Maxwell Boltzmann Research Articles

Structure-Dependent Doppler Broadening Using a Generalized Thermal Scattering Law

The thermal scattering law (TSL), i.e., S(α,β), represents the momentum and energy exchange phase space for a material. The incoherent and coherent components of the TSL correlate an atom’s trajectory with itself and/or with other atoms in the lattice structure. This structural information is especially important for low energies where the wavelength of neutrons is on the order of the lattice interatomic spacing. Both thermal neutron scattering as well as low energy resonance broadening involve processes where incoming neutron responses are lattice dependent. Traditionally, Doppler broadening for absorption resonances approximates these interactions by assuming a Maxwell–Boltzmann distribution for the neutron velocity. For high energies and high temperatures, this approximation is reasonable. However, for low temperatures or low energies, the lattice structure binding effects will influence the velocity distribution. Using the TSL to determine the Doppler broadening directly introduces the material structure into the calculation to most accurately capture the momentum and energy space. Typically, the TSL is derived assuming cubic lattice symmetry. This approximation collapses the directional lattice information, including the polarization vectors and associated energies, into an energy-dependent function called the density of states. The cubic approximation, while valid for highly symmetric and uniformly bonded materials, is insufficient to capture the true structure. In this work, generalized formulation for the exact, lattice-dependent TSL is implemented within the Full Law Analysis Scattering System Hub (FLASSH) using polarization vectors and associated energies as fundamental input. These capabilities are utilized to perform the generalized structure Doppler broadening analysis for UO2.

A study on Raga characterization in Indian classical music in the light of MB and BE distribution

Raga characterization in Indian classical music is an important aspect of music learning in this country. But the methods usually followed are mostly qualitative. In this study, we intend to quantify such abstractness using measurable parameters. To study musical information congregation quantifiably, we introduce methods based on well-known concepts used in Statistical Physics, namely Maxwell-Boltzmann (MB) and Bose-Einstein (BE) distribution. In this present study, these distributions have been applied on the chosen acoustic signals to find new parameters (equivalent to ‘temperature’ in physical systems) which can distinguish between different features of different ragas (containing the same notes) in Indian classical music. Music clips chosen were the ‘Alap’ part of these three different ragas (Marwa, Puriya, Sohini) sung by a legendary classical music maestro. All of the chosen three ragas are based on the following same note structure: Sa, komal Re, shuddh Ga, tivra Ma, shuddh Dha, shuddh Ni. To apply MB statistics to music, it is assumed that different notes with different occurrence frequencies are at different energy levels, the distribution of which follows the MB distribution pattern. In case of BE statistics, a rank-frequency distribution of the time durations of various notes of different ragas is studied. The resulting analysis gives rise to a number of parameters that help to categorize the individual characteristics of ragas. The methods studied here are novel in the music research field and can prove to be useful in the fields of music and speech as quantifying parameters for style identification.

Distribution of random motion at renewal instants in three-dimensional space

In physics, chemistry, and mathematics, the process of Brownian motion is often identified with the Wiener process that has infinitesimal increments. Recently, many models of Brownian motion with finite velocity have been intensively studied. We consider one of such models, namely, a generalization of the Goldstein–Kac process to the three-dimensional case with the Erlang-2 and Maxwell–Boltzmann distributions of velocities alternations. Despite the importance of having a three-dimensional isotropic random model for the motion of Brownian particles, numerous research efforts did not lead to an expression for the probability of the distribution of the particle position, the motion of which is described by the threedimensional telegraph process. The case where a particle carries out its movement along the directions determined by the vertices of a regular n + 1-hedron in the n-dimensional space was studied in [13], and closed-form results for the distribution of the particle position were obtained. Here, we obtain expressions for the distribution function of the norm of the vector that defines particle’s position at renewal instants in semi-Markov cases of the Erlang-2 and Maxwell–Boltzmann distributions and study its properties. By knowing this distribution, we can determine the distribution of particle positions, since the motion of a particle is isotropic, i.e., the direction of its movement is uniformly distributed on the unit sphere in ℝ3. Our results may be useful in studying the properties of an ideal gas.

Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes.

We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell H-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship , and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the H-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the H-functionals are in agreement with the correspondence principle. We discuss how the H-functionals can be identified with the system’s entropy and analyze the relaxation processes of out-of-equilibrium systems.

