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Journal articles on the topic 'Boltzmann Distributions'

Author: Grafiati

Published: 25 May 2024

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1

Khamnei, Hossein Jabbari, Sajad Nikannia, Masood Fathi, and Shahryar Ghorbani. "Modeling income distribution: An econophysics approach." Mathematical Biosciences and Engineering 20, no. 7 (2023): 13171–81. http://dx.doi.org/10.3934/mbe.2023587.

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<abstract><p>This study aims to develop appropriate models for income distribution in Iran using the econophysics approach for the 2006–2018 period. For this purpose, the three improved distributions of the Pareto, Lognormal, and Gibbs-Boltzmann distributions are analyzed with the data extracted from the target household income expansion plan of the statistical centers in Iran. The research results indicate that the income distribution in Iran does not follow the Pareto and Lognormal distributions in most of the study years but follows the generalized Gibbs-Boltzmann distribution function in all study years. According to the results, the generalized Gibbs-Boltzmann distribution also properly fits the actual data distribution and could clearly explain the income distribution in Iran. The generalized Gibbs-Boltzmann distribution also fits the actual income data better than both Pareto and Lognormal distributions.</p></abstract>

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2

Hornack, Fred M. "Visualizing Boltzmann-like distributions." Journal of Chemical Education 65, no. 1 (January 1988): 24. http://dx.doi.org/10.1021/ed065p24.

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3

Orland, Henri. "Accelerated Sampling of Boltzmann Distributions." Journal of the Physical Society of Japan 78, no. 10 (October 15, 2009): 103002. http://dx.doi.org/10.1143/jpsj.78.103002.

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4

Liu, Fu-Hu, Ya-Qin Gao, and Hua-Rong Wei. "On Descriptions of Particle Transverse Momentum Spectra in High Energy Collisions." Advances in High Energy Physics 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/293873.

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The transverse momentum spectra obtained in the frame of an isotropic emission source are compared in terms of Tsallis, Boltzmann, Fermi-Dirac, and Bose-Einstein distributions and the Tsallis forms of the latter three standard distributions. It is obtained that, at a given set of parameters, the standard distributions show a narrower shape than their Tsallis forms which result in wide and/or multicomponent spectra with the Tsallis distribution in between. A comparison among the temperatures obtained from the distributions is made with a possible relation to the Boltzmann temperature. An example of the angular distributions of projectile fragments in nuclear collisions is given.

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Treumann, Rudolf A., and Wolfgang Baumjohann. "Generalised partition functions: inferences on phase space distributions." Annales Geophysicae 34, no. 6 (June 2, 2016): 557–64. http://dx.doi.org/10.5194/angeo-34-557-2016.

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Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

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Lin, Hejie, and Tsung-Wu Lin. "Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Analytical Integration." International Journal of Statistics and Probability 10, no. 3 (April 27, 2021): 135. http://dx.doi.org/10.5539/ijsp.v10n3p135.

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The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. The Maxwell-Boltzmann speed distribution of mixed particles is based on kinetic theory; however, it has never been derived from a mechanical point of view. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles based on probability analysis and Newton’s law of motion. This paper requires the probability density function (PDF) ψ^ab(u_a; v_a, v_b) of the speed u_a  of the particle with mass M_a  after the collision of two particles with mass M_a  in speed v_a  and mass M_b  in speed v_b . The PDF ψ^ab(u_a; v_a, v_b)  in integral form has been obtained before. This paper further performs the exact integration from the integral form to obtain the PDF ψ^ab(u_a; v_a, v_b)  in an evaluated form, which is used in the following equation to get new distribution P_new^a(u_a)  from old distributions P_old^a(v_a) and P_old^b(v_b). When P_old^a(v_a) and P_old^b(v_b)  are the Maxwell-Boltzmann speed distributions, the integration P_new^a(u_a)  obtained analytically is exactly the Maxwell-Boltzmann speed distribution. P_new^a(u_a)=∫_0^∞ ∫_0^∞ ψ^ab(u_a;v_a,v_b) P_old^a(v_a) P_old^b(v_b) dv_a dv_b,    a,b = 1 or 2 The mechanical proof of the Maxwell-Boltzmann speed distribution presented in this paper reveals the unsolved mechanical mystery of the Maxwell-Boltzmann speed distribution since it was proposed by Maxwell in 1860. Also, since the validation is carried out in an analytical approach, it proves that there is no theoretical limitation of mass ratio to the Maxwell-Boltzmann speed distribution. This provides a foundation and methodology for analyzing the interaction between particles with an extreme mass ratio, such as gases and neutrinos.

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7

Liu, Jeremy, Ke-Thia Yao, and Federico Spedalieri. "Dynamic Topology Reconfiguration of Boltzmann Machines on Quantum Annealers." Entropy 22, no. 11 (October 24, 2020): 1202. http://dx.doi.org/10.3390/e22111202.

