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The paper is devoted to presenting a novel closed-form representation of the resonant thermonuclear functions and the nonrelativistic Voigt function, which are essential tools in nuclear physics. Understanding thermonuclear fusion reaction rates within solar analogs is crucial for understanding stellar evolution and energy production mechanisms. Initially, this paper focuses on evaluating fusion reaction rates, particularly emphasizing resonant reactions, which play pivotal roles in stellar evolution phases. A key challenge lies in solving reaction rate integrals in closed form. The Buschman–Fox H -function of two variables is employed to address this issue. Conventionally, it is assumed that the plasma particles' velocity follows the Maxwell–Boltzmann distribution. However, it is acknowledged that particles may deviate from this assumed equilibrium state in actual scenarios, leading to nonequilibrium situations. The study also aims to address these nonequilibrium situations by utilizing appropriate velocity models from the existing literature. Utilizing the Mellin transform technique, we achieve the closed-form representation of the resonant reaction rate integral. Furthermore, we address the nonrelativistic Voigt profile and, in particular, Voigt function. The Voigt profile, resulting from the convolution of Gaussian and Lorentzian distributions, effectively captures the intricate shapes of spectral lines encountered in spectroscopy. Apart from its significance in spectroscopy, the Voigt function finds application in various areas such as plasma nuclear studies, acoustics, and radiation transfer. Many approximations of the Voigt function can be found in the literature, yet currently, there is no existing closed-form expression. This paper also sets out to fill this gap by deriving the exact closed-form expressions for the Voigt function and its conjugate in terms of Buschman–Fox H -function, employing the Mellin convolution theorem. This paper marks the first instance in the literature where the applications of Buschman Fox's H -function has been documented. [ABSTRACT FROM AUTHOR]

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer's G-function and Fox's H-functions and their utilization in mathematical physics are the prime focus of this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a new probability model called as Maxwell-Boltzmann-Exponential (MBE) distribution. The MBE distribution arises as a mixture distribution of the Maxwell-Boltzmann and exponential distributions. The statistical properties of the distributions are studied and obtained in closed-form expressions. Three methodologies are assessed and compared for the estimation of parameters in the MBE distribution. The MBE regression model is defined, with the proposed regression model being an alternative to the gamma regression model for response variables that are extremely right-skewed and bimodal. Two real data sets are used to demonstrate the applicability of the proposed models against the existing models. [ABSTRACT FROM AUTHOR]

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper.

The energy distribution of particles in a gaseous system is well understood through the implementation of a statistical tool, namely, the Maxwell–Boltzmann distribution function in the velocity–space coordinate system. The Maxwell–Boltzmann distribution function is utilized to investigate the velocity distribution of plasma particles like electrons, assuming that their collision frequency does not depend on the velocity. However, there is a swift transition in converting the Maxwell–Boltzmann distribution function to the Druyvesteyn distribution function for the case where a collision frequency is directly proportional to the velocity. Our aim is to incorporate the frequency components to investigate the Maxwell–Boltzmann and Druyvesteyn distribution functions. Employing the equation of motion, we observe that the collisional electron velocity is equal to the equilibrium electron velocity ∼eE/meω multiplied by the collisional frequency over the external source frequency β = ν/ω corresponding to the externally applied electric field. We investigate the difference in the Druyvesteyn distribution function between sheath and pre-sheath regions, when a stream of electrons is traversing or effusing through the part of a pre-sheath region corresponding to the dimension of the order of mean free path. Velocity and corresponding energy distribution functions are compared for non-effusion and effusion cases in the collisional and non-collisional regimes. The Maxwell–Boltzmann and Druyvesteyn velocity and energy distributions are competitive when the collisional frequency is twice the frequency of the applied electric field. [ABSTRACT FROM AUTHOR]

