of the Laplace transform. From this formula, several results can be obtained for the thermodynamics of classical systems composed of a few degrees of freedom, while at the same time recovering the well-known behavior in the thermodynamic limit.

In this brief note, the configurational density of states of a system of particles interacting via power-law pair potentials is computed exactly. The result is consistent with a constant microcanonical heat capacity. The well-known form of the virial theorem for this class of systems is recovered using only the obtained configurational density of states, and shown to be valid beyond the canonical and microcanonical ensembles, in general steady states.

In this article we derive a useful expectation identity using the language of quantum statistical mechanics, where density matrices represent the state of knowledge about the system. This identity allows to establish relations between different quantum observables depending on a continuous parameter gamma is an element of R. Such a parameter can be contained in the observables itself (e.g. perturbative parameter) or may appear as a Lagrange multiplier (inverse temperature, chemical potential, etc.) in the density matrix, excluding parameters that modify the underlying Hilbert space. In this way, using both canonical and grand canonical density matrices along with certain quantum observables (Hamiltonian, number operator, the density matrix itself, etc.) we found new identities in the field, showing not only its derivation but also their meaning. Additionally, we found that some theorems of traditional quantum statistics and quantum chemistry, such as the thermodynamical fluctuation-dissipation theorem, the Ehrenfest, and the Hellmann- Feynman theorems, among others, are particular instances of our aforementioned quantum expectation identity. At last, using a generalized density matrix arising from the Maximum- Entropy principle, we derive generalized quantum expectation identities: these generalized identities allow us to group all the previous cases in a unitary scheme.

The generalized equipartition theorem known as the conjugate variables theorem (Phys. Rev. E 86, 051136 [2012]), originally obtained in the context of statistical inference of continuous random variables, is extended in this work to the case of discrete variables. Using this new set of theorems we derive novel thermodynamic identities for the canonical ensemble connecting temperature with measurable observables.

In this work, we address two main objectives. The first one is to provide a rigorous foundation to the maximum entropy principle in statistical physics, by making use of the Fenchel-Rockafellar duality. The second objective is to discuss the well-foundedness of the so-called escort distributions in the context of non-extensive entropy maximization. The duality treatment of maximum entropy confirms the non-rigorous results obtained via the usual variational calculus, however, the use of escort distributions yields undefined behavior when used consistently, and only leads to the desired results when used in an ad-hoc manner.

Abstract Correlations are essential for the description of the properties of complex systems, particularly from the point of view of statistical mechanics. In this work we discuss the existence of correlations between parts of a non-equilibrium, composite system in a steady state, and its relation with the concept of temperature fluctuations. For this purpose, we review a recently proposed definition of steady-state temperature, namely the fundamental inverse temperature, together with a descriptor of temperature uncertainty, the inverse temperature covariance, showing that both of these quantities are invariants upon the choice of subsystems.