Law of large numbers unifying Maxwell–Boltzmann, Bose–Einstein and Zipf–Mandelbort distributions, and related fluctuations

We consider a system composed of a fixed number of particles with total energy smaller or equal to some prescribed value. The particles are non-interacting, indistinguishable and distributed over fixed number of energy levels. The energy levels are degenerate and degeneracy is a function of the number of particles. Three cases of the degeneracy function is considered. It can increase with either the same rate as the number of particles or slower, or faster. We find useful properties of the entropy of the system and solve related entropy optimization problem. It turned out, there are several solutions. Depending on the magnitude of total energy, the maximum of the entropy can be in the interior of system’s state space or on the boundary. On the boundary it can have further three cases depending on the degeneracy function. The main result, Law of Large Numbers yields the most probable state of the system, which equals to the point of maximum of the entropy. This point can be either Maxwell–Boltzmann statistics or Bose–Einstein statistics, or Zipf–Mandelbort law. We also find the limiting laws for the fluctuations. These laws are different for different cases of the entropy’s maximum. They can be mixture of Normal, Exponential and Discrete distributions. Explicit rates of convergence of moment generating functions are provided for all the theorems. The overview of possible applications is provided in the last section.

Trellis Shaping for Fiber Nonlinearity Mitigation in Coherent Optical OFDM Systems

In this article, trellis shaping technique is numerically and experimentally investigated to improve the performance of optical orthogonal frequency division multiplexing (OFDM) signal. Different from other constellation shaping techniques, trellis shaping can be designed to reduce the correlation value of OFDM data sequence and average power simultaneously in one OFDM symbol. It means that optical OFDM signals generated by trellis shaping can improve the performance under both linear and nonlinear channels. For comparison with trellis shaping technique, we also consider optimal Maxwell–Boltzmann (MB) shaping and uniform signaling techniques at spectral efficiency (SE) of 10 and 14 bits/4D-sym. Numerical simulations show that trellis shaping can provide certain shaping gain over uniform signaling under linear additive white Gaussian noise (AWGN) channel. The fiber transmission results indicate that OFDM signals with trellis shaping have higher effective signal-to-noise ratio (SNR) than those with MB shaping and uniform signaling. It means that trellis shaping technique has the ability to mitigate the fiber nonlinear noise. Considering both linear and nonlinear noise in the fiber link, trellis shaping can extend the transmission distance by $\sim$ 110 km and $\sim$ 200 km over MB shaping and uniform signaling at SE of 10 bits/4D-sym. At SE of 14 bits/4D-sym, trellis shaping provides similar performance as MB shaping but $\sim$ 120-km extension reach in comparison with uniform signaling. The performances of OFDM signals with trellis shaping, MB shaping and uniform signaling are also experimentally investigated in a four-channel wavelength division multiplexing (WDM) fiber transmission systems. The experimental results show that trellis shaping can also mitigate the nonlinear distortions induced by the clipping effects at the transmitter side. Compared with MB shaping, trellis shaping extends the transmission reach by $\sim$ 240 km and $\sim$ 90 km at SE of 10 and 14 bits/4D-sym, respectively.

Determination of Neutron Flux Viewed from the Maxwell-Boltzmann’s Statistical Distribution Using Numerical Approach

This article aims to investigate the computational distribution of neutron flux using Maxwell-Boltzmann statistics. The reactor geometry used is a cylindrical shape. Identify the flux neutron only can when a neutron collides. Data analysis uses numerical analysis with the MATLAB 2013a program. The graph generated from a cylinder geometry is a histogram graph with the Poisson distribution by the Maxwell-Boltzmann distribution. There is a difference in the value of the neutron flux due to statistical uncertainty using a numerical approach. The purpose of writing this article is to determine the computational distribution of neutron fluxes in a cylindrical geometry reactor using Maxwell Boltzmann statistics. By writing this article it is hoped that it will become a reference material for further research or studies on computing the distribution of neutrons in other cases.

Exponential Parsimony to Generate Herd Immunity Against SARS- CoV2 Cybernetically Restricting Doxastic and Desiderative Logics

We Hypothesize that It should be intuitively inferred that the behavior of the pandemics corresponds to the prediction of the Maxwell Boltzmann distribution, which is a direct function of population density. In a higher organizational metalevel, it is analogous to the Brownian movement effect, in which the possibility of being observed is determined by the observational power of the instrument. In this case, it corresponds to the sensibility and sensitivity of the probe as well as the prevalence of the arbitrary states. The states are formally considered as a partition in probability theory; both in the Kolmogorov and frequentist approach. All these become subsumed to the Bayes theorem applied to recursive conditional probability.