The paper discusses the implementation of enhanced Gaussian noise (EGN) model-based probabilistic shaping (PS) in long-haul optical transmission systems to mitigate the effects of non-linearities that can occur when using different input distributions for constellation shaping. This approach aims to increase the achievable information rate (AIR) while maintaining a reasonable signal-to-noise ratio (SNR) at optimal launch power. The authors propose that this approach can improve the efficiency and reliability of long-haul optical transmission systems by reducing the impact of fiber non-linearities. The proposed method aims to maximize mutual information in a coherent optical system without relying on digital signal processors. This is achieved by optimizing PS using sequential quadratic programming in a back-to-back configuration with 16-Quadrature Amplitude Modulation. The objective function is set to minimize the second and fourth order moments of input, resulting in reduced higher order noise coefficients and minimal non-linear interference noise at higher values of SNR. The EGN model is used to optimize the PS, resulting in an SNR value of 14.52 dBm and a minimal SNR penalty of 0.15 dBm towards the uniform probability distribution. The optimized EGN model returns μ ^ 4 and μ ^ 6 values of 1.41 and 2.24 respectively, which further contribute to reducing the impact of higher order noise coefficients on the system. [ABSTRACT FROM AUTHOR]

Consideration has been given to the well-known argument that in a conservative mechanical system existing in an equilibrium state, entropy reaches a maximum. First, using the example of a system of material points modeling a rarefied gas, it has been shown that a maximum of entropy defined through the number of microstates is achieved on a solution that cannot be approximated by an exponential distribution corresponding to the Maxwell–Boltzmann distribution. There are many such solutions and each of them satisfies both the condition of invariability of the number of points and the condition of constancy of the system′s total energy. Next, consideration has been given to an alternative definition of entropy through the density of distribution of the probability of a mechanical system being in an assigned volume of phase space. It has been shown that a conservative mechanical system of general type assigned by its Hamiltonian, in selecting various sets in the phase space for seeking an extremum, has an entropy maximum on various distributions in the case of one and the same energy. A conclusion has been drawn on the occurrence of a Maxwell–Boltzmann distribution in conservative systems of classical mechanics for reasons unrelated to entropy maximization. [ABSTRACT FROM AUTHOR]

This paper presents a new distribution, the product of the mixture between Maxwell-Boltzmann and a particular case of the generalized gamma distributions. The resulting distribution, called the Scale Mixture Maxwell-Boltzmann, presents greater kurtosis than the recently introduced slash Maxwell-Boltzmann distribution. We obtained closed-form expressions for its probability density and cumulative distribution functions. We studied some of its properties and moments, as well as its skewness and kurtosis coefficients. Parameters were estimated by the moments and maximum likelihood methods, via the Expectation-Maximization algorithm for the latter case. A simulation study was performed to illustrate the parameter recovery. The results of an application to a real data set indicate that the new model performs very well in the presence of outliers compared with other alternatives in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a new distribution, the product of the mixture between Maxwell-Boltzmann and a particular case of the generalized gamma distributions. The resulting distribution, called the Scale Mixture Maxwell-Boltzmann, presents greater kurtosis than the recently introduced slash Maxwell-Boltzmann distribution. We obtained closed-form expressions for its probability density and cumulative distribution functions. We studied some of its properties and moments, as well as its skewness and kurtosis coefficients. Parameters were estimated by the moments and maximum likelihood methods, via the Expectation-Maximization algorithm for the latter case. A simulation study was performed to illustrate the parameter recovery. The results of an application to a real data set indicate that the new model performs very well in the presence of outliers compared with other alternatives in the literature.

The kink in the source-drain current of GaN devices has been not clarified. Based on the Maxwell-Boltzmann distribution of hot channel electrons, the generation of excess carriers in the L valleys of GaN and the origin of the kink had been modeled. This proposed model can predict how the kink depends on the gate voltage in experiments. It provides the reason why the kink partially disappears or disappears by sweeping the source-drain voltage from high voltage to low voltage, disappears under illumination, is more obvious at a lower temperature, and can be affected by different traps. All above experimental phenomena are associated with excess carriers in the L valleys of gain due to the hot channel carrier via internal transport. This model gives physical insight into how physical parameters of GaN devices affect the kink in the source-drain current. Thus, it can be used to optimize the physical parameters of GaN devices for reducing the kink.