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Boltzmann machines have useful roles in deep learning applications, such as generative data modeling, initializing weights for other types of networks, or extracting efficient representations from high-dimensional data. Most Boltzmann machines use restricted topologies that exclude looping connectivity, as such connectivity creates complex distributions that are difficult to sample. We have used an open-system quantum annealer to sample from complex distributions and implement Boltzmann machines with looping connectivity. Further, we have created policies mapping Boltzmann machine variables to the quantum bits of an annealer. These policies, based on correlation and entropy metrics, dynamically reconfigure the topology of Boltzmann machines during training and improve performance.

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8

KANIADAKIS, G. "PHYSICAL ORIGIN OF THE POWER-LAW TAILED STATISTICAL DISTRIBUTIONS." Modern Physics Letters B 26, no. 10 (April 8, 2012): 1250061. http://dx.doi.org/10.1142/s0217984912500613.

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Starting from the BBGKY hierarchy, describing the kinetics of nonlinear particle system, we obtain the relevant entropy and stationary distribution function. Subsequently, by employing the Lorentz transformations we propose the relativistic generalization of the exponential and logarithmic functions. The related particle distribution and entropy represents the relativistic extension of the classical Maxwell–Boltzmann distribution and of the Boltzmann entropy, respectively, and define the statistical mechanics presented in [Phys. Rev. E66 (2002) 056125] and [Phys. Rev. E72 (2005) 036108]. The achievements of the present effort, support the idea that the experimentally observed power-law tailed statistical distributions in plasma physics, are enforced by the relativistic microscopic particle dynamics.

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White, J. R., W. Johns, C. J. Fontes, N. M. Gill, N. R. Shaffer, and C. E. Starrett. "Charge state distributions in dense plasmas." Physics of Plasmas 29, no. 4 (April 2022): 043301. http://dx.doi.org/10.1063/5.0084109.

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Charge state distributions in hot, dense plasmas are a key ingredient in the calculation of spectral quantities like the opacity. However, they are challenging to calculate, as models like Saha–Boltzmann become unreliable for dense, quantum plasmas. Here, we present a new variational model for the charge state distribution, along with a simple model for the energy of the configurations that includes the orbital relaxation effect. Comparison with other methods reveals generally good agreement with average atom-based calculations, the breakdown of the Saha–Boltzmann method, and mixed agreement with a chemical model. We conclude that the new model gives a relatively inexpensive, but reasonably high fidelity method of calculating the charge state distribution in hot dense plasmas, in local thermodynamic equilibrium.

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10

Castillo, Jaime S., Katherine P. Gaete, Héctor A. Muñoz, Diego I. Gallardo, Marcelo Bourguignon, Osvaldo Venegas, and Héctor W. Gómez. "Scale Mixture of Maxwell-Boltzmann Distribution." Mathematics 11, no. 3 (January 18, 2023): 529. http://dx.doi.org/10.3390/math11030529.

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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.

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Gomez, Ignacio S., Bruno G. da Costa, and Maike A. F. dos Santos. "Majorization and Dynamics of Continuous Distributions." Entropy 21, no. 6 (June 14, 2019): 590. http://dx.doi.org/10.3390/e21060590.

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In this work we show how the concept of majorization in continuous distributions can be employed to characterize mixing, diffusive, and quantum dynamics along with the H-Boltzmann theorem. The key point lies in that the definition of majorization allows choosing a wide range of convex functions ϕ for studying a given dynamics. By choosing appropriate convex functions, mixing dynamics, generalized Fokker–Planck equations, and quantum evolutions are characterized as majorized ordered chains along the time evolution, being the stationary states the infimum elements. Moreover, assuming a dynamics satisfying continuous majorization, the H-Boltzmann theorem is obtained as a special case for ϕ ( x ) = x ln x .

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12

Frausto-Solis, Juan, Ernesto Liñán-García, Juan Paulo Sánchez-Hernández, J. Javier González-Barbosa, Carlos González-Flores, and Guadalupe Castilla-Valdez. "Multiphase Simulated Annealing Based on Boltzmann and Bose-Einstein Distribution Applied to Protein Folding Problem." Advances in Bioinformatics 2016 (June 20, 2016): 1–16. http://dx.doi.org/10.1155/2016/7357123.

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A new hybrid Multiphase Simulated Annealing Algorithm using Boltzmann and Bose-Einstein distributions (MPSABBE) is proposed. MPSABBE was designed for solving the Protein Folding Problem (PFP) instances. This new approach has four phases: (i) Multiquenching Phase (MQP), (ii) Boltzmann Annealing Phase (BAP), (iii) Bose-Einstein Annealing Phase (BEAP), and (iv) Dynamical Equilibrium Phase (DEP). BAP and BEAP are simulated annealing searching procedures based on Boltzmann and Bose-Einstein distributions, respectively. DEP is also a simulated annealing search procedure, which is applied at the final temperature of the fourth phase, which can be seen as a second Bose-Einstein phase. MQP is a search process that ranges from extremely high to high temperatures, applying a very fast cooling process, and is not very restrictive to accept new solutions. However, BAP and BEAP range from high to low and from low to very low temperatures, respectively. They are more restrictive for accepting new solutions. DEP uses a particular heuristic to detect the stochastic equilibrium by applying a least squares method during its execution. MPSABBE parameters are tuned with an analytical method, which considers the maximal and minimal deterioration of problem instances. MPSABBE was tested with several instances of PFP, showing that the use of both distributions is better than using only the Boltzmann distribution on the classical SA.

