Search Results (25)

The Maxwell–Boltzmann (MB) distribution is important because it provides the statistical foundation for connecting microscopic particle motion to macroscopic gas properties by statistically describing molecular speeds and energies, making it essential for understanding and predicting the behavior of classical ideal gases. This study advances the statistical modeling of lifetime distributions by developing a comprehensive reliability analysis of the MB distribution under an improved adaptive progressive censoring framework. The proposed scheme strategically enhances experimental flexibility by dynamically adjusting censoring protocols, thereby preserving more information from test samples compared to conventional designs. Maximum likelihood estimation, interval estimation, and Bayesian inference are rigorously derived for the MB parameters, with asymptotic properties established to ensure methodological soundness. To address computational challenges, Markov chain Monte Carlo algorithms are employed for efficient Bayesian implementation. A detailed exploration of reliability measures—including hazard rate, mean residual life, and stress–strength models—demonstrates the MB distribution’s suitability for complex reliability settings. Extensive Monte Carlo simulations validate the efficiency and precision of the proposed inferential procedures, highlighting significant gains over traditional censoring approaches. Finally, the utility of the methodology is showcased through real-world applications to physics and engineering datasets, where the MB distribution coupled with such censoring yields superior predictive performance. This genuine examination is conducted through two datasets (including the failure times of aircraft windshields, capturing degradation under extreme environmental and operational stress, and mechanical component failure times) that represent recurrent challenges in industrial systems. This work contributes a unified statistical framework that broadens the applicability of the Maxwell–Boltzmann model in reliability contexts and provides practitioners with a powerful tool for decision making under censored data environments. Full article

The Maxwell–Boltzmann (MB) distribution is fundamental in statistical physics, providing an exact description of particle speed or energy distributions. In this study, a discrete formulation derived via the survival function discretization technique extends the MB model’s theoretical strengths to realistically handle lifetime and reliability data recorded in integer form, enabling accurate modeling under inherently discrete or censored observation schemes. The proposed discrete MB (DMB) model preserves the continuous MB’s flexibility in capturing diverse hazard rate shapes, while directly addressing the discrete and often censored nature of real-world lifetime and reliability data. Its formulation accommodates right-skewed, left-skewed, and symmetric probability mass functions with an inherently increasing hazard rate, enabling robust modeling of negatively skewed and monotonic-failure processes where competing discrete models underperform. We establish a comprehensive suite of distributional properties, including closed-form expressions for the probability mass, cumulative distribution, hazard functions, quantiles, raw moments, dispersion indices, and order statistics. For parameter estimation under Type-II censoring, we develop maximum likelihood, Bayesian, and bootstrap-based approaches and propose six distinct interval estimation methods encompassing frequentist, resampling, and Bayesian paradigms. Extensive Monte Carlo simulations systematically compare estimator performance across varying sample sizes, censoring levels, and prior structures, revealing the superiority of Bayesian–MCMC estimators with highest posterior density intervals in small- to moderate-sample regimes. Two genuine datasets—spanning engineering reliability and clinical survival contexts—demonstrate the DMB model’s superior goodness-of-fit and predictive accuracy over eleven competing discrete lifetime models. Full article

The Unit Inverse Maxwell–Boltzmann (UIMB) distribution is introduced as a novel single-parameter model for data constrained within the unit interval ($0,1$), derived through an exponential transformation of the Inverse Maxwell–Boltzmann distribution. Designed to address the limitations of traditional unit-interval distributions, the UIMB model exhibits flexible density shapes and hazard rate behaviors, including right-skewed, left-skewed, unimodal, and bathtub-shaped patterns, making it suitable for applications in reliability engineering, environmental science, and health studies. This study derives the statistical properties of the UIMB distribution, including moments, quantiles, survival, and hazard functions, as well as stochastic ordering, entropy measures, and the moment-generating function, and evaluates its performance through simulation studies and real-data applications. Various estimation methods, including maximum likelihood, Anderson–Darling, maximum product spacing, least-squares, and Cramér–von Mises, are assessed, with maximum likelihood demonstrating superior accuracy. Simulation studies confirm the model’s robustness under normal and outlier-contaminated scenarios, with MLE showing resilience across varying skewness levels. Applications to manufacturing and environmental datasets reveal the UIMB distribution’s exceptional fit compared to competing models, as evidenced by lower information criteria and goodness-of-fit statistics. The UIMB distribution’s computational efficiency and adaptability position it as a robust tool for modeling complex unit-interval data, with potential for further extensions in diverse domains. Full article