Quantitative comprehension the impact of X-light on current gain of lateral PNP bipolar transistor

Bias emission voltage and dose-dependent current gain are the key parameters to detect the physical performance of bipolar transistors. Firstly, the effect of X-light with energy 22.1 and 25.0 keV from 109Cd on the current gain of lateral PNP bipolar transistors is measured. The substantial difference of current gain with the variation of bias and emission voltage is observed. Then, on basis of the outcomes of the nonlinear curve fitting approach, the relationship between bias voltage and emission-related current gain is successfully explained by using the model of combination Maxwell Boltzmann statistical with Bose-Einstein probability. The simulation results are in good agreement with the observed curves (current gain vs. voltage). The minimal correlation coefficient between experimental data and computed data is 0.998. In all cases considered, the mean relative error between the measured value and the model value is not more than 5.0%. Finally, the mathematical link between the parameters of current gain and voltage and the X-light dose is obtained. Therefore, using only one function, the current gain can be anticipated by the voltage and dose between bias and emission. The influence of X-light on the current gain of transverse PNP bipolar transistors is firstly discussed quantitatively.

Models of Particles of the Michie–King Type

In this paper we study the model of the Michie–King distribution of the self-gravitating system of diffusive particles. This leads to the pressure function of the Michie–King type, being the alternative to the Maxwell–Boltzmann and Fermi–Dirac statistics. The most recent considerations of such systems were suggested by de Vega et al. and Chavanis et al. We consider the existence vs blow-up depending on mass M and the temperature $$\theta $$ of the system. In the two dimensional case $$8\pi $$ is the threshold value for the $$M/\theta $$ for the classical Michie–King model exactly like in the Maxwell–Boltzmann case. Related systems appear in models of chemotaxis and description of 2-D Euler vortices.

Energetics of mesoscale cell turbulence in two-dimensional monolayers

Investigation of energy mechanisms at the collective cell scale is a challenge for understanding various biological processes, such as embryonic development and tumor metastasis. Here we investigate the energetics of self-sustained mesoscale turbulence in confluent two-dimensional (2D) cell monolayers. We find that the kinetic energy and enstrophy of collective cell flows in both epithelial and non-epithelial cell monolayers collapse to a family of probability density functions, which follow the q-Gaussian distribution rather than the Maxwell–Boltzmann distribution. The enstrophy scales linearly with the kinetic energy as the monolayer matures. The energy spectra exhibit a power-decaying law at large wavenumbers, with a scaling exponent markedly different from that in the classical 2D Kolmogorov–Kraichnan turbulence. These energetic features are demonstrated to be common for all cell types on various substrates with a wide range of stiffness. This study provides unique clues to understand active natures of cell population and tissues.

Huffman-Coded Sphere Shaping for Extended-Reach Single-Span Links

Huffman-coded sphere shaping (HCSS) is an algorithm for finite-length probabilistic constellation shaping, which provides nearly optimal energy efficiency at low implementation complexity. In this paper, we experimentally study the nonlinear performance of HCSS employing dual-polarization 64-ary quadrature amplitude modulation (DP-64QAM) in an extended-reach single-span link comprising 200 km of standard single-mode fiber (SSMF). We investigate the effects of shaping sequence length, dimensionality of symbol mapping, and shaping rate. We determine that the naive approach of Maxwell–Boltzmann distribution matching — which is optimal in the additive white Gaussian noise channel — provides a maximum achievable information rate (AIR) gain of 0.18 bits/4D-symbol with respect to uniform signaling at optimum launch power in the infinite length regime. Conversely, HCSS can achieve a gain of 0.37 bits/4D-symbol over uniform signaling using amplitude sequence length of 32, which may be implemented without multiplications, using integer comparison and addition operations only. Coded system performance, with a net data rate of approximately 425 Gb/s for both shaped and uniform inputs, is also analyzed.

A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas

In recent years, the implications of the generalized (GUP) and extended (EUP) uncertainty principles on Maxwell–Boltzmann distribution have been widely investigated. However, at high energy regimes, the validity of Maxwell–Boltzmann statistics is under debate and instead, the Jüttner distribution is proposed as the distribution function in relativistic limit. Motivated by these considerations, in the present work, our aim is to study the effects of GUP and EUP on a system that obeys the Jüttner distribution. To achieve this goal, we address a method to get the distribution function by starting from the partition function and its relation with thermal energy which finally helps us in finding the corresponding energy density states.

Complexity Measures for Maxwell–Boltzmann Distribution

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.

Monte Carlo Simulation of a Modified Chi Distribution with Unequal Variances in the Generating Gaussians. A Discrete Methodology to Study Collective Response Times

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.