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13

LEVY, MOSHE, and SORIN SOLOMON. "POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS." International Journal of Modern Physics C 07, no. 04 (August 1996): 595–601. http://dx.doi.org/10.1142/s0129183196000491.

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Multiplicative random processes in (not necessarily equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the elementary variables. In terms of the original variables this gives a power-law distribution. This mechanism implies certain relations between the constraints of the system, the power of the distribution and the dispersion law of the fluctuations. These predictions are validated by Monte Carlo simulations and experimental data. We speculate that stochastic multiplicative dynamics might be the natural origin for the emergence of criticality and scale hierarchies without fine-tuning.

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14

Sebastian, Nicy, Arak M. Mathai, and Hans J. Haubold. "Entropy Optimization, Maxwell–Boltzmann, and Rayleigh Distributions." Entropy 23, no. 6 (June 15, 2021): 754. http://dx.doi.org/10.3390/e23060754.

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In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived. These include multivariate and matrix-variate Maxwell–Boltzmann and Rayleigh densities in the real and complex domains, multivariate Student-t, Cauchy, matrix-variate type-1 beta, type-2 beta, and gamma densities and their generalizations.

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15

Qian, Hong, and John A. Schellman. "Transformed Poisson−Boltzmann Relations and Ionic Distributions†." Journal of Physical Chemistry B 104, no. 48 (December 2000): 11528–40. http://dx.doi.org/10.1021/jp994168m.

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16

Zaninetti, Lorenzo. "New Probability Distributions in Astrophysics: III. The Truncated Maxwell-Boltzmann Distribution." International Journal of Astronomy and Astrophysics 10, no. 03 (2020): 191–202. http://dx.doi.org/10.4236/ijaa.2020.103010.

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17

Zaninetti, Lorenzo. "New Probability Distributions in Astrophysics: IV. The Relativistic Maxwell-Boltzmann Distribution." International Journal of Astronomy and Astrophysics 10, no. 04 (2020): 302–18. http://dx.doi.org/10.4236/ijaa.2020.104016.

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18

Liu, Fu-Hu, Ya-Hui Chen, Hua-Rong Wei, and Bao-Chun Li. "Transverse Momentum Distributions of Final-State Particles Produced in Soft Excitation Process in High Energy Collisions." Advances in High Energy Physics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/965735.

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Transverse momentum distributions of final-state particles produced in soft process in proton-proton (pp) and nucleus-nucleus (AA) collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies are studied by using a multisource thermal model. Each source in the model is treated as a relativistic and quantum ideal gas. Because the quantum effect can be neglected in investigation on the transverse momentum distribution in high energy collisions, we consider only the relativistic effect. The concerned distribution is finally described by the Boltzmann or two-component Boltzmann distribution. Our modeling results are in agreement with available experimental data.

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19

Belgacem, Chokri Hadj. "Comparative study between exponential Boltzmann and Lambert Boltzmann distributions for heat capacity calculation." International Journal of Modern Physics B 33, no. 14 (June 10, 2019): 1950140. http://dx.doi.org/10.1142/s0217979219501406.

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The Stirling’s estimation to [Formula: see text](N!) is typically introduced to students as a step in the derivation of the statistical expression for the heat capacity. However, naïve application of this estimation leads to wrong conclusions. In this paper, firstly, the heat capacity of some semiconductor compounds was calculated using exponential Boltzmann distribution and compared with experimental data. It has shown a disagreement between experimental results and those calculated. Secondly, by applying the more exact Stirling formula, an analytical formulation of Boltzmann statistics using Lambert W function is shown to be a very good model and proves an excellent agreement between calculated and experimental data for heat capacity over the entire temperature range.

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20

Corral, Álvaro, and Montserrat García del Muro. "From Boltzmann to Zipf through Shannon and Jaynes." Entropy 22, no. 2 (February 5, 2020): 179. http://dx.doi.org/10.3390/e22020179.