Kappa distributions, their statistical framework, and their thermodynamic origin describe systems with correlations among their particle energies, residing in stationary states out of classical thermal equilibrium/space plasmas, from solar wind to the outer heliosphere, are such systems. We show how correlations from long-range interactions compete with collisions to define the specific shape of particle velocity distributions, using a simple numerical experiment with collisions and a variable amount of correlation among the particles. When the correlations are turned off, collisions drive any initial distribution to evolve toward equilibrium and a Maxwell–Boltzmann (MB) distribution. However, when some correlation is introduced, the distribution evolves toward a different stationary state defined by a kappa distribution with some finite value of the thermodynamic kappa $\kappa$ (where $\kappa \to \infty$ corresponds to a MB distribution). Furthermore, the stronger the correlations, the lower the $\kappa$ value. This simple numerical experiment illuminates the role of correlations in forming stationary state particle distributions, which are described by kappa distributions, as well as the physical interpretation of correlations from long-range interactions and how they are related to the thermodynamic kappa. Full article

In the vast statistical literature, there are numerous probability distribution models that can model data from real-world phenomena. New probability models, nevertheless, are still required in order to represent data with various spread behaviors. It is a known fact that there is a great need for new models with limited support. In this study, a flexible probability model called the unit Maxwell-Boltzmann distribution, which can model data values in the unit interval, is derived by selecting the Maxwell-Boltzmann distribution as a base-line model. The important characteristics of the derived distribution in terms of statistics and mathematics are investigated in detail in this study. Furthermore, the inference problem for the mentioned distribution is addressed from the perspectives of maximum likelihood, method of moments, least squares, and maximum product space, and different estimators are obtained for the unknown parameter of the distribution. The derived distribution outperforms competitive models according to different fit tests and information criteria in the applications performed on four actual air pollutant concentration data sets, indicating that it is an effective model for modeling air pollutant concentration data. Full article

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. Full article

Today, the reliability or quality practitioner always aims to shorten testing duration and reduce testing costs without neglecting efficient statistical inference. So, a generalized progressively Type-II hybrid censored mechanism has been developed in which the experimenter prepays for usage of the testing facility for T units of time. This paper investigates the issue of estimating the model parameter, reliability, and hazard rate functions of the Maxwell–Boltzmann distribution in the presence of generalized progressive Type-II hybrid censored data by making use of the likelihood and Bayesian inferential methods. Using an inverse gamma prior distribution, the Bayes estimators of the same unknown parameters with respect to the most commonly squared-error loss are derived. Since the joint likelihood function is produced in complex form, following the Monte-Carlo Markov-chain idea, the Bayes’ point estimators as well as the Bayes credible and highest posterior density intervals cannot be derived analytically, but they may be examined numerically. Via the normal approximation of the acquired maximum likelihood and log-maximum-likelihood estimators, the approximate confidence interval bounds of the unknown quantities are derived. Via comprehensive numerical comparisons, with regard to simulated root mean squared-error, mean relative absolute bias, average confidence length, and coverage probability, the actual behavior of the proposed estimation methodologies is examined. To illustrate how the offered methodologies may be used in real circumstances, two different applications, representing the failure time points of aircraft windscreens as well as the daily average wind speed in Cairo during 2009, are explored. Numerical evaluations recommend utilizing a Bayes model via the Metropolis-Hastings technique to produce samples from the posterior distribution to estimate any parameter of the Maxwell–Boltzmann distribution when collecting data from a generalized progressively Type-II hybrid censored mechanism. Full article

Two novel methods for heart disease prediction, which use the kurtosis of the features and the Maxwell–Boltzmann distribution, are presented. A Majority Voting approach is applied, and two base classifiers are derived through statistical weight calculation. First, exploitation of attribute kurtosis and attribute Kolmogorov–Smirnov test (KS test) result is done by plugging the base categorizer into a Bagging Classifier. Second, fitting Maxwell random variables to the components and summating KS statistics are used for weight assignment. We have compared state-of-the-art methods to the proposed classifiers and reported the results. According to the findings, our Gaussian distribution and kurtosis-based Majority Voting Bagging Classifier (GKMVB) and Maxwell Distribution-based Majority Voting Bagging Classifier (MKMVB) outperform SVM, ANN, and Naive Bayes algorithms. In this context, which also indicates, especially when we consider that the KS test and kurtosis hack is intuitive, that the proposed routine is promising. Following the state-of-the-art, the experiments were conducted on two well-known datasets of Heart Disease Prediction, namely Statlog, and Spectf. A comparison of Optimized Precision is made to prove the effectiveness of the methods: the newly proposed methods attained 85.6 and 81.0 for Statlog and Spectf, respectively (while the state of the heart attained 83.5 and 71.6, respectively). We claim that the Majority Voting family of classifiers is still open to new developments through appropriate weight assignment. This claim is obvious, especially when its simple structure is fused with the Ensemble Methods’ generalization ability and success. Full article

In nuclear reactors, tracking the loss and production of neutrons is crucial for the safe operation of such devices. In this regard, the microscopic cross section with the Doppler broadening function is a way to represent the thermal agitation movement in a reactor core. This function usually considers the Maxwell–Boltzmann statistics for the velocity distribution. However, this distribution cannot be applied on every occasion, i.e., in conditions outside the thermal equilibrium. In order to overcome this potential limitation, Kaniadakis entropy has been used over the last seven years to generate generalised nuclear data. This short review article summarises what has been conducted so far and what has to be conducted yet. Full article