Green synthesis of palladium mixed with PdO nanoparticles by chemical bath deposition

In this paper, we present preliminary experimental results in the synthesis of Pd mixed with PdO nanoparticles, grown by chemical bath deposition at ~80 °C. These samples were subjected to further thermal annealing treatment at ~1000 °C in air atmosphere. The properties of these compounds were investigated using the scanning electron microscopy (SEM), Fourier-transform infrared (FTIR), X-ray diffraction (XRD), optical absorption and photoluminescence (PL) techniques. The SEM images for the grown sample show small dense and compact crystals. The FTIR spectra shows absorption bands associated with the OH− and CO32− ions. XRD diffractograms identified a face-centred cubic structure for metallic Pd, and traces of PdO in the tetragonal structure. The grain size ranges between ~33.91 nm and 86.92 nm, and ~33.96 nm–86.94 nm, respectively. The bandgap energy (~3.7 eV) showed a shift towards higher energy linked with the quantum confinement effect. PL spectrum displayed eight emission bands at the UV–Vis range (~363–524 nm). The time-resolved (delayed) PL spectra were achieved ~13.0 ns after excitation. PL signals presented two blue emission bands in the Vis region, and they were associated with native crystalline defects. These blue emission bands are explained by the vacancies and interstitial defects, mostly associated with surfaces or grain boundaries. Likewise, emission bands were investigated, applying the Maxwell–Boltzmann theoretical model and contrasting them with the experimentally registered signals. The PL dependence on trap density and the surface recombination velocity is discussed in the light of calculated lifetimes, PdA1 ~2.04 cm3 s−1 and PdA2 ~9.65 cm3 s−1.

Deterministic quantum mechanics: The role of the Maxwell–Boltzmann distribution

AbstractTo 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.

Modeling the Viscoelastoplastic Deformation of Flexible Reinforced Plates with Weak Resistance to Transverse Shear

Abstract—Based on the time step algorithm, we have developed a mathematical model of the viscoelastoplastic deformation of flexible plates cross-reinforced in planes parallel to the middle plane. The strains of the plate composite components are assumed to be small and are decomposed into elastic and plastic components. The viscoelastic behavior of the composite materials is described by the Maxwell–Boltzmann body relations. The inelastic deformation is represented by the equations of the theory of a plastic flow with isotropic hardening. The normal stresses in the transverse direction are approximated linearly over the plate thickness. For this reason, the linear strains in the transverse direction and their rates are excluded from the governing equations for the composite components. The weakened resistance of fibrous plates to transverse shear is taken into account within the nonclassical Reddy bending theory. The geometric nonlinearity of the problem is considered in the Karman approximation. The formulated initial-boundary value problems are solved numerically with the application of an explicit cross-type scheme. A workaround is used to obtain a stable numerical scheme: the stresses in the Maxwell–Boltzmann viscoelastic relations at the current discrete time are expressed via the stress rate by the trapezoid formula with a backward step. We have investigated the elastoplastic and viscoelastoplastic bending dynamic behavior of relatively thick, orthogonally reinforced fiberglass rectangular plates subjected to blast loads. A change in the reinforcement structure is shown to lead to a change in the residual deflection of the structure. We demonstrate that the amplitude of reinforced-plate oscillations in the vicinity of the initial time exceeds significantly the residual deflection.

Covariant momentum map thermodynamics for parametrized field theories

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

Comments on “A New Derivation of the Law of the Junctions”

Contribution: This brief comment highlights some crucial assumptions behind the “law of the junction” that are overlooked by the above paper and argues that the proposed derivation is not actually a “new” derivation at all. Background: The “law of the junction” is one of the most significant and useful results within the field of solid-state devices. The above paper is likely to confuse readers, particularly those who are undergraduate electrical engineering students studying semiconductor device physics for the first time. This is especially so because of the abstract nature of the underlying quantum mechanics framework and solid-state physics models (subjects which the typical student at that level lacks a substantial background in) as well as the plethora of tedious equations in the curriculum. Research Questions: What core physical concepts are essential to a fundamental yet intuitive understanding of the law of the junction? Methodology: Several key features of how semiconductor junctions behave under bias are explained. References to well-known textbooks are provided where appropriate. Findings: The above paper’s primary mistake is its assertion that its derivation does not rely on the assumption of thermal equilibrium. However, the law of the junction is equivalent to a calculation of depletion-edge minority carrier concentrations using Maxwell–Boltzmann statistics—a distribution which only holds under thermal equilibrium conditions. More rigorously, in a nondegenerate semiconductor, Fermi–Dirac statistics (which governs electrons) reduces to Boltzmann statistics only when the electrochemical potential is spatially uniform, a condition equivalent to having no net flow of thermal energy—the very definition of thermal equilibrium.