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The word-frequency distribution provides the fundamental building blocks that generate discourse in natural language. It is well known, from empirical evidence, that the word-frequency distribution of almost any text is described by Zipf’s law, at least approximately. Following Stephens and Bialek (2010), we interpret the frequency of any word as arising from the interaction potentials between its constituent letters. Indeed, Jaynes’ maximum-entropy principle, with the constrains given by every empirical two-letter marginal distribution, leads to a Boltzmann distribution for word probabilities, with an energy-like function given by the sum of the all-to-all pairwise (two-letter) potentials. The so-called improved iterative-scaling algorithm allows us finding the potentials from the empirical two-letter marginals. We considerably extend Stephens and Bialek’s results, applying this formalism to words with length of up to six letters from the English subset of the recently created Standardized Project Gutenberg Corpus. We find that the model is able to reproduce Zipf’s law, but with some limitations: the general Zipf’s power-law regime is obtained, but the probability of individual words shows considerable scattering. In this way, a pure statistical-physics framework is used to describe the probabilities of words. As a by-product, we find that both the empirical two-letter marginal distributions and the interaction-potential distributions follow well-defined statistical laws.

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SUZUKI, N., and M. BIYAJIMA. "TRANSVERSE MOMENTUM DISTRIBUTION WITH RADIAL FLOW IN RELATIVISTIC DIFFUSION MODEL." International Journal of Modern Physics E 16, no. 01 (January 2007): 133–47. http://dx.doi.org/10.1142/s0218301307005582.

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Large transverse momentum distributions of identified particles observed at the Relativistic Heavy Ion Collider (RHIC) are analyzed by a relativistic stochastic model in the three dimensional (non-Euclidean) rapidity space. A distribution function obtained from the model is Gaussian-like in radial rapidity. It can well describe observed transverse momentum pT distributions. Estimation of radial flow is made from the analysis of pT distributions for [Formula: see text] in Au + Au Collisions. Temperatures are estimated from observed large pT distributions under the assumption that the distribution function approaches to the Maxwell–Boltzmann distribution in the lower momentum limit. The power-law behavior of large pT distribution is also derived from the model.

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22

Flannigan, David J., and Kenneth S. Suslick. "Non-Boltzmann Population Distributions during Single-Bubble Sonoluminescence." Journal of Physical Chemistry B 117, no. 49 (October 7, 2013): 15886–93. http://dx.doi.org/10.1021/jp409222x.

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23

Baker, Nathan A., and J. Andrew McCammon. "Non-Boltzmann Rate Distributions in Stochastically Gated Reactions." Journal of Physical Chemistry B 103, no. 4 (January 1999): 615–17. http://dx.doi.org/10.1021/jp984151o.

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24

Visscher, P. B., and D. M. Apalkov. "Non-Boltzmann energy distributions in spin-torque devices." Journal of Applied Physics 99, no. 8 (April 15, 2006): 08G513. http://dx.doi.org/10.1063/1.2165785.

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25

Kugel, Roger W., and Paul A. Weiner. "Energy Distributions in Small Populations: Pascal versus Boltzmann." Journal of Chemical Education 87, no. 11 (November 2010): 1200–1205. http://dx.doi.org/10.1021/ed1003838.

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26

Mathai, A. M., and Serge B. Provost. "Generalized Boltzmann factors induced by Weibull-type distributions." Physica A: Statistical Mechanics and its Applications 392, no. 4 (February 2013): 545–51. http://dx.doi.org/10.1016/j.physa.2012.10.030.

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27

Smaal, Nicholas, and José Roberto C. Piqueira. "Complexity Measures for Maxwell–Boltzmann Distribution." Complexity 2021 (January 23, 2021): 1–6. http://dx.doi.org/10.1155/2021/9646713.

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This work presents a discussion about the application of the Kolmogorov; López-Ruiz, Mancini, and Calbet (LMC); and Shiner, Davison, and Landsberg (SDL) complexity measures to a common situation in physics described by the Maxwell–Boltzmann distribution. The first idea about complexity measure started in computer science and was proposed by Kolmogorov, calculated similarly to the informational entropy. Kolmogorov measure when applied to natural phenomena, presents higher values associated with disorder and lower to order. However, it is considered that high complexity must be associated to intermediate states between order and disorder. Consequently, LMC and SDL measures were defined and used in attempts to model natural phenomena but with the inconvenience of being defined for discrete probability distributions defined over finite intervals. Here, adapting the definitions to a continuous variable, the three measures are applied to the known Maxwell–Boltzmann distribution describing thermal neutron velocity in a power reactor, allowing extension of complexity measures to a continuous physical situation and giving possible discussions about the phenomenon.

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Pavlov, A. V., and K. I. Oyama. "The role of vibrationally excited nitrogen and oxygen in the ionosphere over Millstone Hill during 16-23 March, 1990." Annales Geophysicae 18, no. 8 (August 31, 2000): 957–66. http://dx.doi.org/10.1007/s00585-000-0957-2.