In this article, a sheet model is utilized to model and simulate hot electron transport properties in vacuum gap. Based on this model, emitted electrons with constant initial velocities and Maxwell–Boltzmann distribution are considered. Numerical analyses of the potential, electric field, current density, velocity, and phase-space distribution are also performed. The accuracy of this model is verified by comparing with analytical results. Furthermore, we studied the velocity distribution of the emitted electrons at any position in the interelectrode gap and observed a significant change in the distribution. [ABSTRACT FROM AUTHOR]

The Maxwell–Boltzmann distribution is a hallmark of statistical physics in thermodynamic equilibrium linking the probability density of a particle’s kinetic energies to the temperature of the system that also determines its configurational fluctuations. This unique relation is lost for Hot Brownian Motion, e.g., when the Brownian particle is constantly heated to create an inhomogeneous temperature in the surrounding liquid. While the fluctuations of the particle in this case can be described with an effective temperature, it is not unique for all degrees of freedom and suggested to be different at different timescales. In this work, we report on our progress to measure the effective temperature of Hot Brownian Motion in the ballistic regime. We have constructed an optical setup to measure the displacement of a heated Brownian particle with a temporal resolution of 10 ns giving a corresponding spatial resolution of about 23 pm for a 0.92 μm PMMA particle in water. Using a gold-coated polystyrene (AuPS) particle of 2.15 μm diameter we determine the mean squared displacement of the particle over more than six orders of magnitude in time. Our data recovers the trends for the effective temperature at long timescales, yet shows also clear effects in the region of hydrodynamic long time tails.

An ideal gas is a gas in the form of a particle or molecule. Its velocity and kinetic energy are interesting topics in several studies in physical chemistry. This research aims to evaluate the average run length based on the exponentially weighted moving average statistic for its molecular velocity and kinetic energy of Maxwell-Boltzmann distribution. Derivation of the integral equation, which is equal to the average run length and numerical method of the integral equation, was applied to evaluate the average run length of a gas molecule's molecular velocity and kinetic energy. The Trapezoidal rule as numerical method and its error was analyzed for approximation of average run length. The findings showed that the average run length of molecular velocity decreased in the higher temperature with the given mass of the molecule. Moreover, there was a decrease in the average run length of molecular kinetic energy in the higher temperature. [ABSTRACT FROM AUTHOR]

To be accepted by the community, the claim that nuclear motion has to be treated classically must be tested for all kinds of phenomena. For the moment we claim that in a quantum chemical calculation, a classical description of nuclear motion is superior to the use of the Schrödinger equation, and investigate how far we get with this statement. In the present paper we address the question what nuclear quantum statistics means in this context. We will show that the Maxwell–Boltzmann velocity distribution evolves quickly in any molecular dynamics simulation and this guarantees the physically correct behavior of molecular systems. Using first‐principles molecular dynamics simulations, or more precisely Car–Parrinello molecular dynamics, we investigate what this means for Bose‐Einstein condensates and for Cooper pairs. It turns out that our approach can explain all relevant phenomena. As a consequence, we can introduce a deterministic formulation of quantum mechanics and can get rid of all the paradoxa in traditional quantum mechanics. The basic idea is to treat electrons and nuclei differently. [ABSTRACT FROM AUTHOR]

Energy transfer between gases and solid surfaces plays a vital role in many technologies. Thermal accommodation coefficient (TAC), which specifies the energy transferred between a surface composed of thermally-vibrating atoms and a colliding gas molecule, can be calculated by sampling the incident velocities of gas molecules by the Maxwell-Boltzmann (MB) distribution. However, the MB distribution describes gas behavior in free space rather than near a solid surface. In this work, we calculated the TAC between helium and graphite by a modified Maxwell-Boltzmann distribution accounting for the fast atom effect where gases of higher out-of-plane velocity have a larger chance of reaching the solid surface at the same period. A significant reduction in TAC is found by correcting the MB distribution with the modified-MB distribution, 32.7% for helium at 500 K and 10.74% at 300 K. Further analysis shows that the out-of-plane velocity component dominates the scattering process and energy transfer. This result underlines the importance and necessity to correct the MB distribution to incorporate the fast atom effect. Moreover, the lower and upper boundaries of the scattering events were depicted. These findings can provide a fundamental understanding of gas-surface energy transfer and guidelines for molecular dynamics simulations for TAC. Image 1 [ABSTRACT FROM AUTHOR]