This paper focuses mainly on the problem of computing the ${\gamma }^{\mathrm{th}}$, $\gamma >0$, moment of a random variable $Y_n:={\sum }_{i=1}^n{\alpha }_i{X}_i$ in which the ${\alpha }_i$’s are positive real numbers and the $X_i$’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of $Y_n$, especially when not all ${\alpha }_i$’s are equal. We analytically solve this problem by showing that the ${\gamma }^{\mathrm{th}}$ moment of $Y_n$ can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the ${\gamma }^{\mathrm{th}}$ moment of $Y_n$ when $X_i$ is a combination of statistically independent $Z_i^2$ and $G_i$ in which the $Z_i$’s are distributed according to normal or Maxwell–Boltzmann distributions and the $G_i$’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding ${\gamma }^{\mathrm{th}}$ conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests. Full article

The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases. Full article

Kaniadakis statistics is a widespread paradigm to describe complex systems in the relativistic realm. Recently, gravitational and cosmological scenarios based on Kaniadakis ($\kappa$-deformed) entropy have been considered, leading to generalized models that predict a richer phenomenology comparing to their standard Maxwell–Boltzmann counterparts. The purpose of the present effort is to explore recent advances and future challenges of Gravity and Cosmology in Kaniadakis statistics. More specifically, the first part of the work contains a review of $\kappa$-entropy implications on Holographic Dark Energy, Entropic Gravity, Black hole thermodynamics and Loop Quantum Gravity, among others. In the second part, we focus on the study of Big Bang Nucleosynthesis in Kaniadakis Cosmology. By demanding consistency between theoretical predictions of our model and observational measurements of freeze-out temperature fluctuations and primordial abundances of ${}^{4}He$ and D, we constrain the free $\kappa$-parameter, discussing to what extent the Kaniadakis framework can provide a successful description of the observed Universe. Full article

In the last seven years, Kaniadakis statistics, or $\kappa$-statistics, have been applied in reactor physics to obtain generalized nuclear data, which can encompass, for instance, situations that lie outside thermal equilibrium. In this sense, numerical and analytical solutions were developed for the Doppler broadening function using the $\kappa$-statistics. However, the accuracy and robustness of the developed solutions contemplating the κ distribution can only be appropriately verified if applied inside an official nuclear data processing code to calculate neutron cross-sections. Hence, the present work inserts an analytical solution for the deformed Doppler broadening cross-section inside the nuclear data processing code FRENDY, developed by the Japan Atomic Energy Agency. To do that, we applied a new computational method called the Faddeeva package, developed by MIT, to calculate error functions present in the analytical function. With this deformed solution inserted in the code, we were able to calculate, for the first time, deformed radiative capture cross-section data for four different nuclides. The usage of the Faddeeva package brought more accurate results when compared to other standard packages, reducing the percentage errors in the tail zone in relation to the numerical solution. The deformed cross-section data agreed with the expected behavior compared to the Maxwell–Boltzmann. Full article

Relying on the quantum tunnelling concept and Maxwell–Boltzmann–Gibbs statistics, Gamow shows that the star-burning process happens at temperatures comparable to a critical value, called the Gamow temperature ($\mathtt{T}$) and less than the prediction of the classical framework. In order to highlight the role of the equipartition theorem in the Gamow argument, a thermal length scale is defined, and then the effects of non-extensivity on the Gamow temperature have been investigated by focusing on the Tsallis and Kaniadakis statistics. The results attest that while the Gamow temperature decreases in the framework of Kaniadakis statistics, it can be bigger or smaller than $\mathtt{T}$ when Tsallis statistics are employed. Full article

We present in this review our recent theoretical studies on atomic processes subject to the plasma environment including the $\alpha$ and $\beta$ emissions and the ground state photoabsorption of the one- and two-electron atoms and ions. By carefully examining the spatial and temporal criteria of the Debye–Hückel (DH) approximation based on the classical Maxwell–Boltzmann statistics, we were able to represent the plasma effect with a Debye–Hückel screening potential $V_{DH}$ in terms of the Debye length D, which is linked to the ratio between the plasma density N and its temperature $kT$. Our theoretical data generated with $V_{DH}$ from the detailed non-relativistic and relativistic multiconfiguration atomic structure calculations compare well with the limited measured results from the most recent experiments. Starting from the quasi-hydrogenic picture, we were able to show qualitatively that the energy shifts of the emission lines could be expressed in terms of a general expression as a function of a modified parameter, i.e., the reduced Debye length $\lambda$. The close agreement between theory and experiment from our study may help to facilitate the plasma diagnostics to determine the electron density and the temperature of the outside plasma. Full article