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Abstract. We present a comparison of the observed behavior of the F region ionosphere over Millstone Hill during the geomagnetically quiet and storm period on 16-23 March, 1990, with numerical model calculations from the time-dependent mathematical model of the Earth's ionosphere and plasmasphere. The effects of vibrationally excited N2(v) and O2(v) on the electron density and temperature are studied using the N2(v) and O2(v) Boltzmann and non-Boltzmann distribution assumptions. The deviations from the Boltzmann distribution for the first five vibrational levels of N2(v) and O2(v) were calculated. The present study suggests that these deviations are not significant at vibrational levels v = 1 and 2, and the calculated distributions of N2(v) and O2(v) are highly non-Boltzmann at vibrational levels v > 2. The N2(v) and O2(v) non-Boltzmann distribution assumption leads to the decrease of the calculated daytime NmF2 up to a factor of 1.44 (maximum value) in comparison with the N2(v) and O2(v) Boltzmann distribution assumption. The resulting effects of N2(v > 0) and O2(v > 0) on the NmF2 is the decrease of the calculated daytime NmF2 up to a factor of 2.8 (maximum value) for Boltzmann populations of N2(v) and O2(v) and up to a factor of 3.5 (maximum value) for non-Boltzmann populations of N2(v) and O2(v) . This decrease in electron density results in the increase of the calculated daytime electron temperature up to about 1040-1410 K (maximum value) at the F2 peak altitude giving closer agreement between the measured and modeled electron temperatures. Both the daytime and nighttime densities are not reproduced by the model without N2(v > 0) and O2(v > 0) , and inclusion of vibrationally excited N2 and O2 brings the model and data into better agreement. The effects of vibrationally excited O2 and N2 on the electron density and temperature are most pronounced during daytime.Key words: Ionosphere (ion chemistry and composition; ionosphere-atmosphere interactions; ionospheric disturbances)

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Bulso, Nicola, and Yasser Roudi. "Restricted Boltzmann Machines as Models of Interacting Variables." Neural Computation 33, no. 10 (September 16, 2021): 2646–81. http://dx.doi.org/10.1162/neco_a_01420.

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Abstract We study the type of distributions that restricted Boltzmann machines (RBMs) with different activation functions can express by investigating the effect of the activation function of the hidden nodes on the marginal distribution they impose on observed binary nodes. We report an exact expression for these marginals in the form of a model of interacting binary variables with the explicit form of the interactions depending on the hidden node activation function. We study the properties of these interactions in detail and evaluate how the accuracy with which the RBM approximates distributions over binary variables depends on the hidden node activation function and the number of hidden nodes. When the inferred RBM parameters are weak, an intuitive pattern is found for the expression of the interaction terms, which reduces substantially the differences across activation functions. We show that the weak parameter approximation is a good approximation for different RBMs trained on the MNIST data set. Interestingly, in these cases, the mapping reveals that the inferred models are essentially low order interaction models.

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30

Meir, Ronny. "ON DERIVING DETERMINISTIC LEARNING RULES FROM STOCHASTIC SYSTEMS." International Journal of Neural Systems 02, no. 04 (January 1991): 283–89. http://dx.doi.org/10.1142/s012906579100025x.

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We discuss the derivation of deterministic learning rules from an underlying stochastic system. We focus on the symmetrically connected Boltzmann machine and show how various approximations give rise to different learning algorithms. In particular, we show how to derive a symmetrized form of the recurrent back propagation learning algorithm from the Boltzmann machine. We also discuss the connection between the different deterministic learning algorithms focusing on the probability distributions from which they originate. It will also be shown that inspite of the fact that two probability distributions have the same moments to any finite order, they give rise to two distinct learning algorithms.

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Landsman, Zinoviy, Udi Makov, and Tomer Shushi. "A New Class of Distributions Based on Hurwitz Zeta Function with Applications for Risk Management." Open Statistics & Probability Journal 7, no. 1 (December 27, 2016): 53–62. http://dx.doi.org/10.2174/1876527001607010053.

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This paper constructs a new family of distributions, which is based on the Hurwitz zeta function, which includes novel distributions as well important known distributions such as the normal, gamma, Weibull, Maxwell-Boltzmann and the exponential power distributions. We provide the n-th moment, the Esscher transform and premium and the tail conditional moments for this family.

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Jang, Dukjae, Youngshin Kwon, Kyujin Kwak, and Myung-Ki Cheoun. "Big Bang nucleosynthesis in a weakly non-ideal plasma." Astronomy & Astrophysics 650 (June 2021): A121. http://dx.doi.org/10.1051/0004-6361/202038478.

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We propose a correction of the standard Big Bang nucleosynthesis (BBN) scenario to resolve the primordial lithium problem by considering a possibility that the primordial plasma can deviate from the ideal state. In the standard BBN, the primordial plasma is assumed to be ideal, with particles and photons satisfying the Maxwell-Boltzmann and Planck distribution, respectively. We suggest that this assumption of the primordial plasma being ideal might oversimplify the early Universe and cause the lithium problem. We find that a deviation of photon distribution from the Planck distribution, which is parameterised with the help of Tsallis statistics, can resolve the primordial lithium problem when the particle distributions of the primordial plasma still follow the Maxwell-Boltzmann distribution. We discuss how the primordial plasma can be weakly non-ideal in this specific fashion and its effects on the cosmic evolution.