This interdisciplinary work proposes to use astronomical objects as a ludic tool to arouse student's curiosity, provide opportunities and enhance the teaching-learning process in Physics II and Statistical Mechanics classes. Its purpose is to discuss whether the Maxwell-Boltzmann velocity distribution law for ideal gases also describes the behavior of star velocities in star clusters. For this, the student's investigative thoughts it will be triggered by star velocity data, obtained through observations from the Blanco telescope and KPNO (Kitt Peak National Observatory), provided in the literature for the Omega Centauri globular cluster and the Pleiades open star cluster. It can be concluded that in dynamically relaxed globular clusters, although physically very different from ideal gases, stars follow Maxwell-Boltzmann's law of velocity distribution. This is because, in fact, any classical many-body system that satisfies the two assumptions used in the computations of velocity distribution will satisfy the Maxwell-Boltzmann distribution, regardless of whether the interaction between the bodies is of short or long-range and whether they are microscopic or macroscopic objects. Thus, by demonstrating that star clusters follow ideal gas-like laws, it became possible to use the ludic of astronomy to encourage students in the process of learning the content of the kinetic theory of gases. [ABSTRACT FROM AUTHOR]

• The calculated Deformed Doppler broadening function was considered. • The 4-pole Padé approximation was employed. • The considered Tsallis distribution was used. • An effective medium temperature model is proposed. This paper presents a method for establishing an equivalence relationship between the Doppler Broadening Function based on the Maxwell-Boltzmann Distribution and the Deformed Doppler Broadening Function, according to the generalised Tsallis Distribution, through an effective temperature T eff of the medium. The proposed method allows a precise calculation of the Maxwellian Doppler Broadening Function, which reproduces the value of the Tsallis Doppler Broadening Function (TDBF), where ξ is associated to the temperature of medium T and ξ ∼ q relates to the effective temperature T eff of the medium. In this paper, we calculate ξ ∼ q for typical values of the deformation factor q together with a parameterization example for ξ ∼ q as a result of the ordered pair x , ξ , using a genetic algorithm technique. The results obtained were accurate, with a percentage deviation under 2%. [ABSTRACT FROM AUTHOR]

In nuclear fusion reactions taking place in stellar interiors, if there is a deviation from hydrostatic equilibrium, then one can think of an alternative velocity distribution instead of the Maxwell–Boltzmann distribution. The paper is devoted to study the extension of non-resonant reaction rate integrals in standard, cut-off, depleted, and screened cases via Mittag-Leffler distribution. The extended non-resonant reaction rate integrals are represented in the form of Fox's H -function, generalized Wright function. A comparison of the Mittag-Leffler non-resonant energy distributions with the standard Maxwell–Boltzmann energy distribution and the pathway energy distribution is also done. [ABSTRACT FROM AUTHOR]

The Chi distribution is a continuous probability distribution of a random variable obtained from the positive square root of the sum of k squared variables, each coming from a standard Normal distribution (mean = 0 and variance = 1). The variable k indicates the degrees of freedom. The usual expression for the Chi distribution can be generalised to include a parameter which is the variance (which can take any value) of the generating Gaussians. For instance, for k = 3, we have the case of the Maxwell-Boltzmann (MB) distribution of the particle velocities in the Ideal Gas model of Physics. In this work, we analyse the case of unequal variances in the generating Gaussians whose distribution we will still represent approximately in terms of a Chi distribution. We perform a Monte Carlo simulation to generate a random variable which is obtained from the positive square root of the sum of k squared variables, but this time coming from non-standard Normal distributions, where the variances can take any positive value. Then, we determine the boundaries of what to expect when we start from a set of unequal variances in the generating Gaussians. In the second part of the article, we present a discrete model to calculate the parameter of the Chi distribution in an approximate way for this case (unequal variances). We also comment on the application of this simple discrete model to calculate the parameter of the MB distribution (Chi of k = 3) when it is used to represent the reaction times to visual stimuli of a collective of individuals in the framework of a Physics inspired model we have published in a previous work.