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Lin, Hejie, and Tsung-Wu Lin. "Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Numerical Iterations." International Journal of Statistics and Probability 10, no. 4 (June 1, 2021): 21. http://dx.doi.org/10.5539/ijsp.v10n4p21.

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The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles using computer-generated data based on Newton’s law of motion. To achieve this, this paper derives the probability density function ψ^ab(u_a;v_a,v_b)  of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b. The function ψ^ab(u_a;v_a,v_b)  is obtained through a unique procedure that considers (1) the randomness of the relative direction before a collision by an angle α. (2) the randomness of the direction after the collision by another independent angle. The function ψ^ab(u_a;v_a,v_b) is used in the equation below for the numerical iterations to get new distributions P_new^a(u_a) from old distributions P_old^a(v_a), and repeat with P_old^a(v_a)=P_new^a(v_a), where n_a is the fraction of particles with mass M_a.   P_new^1(u_1)=n_1 ∫_0^∞ ∫_0^∞ ψ^11(u_1;v_1,v’_1) P_old^1(v_1) P_old^1(v’_1) dv_1 dv’_1                           +n_2 ∫_0^∞ ∫_0^∞ ψ^12(u_1;v_1,v_2) P_old^1(v_1) P_old^2(v_2) dv_1 dv_2 P_new^2(u_2)=n_1 ∫_0^∞ ∫_0^∞ ψ^21(u_2;v_2,v_1) P_old^2(v_2) P_old^1(v_1) dv_2 dv_1                           +n_2 ∫_0^∞ ∫_0^∞ ψ^22(u_2;v_2,v’_2) P_old^2(v_2) P_old^2(v’_2) dv_2 dv’_2 The final distributions converge to the Maxwell-Boltzmann speed distributions. Moreover, the square of the root-mean-square speed from the final distribution is inversely proportional to the particle masses as predicted by Avogadro’s law.

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34

Rafik, Zeraoulia, Alvaro H. Salas, and David L. Ocampo. "A New Special Function and Its Application in Probability." International Journal of Mathematics and Mathematical Sciences 2018 (November 1, 2018): 1–12. http://dx.doi.org/10.1155/2018/5146794.

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In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chèbyshev polynomials of the first kind and the error function. Also we provide its series representation using Padé approximant. We show a convincing numerical evidence about an accuracy of 10-6 for the approximants in the sense of the quadratic mean norm. A similar approach may be applied to other probability distributions, for example, Maxwell–Boltzmann distribution and normal distribution, such that we show its application using both of those distributions.

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35

Quevedo Cubillos, Hernando, and María N. Quevedo. "Income distribution in the Colombian economy from an econophysics perspective." Cuadernos de Economía 35, no. 69 (September 1, 2016): 691–707. http://dx.doi.org/10.15446/cuad.econ.v35n69.44876.

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Recently, in econophysics, it has been shown that it is possible to analyze economic systems as equilibrium thermodynamic models. We apply statistical thermodynamics methods to analyze income distribution in the Colombian economic system. Using the data obtained in random polls, we show that income distribution in the Colombian economic system is characterized by two specific phases. The first includes about 90% of the interviewed individuals, and is characterized by an exponential Boltzmann-Gibbs distribution. The second phase, which contains the individuals with the highest incomes, can be described by means of one or two power-law density distributions that are known as Pareto distributions.

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36

HUBA, J. D., and J. G. LYON. "A new 3D MHD algorithm: the distribution function method." Journal of Plasma Physics 61, no. 3 (April 1999): 391–405. http://dx.doi.org/10.1017/s0022377899007503.

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A new three-dimensional (3D) magnetohydrodynamic (MHD) algorithm is described. The 3D MHD equations are solved in conservative form using a finite-volume scheme. The hydrodynamic variables in a cell are updated by calculating fluxes across the cell interfaces. The fluxes of mass, momentum and energy across cell interfaces are calculated by integrating a Boltzmann-like distribution function over velocity space. The novel feature of the method is that the distribution function incorporates most of the electromagnetic terms. In addition, the electric field along cell edges, which is used to update the magnetic field at cell faces, is also calculated using the Boltzmann-like distribution function. An important aspect of the method is that it provides a theoretical framework to incorporate additional terms in the 3D MHD equations (e.g. an anisotropic ion stress tensor and anisotropic temperature distributions).

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37

Cushman, Samuel A. "Generalizing Boltzmann Configurational Entropy to Surfaces, Point Patterns and Landscape Mosaics." Entropy 23, no. 12 (December 1, 2021): 1616. http://dx.doi.org/10.3390/e23121616.

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Several methods have been recently proposed to calculate configurational entropy, based on Boltzmann entropy. Some of these methods appear to be fully thermodynamically consistent in their application to landscape patch mosaics, but none have been shown to be fully generalizable to all kinds of landscape patterns, such as point patterns, surfaces, and patch mosaics. The goal of this paper is to evaluate if the direct application of the Boltzmann relation is fully generalizable to surfaces, point patterns, and landscape mosaics. I simulated surfaces and point patterns with a fractal neutral model to control their degree of aggregation. I used spatial permutation analysis to produce distributions of microstates and fit functions to predict the distributions of microstates and the shape of the entropy function. The results confirmed that the direct application of the Boltzmann relation is generalizable across surfaces, point patterns, and landscape mosaics, providing a useful general approach to calculating landscape entropy.

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38

Kryvdyk, V. "Particle Acceleration and Radiation in Magnetospheres of Collapsing Stars." Symposium - International Astronomical Union 195 (2000): 403–6. http://dx.doi.org/10.1017/s0074180900163296.

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Particle dynamics and nonthermal emission therefrom in the magnetospheres of collapsing stars with initial dipole magnetic fields and a certain initial energy distribution of charged particles (power-law, relativistic Maxwell, and Boltzmann distributions) are considered. The radiation fluxes are calculated for various collapsing stars with initial dipole magnetic fields and an initial power-law particle energy distribution in the magnetosphere. The effects can be observed by means of modern instruments.

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39

SHIDA, CLAUDIO S., and VERA B. HENRIQUES. "SMART AND WRONG VERSUS CORRECT MONTE CARLO SIMULATIONS OF KAWASAKI–ISING MODEL." International Journal of Modern Physics C 11, no. 05 (July 2000): 1033–36. http://dx.doi.org/10.1142/s0129183100000870.

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40

Wright, Sam O. M., Stevanus K. Nugroho, Matteo Brogi, Neale P. Gibson, Ernst J. W. de Mooij, Ingo Waldmann, Jonathan Tennyson, et al. "A Spectroscopic Thermometer: Individual Vibrational Band Spectroscopy with the Example of OH in the Atmosphere of WASP-33b." Astronomical Journal 166, no. 2 (July 4, 2023): 41. http://dx.doi.org/10.3847/1538-3881/acdb75.

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Abstract Individual vibrational band spectroscopy presents an opportunity to examine exoplanet atmospheres in detail, by distinguishing where the vibrational state populations of molecules differ from the current assumption of a Boltzmann distribution. Here, retrieving vibrational bands of OH in exoplanet atmospheres is explored using the hot Jupiter WASP-33b as an example. We simulate low-resolution spectroscopic data for observations with the JWST's NIRSpec instrument and use high-resolution observational data obtained from the Subaru InfraRed Doppler instrument (IRD). Vibrational band–specific OH cross-section sets are constructed and used in retrievals on the (simulated) low- and (real) high-resolution data. Low-resolution observations are simulated for two WASP-33b emission scenarios: under the assumption of local thermal equilibrium (LTE) and with a toy non-LTE model for vibrational excitation of selected bands. We show that mixing ratios for individual bands can be retrieved with sufficient precision to allow the vibrational population distributions of the forward models to be reconstructed. A fit for the Boltzmann distribution in the LTE case shows that the vibrational temperature is recoverable in this manner. For high-resolution, cross-correlation applications, we apply the individual vibrational band analysis to an IRD spectrum of WASP-33b, applying an “unpeeling” technique. Individual detection significances for the two strongest bands are shown to be in line with Boltzmann-distributed vibrational state populations, consistent with the effective temperature of the WASP-33b atmosphere reported previously. We show the viability of this approach for analyzing the individual vibrational state populations behind observed and simulated spectra, including reconstructing state population distributions.

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41

MAYER, GUSZTÁV, GÁBOR HÁZI, JÓZSEF PÁLES, ATTILA R. IMRE, BJÖRN FISCHER, and THOMAS KRASKA. "ON THE SYSTEM SIZE OF LATTICE BOLTZMANN SIMULATIONS." International Journal of Modern Physics C 15, no. 08 (October 2004): 1049–60. http://dx.doi.org/10.1142/s0129183104006492.

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In lattice Boltzmann simulations particle groups — represented by scalar velocity distributions — are moved on a finite lattice. The size of these particle groups is not well-defined although it is crucial to assume that they should be big enough for using a continuous distribution. Here we propose to use the liquid–vapor interface as an internal yardstick to scale the system. Comparison with existing experimental data and with molecular dynamics simulation of Lennard–Jones-argon shows that the number of atoms located on one lattice site is in the order of few atoms. This contradicts the initial assumption concerning the number of particles in the group, therefore seems to raise some doubts about the applicability of the lattice Boltzmann method in certain problems whenever interfaces play important role and ergodicity does not hold.