As a continuation of our previous work, where a Maxwell–Boltzmann distribution was found to model a collective’s reaction times, in this work we will carry out a percentile study of the χ distribution for some freedom ranging from k = 2 to k = 10. The most commonly used percentiles in the biomedical and behavioral sciences have been included in the analysis. We seek to provide a look-up table with percentile ratios, taken symmetrically about the median, such that this distribution can be identified in practice in an easy way. We have proven that these ratios do not depend upon the variance chosen for the k generating Gaussians. In general, the χ probability density, generalized to take any value of the variance, represents an ideal gas in a k-dimensional space. We also derive an approximate expression for the median of the generalized χ distribution. In the second part of the results, we will focus on the practical case of k = 3, which represents the ideal gas in physics, and models quite well the reaction times of a human collective. Accurately, we will perform a more detailed scrutiny of the percentiles for the reaction time distribution of a sample of 50 school-aged children (7200 reaction times).

The Maxwell-Boltzmann distribution in its classic form concerns the distribution of perfect gas particles into energy levels. In the paper, it was used to describe the energy distribution of ground grains of the brittle substance as a function of grain size, which allowed describing the particle size distribution of the process product. The particle size distribution of the product was identified with the effect of the grinding process, which is an extremely important parameter for the description of the entire process. The results of calculations were presented and compared with the results of experimental analyses. The experiment was carried out at different values of thermodynamic parameters of working air, which corresponded to different values of grinding energy. Comparison of the results showed that the thesis of applying the laws of statistical physics to the description of complex industrial processes gives positive results. Numerical determination of the particle size distribution of the grinding product is an extremely desirable result of research, because it allows to avoid the tedious and time-consuming experimental determinations necessary to assess the grinding product. [ABSTRACT FROM AUTHOR]

Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell–Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process which also applies to other real-world systems with similar nature. A concrete example is provided to illustrate the application of the methodology where the fundamental diagram of vehicular traffic is modeled as a stochastic process to capture the scattering effect in flow–density relationship. A verification study was conducted on the model using empirical data and the statistical analysis shows that the overall quality of the fitted stochastic process is acceptable. Related existing efforts are referenced to the proposed stochastic fundamental diagram where their similarities and differences are elaborated. Further discussion is carried out on the significance of the stochastic fundamental diagram as well as the proposed methodology with an additional real-world example to illustrate its applications. [ABSTRACT FROM AUTHOR]

We have carried out multimillion atom non-equilibrium molecular dynamics simulations for investigating the effect of crystallographic orientation over the evolution of deformation pathway of single crystal copper under shock compression. Based on symmetry, three different crystallographic directions, 〈 1 0 0 〉 , 〈 1 1 0 〉 and 〈 1 1 1 〉 are selected and taken as shock directions. Shock Hugoniot points has been calculated and compared among these different directions up to ∼ 450 GPa of shock pressure i.e. piston velocity of 3.0 km/s. Orientational anisotropy has been observed for the bulk Cu single crystals shock loaded along these three different directions. Even though this feature may not show up explicitly in experimental investigations which typically measures shock-velocity and density Hugoniot curve, it is apparent from large scale atomistic simulations which measures the temperature Hugoniot curve quite accurately. Differences are observed in the von-Mises strain and stress plot distributions for shock loading of different intensities along the three directions. Large directional dependency is also evident in the evolution mechanism of deformation. Temperature profiles at different piston velocities for the shock front and the shock equilibrated regions shows significantly different and interesting patterns along the three orientations. Maxwell-Boltzmann distribution is observed in the atomic velocities (thereby the temperature profiles also) for both the shock front region as well as the shock equilibrated region for shock loading along all the three directions. [ABSTRACT FROM AUTHOR]

In this paper we use the super-statistics theory to obtain three q-exponential type Boltzmann factors. We apply three types of effective Boltzmann factors to the kinetic theory of an ideal gas. We also compute the most probable speeds, average speeds and averages of squared velocity and discuss the q-deformed equipartition theorem for three types Boltzmann factors. [ABSTRACT FROM AUTHOR]

The present paper is implementing a numerical application of the Boltzmann–Arrhenius–Zhurkov (BAZ) model and relates to the statistic reliability model derived from the Transition State Theory paradigm. It shows how the quantified tool can be applied to determine the associated effective activation energy. The unified multiple stress reliability model for electronic devices is applied to Normally-Off Power GaN transistor technologies to quantify and predict the reliability figures of this electronic type of product when operating under multiple stresses in an embedded system operating under such harsh environment conditions as set for Aerospace, Space, Nuclear, Submarine, Transport or Ground application. [ABSTRACT FROM AUTHOR]