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42

Bernegger, Stefan. "The Swiss Re Exposure Curves and the MBBEFD Distribution Class." ASTIN Bulletin 27, no. 1 (May 1997): 99–111. http://dx.doi.org/10.2143/ast.27.1.563208.

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AbstractA new two-parameter family of analytical functions will be introduced for the modelling of loss distributions and exposure curves. The curve family contains the Maxwell-Boltzmann, the Bose-Einstein and the Fermi-Dirac distributions, which are well known in statistical mechanics. The functions can be used for the modelling of loss distributions on the finite interval [0, 1] as well as on the interval [0, ∞]. The functions defined on the interval [0, 1] are discussed in detail and related to several Swiss Re exposure curves used in practice. The curves can be fitted to the first two moments μ and σ of a loss distribution or to the first moment μ and the total loss probability p.

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43

DeSalvo, Stephen. "Improvements to exact Boltzmann sampling using probabilistic divide-and-conquer and the recursive method." Pure Mathematics and Applications 26, no. 1 (June 27, 2017): 22–45. http://dx.doi.org/10.1515/puma-2015-0020.

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Abstract We demonstrate an approach for exact sampling of certain discrete combinatorial distributions, which is a hybrid of exact Boltzmann sampling and the recursive method, using probabilistic divide-and-conquer (PDC). The approach specializes to exact Boltzmann sampling in the trivial setting, and specializes to PDC deterministic second half in the first non-trivial application. A large class of examples is given for which this method broadly applies, and several examples are worked out explicitly.

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44

Cotăescu, Ion I. "Collective classical motion on hyperbolic spacetimes of any dimensions." Modern Physics Letters A 34, no. 21 (July 9, 2019): 1950165. http://dx.doi.org/10.1142/s0217732319501657.

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The geodesics equations on de Sitter (dS) and anti-de Sitter (AdS) spacetimes of any dimensions, are the starting point for deriving the general form of the Boltzmann equation in terms of conserved quantities. The simple equation for the non-equilibrium Marle and Anderson–Witting models are derived and the distributions of the Boltzmann–Marle model on these manifolds are written down first in terms of conserved quantities and then as functions of canonical variables.

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45

Fleitz, P. A., and C. J. Seliskar. "Characterization of OH Radical 306.4 nm Emission in Argon and Helium Reduced-Pressure ICPs." Applied Spectroscopy 41, no. 4 (May 1987): 679–82. http://dx.doi.org/10.1366/0003702874448760.

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The A → X emission of OH at 306.4 nm has been studied in reduced-pressure inductively coupled argon and helium plasmas. Under a variety of conditions of power and flow rate, a comparison of the A state rotational level distributions shows significant differences in the two plasmas. The rotational level distributions are generally nonlinear but can be quantitatively described as the sum of two separate Boltzmann distributions.

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46

Bahou, Mohammed, and Yuan-Pern Lee. "Photodissociation Dynamics of Vinyl Chloride Investigated with a Pulsed Slit-Jet and Time-Resolved Fourier-Transform Spectroscopy." Australian Journal of Chemistry 57, no. 12 (2004): 1161. http://dx.doi.org/10.1071/ch04117.

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Following photodissociation of vinyl chloride seeded in a He supersonic jet at 193 nm, rotationally resolved infrared emission of HCl (v) are recorded to yield nascent rotational and vibrational distributions. Preliminary results show that the rotational distribution of HCl free from rotational quenching deviates slightly from Boltzmann-type distribution and agrees well with trajectory calculations; a portion of the low-J component observed previously in a flow system is attributed to quenching. The implications for photodissociation dynamics are discussed.

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47

Fonseca, F., and A. Franco. "Study of complex charge distributions in an electrolyte using the Poisson–Boltzmann equation by lattice-Boltzmann method." Microelectronics Journal 39, no. 11 (November 2008): 1224–25. http://dx.doi.org/10.1016/j.mejo.2008.01.061.

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48

Obuchi, Tomoyuki, Hirokazu Koma, and Muneki Yasuda. "Boltzmann-Machine Learning of Prior Distributions of Binarized Natural Images." Journal of the Physical Society of Japan 85, no. 11 (November 15, 2016): 114803. http://dx.doi.org/10.7566/jpsj.85.114803.

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49

Albritton, J. R., E. A. Williams, I. B. Bernstein, and K. P. Swartz. "Nonlocal Electron Heat Transport by Not Quite Maxwell-Boltzmann Distributions." Physical Review Letters 57, no. 15 (October 13, 1986): 1887–90. http://dx.doi.org/10.1103/physrevlett.57.1887.

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50

Glinski, R. J., S. A. Schulz, and J. A. Nuth. "Non-Boltzmann vibrational distributions in homonuclear diatomic molecules and ions." Monthly Notices of the Royal Astronomical Society 288, no. 2 (June 21, 1997): 286–94. http://dx.doi.org/10.1093/mnras/288.2.286.

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