Boltzmann Distribution Function Research Articles

Integration of Hertz–Knudsen–Schrage phase change in phase-field lattice Boltzmann method: Validation and parametric studies

The phase transition between liquid and vapor is of vital importance in daily life and industry. Given the importance of the lattice Boltzmann method (LBM), in particular the phase field method, in the simulation of two-phase flows, a robust LBM phase transition model is essential. This study introduces a novel approach by integrating the widely used, Hertz–Knudsen–Schrage (HKS) phase change rate into a conservative phase-field LBM. The phase-field and momentum equations are solved using the Boltzmann distribution function, whereas the energy equation is solved using the finite difference method. Once the necessary parameters for the calculation of the phase change rate are obtained, the corresponding source terms are incorporated into each equation. The model's validation is performed through a series of benchmark problems, including the one-dimensional Stefan problem, Nusselt's film condensation, bubble detachment, centered droplet evaporation, and sessile droplet evaporation. The results demonstrate favorable agreement between the LBM solution and analytical or empirical data. Furthermore, this study highlights the model's ability to approximate steady-state phenomena with minimal reliance on the phase change coefficient of the HKS theory. It also underscores the model's capacity to accurately capture transient phenomena by appropriately selecting values for this coefficient. In addition, parametric studies are conducted to investigate evaporation problems using the HKS theory for recognizing the effect of superheat, contact angle, and droplet size on evaporation. Finally, this model not only can detect trends and behaviors of phenomena but also can adapt empirical and analytical results with good agreement.

Photocatalytic Activity of BaAl2O4 for Water Purification.

Degradation of dyes under natural light sources is one of the most active research areas in basic science for greener technology. In this context, the photocatalytic activity of semiconductors has received massive attention in solving water treatment-related issues as these possess enormous potential for degrading organic impurities. Here, we report that barium aluminate (BaAl2O4, BAO), which has been extensively studied for photoluminescence applications, is found to be a highly potent candidate for photocatalytic activities. We have explored the degradation of dyes (meant for water purification) by using the photocatalytic properties of pure and Dy- and Yb-codoped BAO. Crystal structure, electron microscopy, and Raman analysis of the autocombustion-synthesized pure and codoped BAO samples revealed significant morphological changes such as increased particle size and stabilization of rod-like structures. UV-vis absorbance measurements confirm the presence of multiple bandgaps in the BAO samples, which is substantiated by X-ray absorption spectroscopy measurements. Photocatalytic degradation studies of methylene blue (MB) dye (with different catalyst concentrations, dopings, and MB dye concentrations) have been carried out by using BAO. The kinetics of the photocatalytic degradation measurements has been explained by the Boltzmann distribution function, and the fastest (in less than 40 min), with more than 99% degradation of MB impurity, is reported here for the first time in BAO compounds. Synthesized BAO samples show excellent cyclic stability, which is essential for their potential applications in environmental remediation. The trade-off between the enhancement of surface area and increased particle size is considered the key parameter for controlling the photocatalytic performance of the BAO catalyst after Dy and Yb codopings.

Maxwell–Boltzmann and Druyvesteyn Distribution Functions Expressing the Particle Velocity and the Energy in Sheath Plasmas

Maxwell–Boltzmann and Druyvesteyn Distribution Functions Expressing the Particle Velocity and the Energy in Sheath Plasmas

Inertial rotational diffusion and magnetic relaxation of the spin system in a strong magnetic field

Analytical expressions for the longitudinal and transverse correlation functions pertaining to inertial magnetic relaxation are obtained by expanding the deterministic magnetization trajectories into a Fourier series and averaging the result over all possible initial conditions applying the stationary Boltzmann distribution function. The longitudinal and transverse components of the magnetic susceptibility tensor are calculated for a system of noninteracting macrospins in a strong uniform external field using Bloch's phenomenological approach, which postulates the exponential evolution of the spin system towards equilibrium, and the Lorentz model of rotational diffusion of the magnetization vector. It is shown that the strength of the external field and the magnitude of the inertia parameters noticeably affect the shape of the susceptibility in the THz (nutation resonance) spectrum region. It is also demonstrated that the simple analytical Lorentz-type expression describes the main features of the complex susceptibility in the ferromagnetic resonance and nutation resonance regions.

The Temperature Dependent Boltzmann Equation with Reservoir of Radiation Photons

Usually, the temperature is introduced by the Boltzmann distribution function under the local equilibrium assumption. In this manuscript, we propose a temperature dependent Boltzmann equation, where the temperature is taken into account by the collision of transport particles and the particles in the reservoir, and we assume the reservoir remain in an equilibrium or local equilibrium state with an external temperature. Different from the usual Boltzmann equation, our temperature dependent Boltzmann equation obtains a new damping force which originates from the collision between transport particle and reservoir and is contrary to the external force applied to the system, meanwhile the relaxation time is also affected by the derivative of the damping force. In our study, the reservoir are consist of the radiations of photons, so the Compton scattering happens between the transport particle and the photon. Finally, we illustrate them by the example of one-dimensional transport process, the damping force and the non-equilibrium temperature defined by us are shown.

Nonextensive hydrodynamics of boost-invariant plasmas

We use quasiparticle anisotropic hydrodynamics to study the non-conformal and non-extensive dynamics of a system undergoing boost-invariant Bjorken expansion. To introduce nonextensivity, we use an underlying Tsallis distribution with a time-dependent nonextensivity parameter q. By taking moments of the quasiparticle Boltzmann equation in the relaxation-time approximation, we obtain dynamical equations which allow us to determine the time evolution of all microscopic parameters including q. We compare numerical solutions for bulk observables obtained using the nonextensive evolution with results obtained using quasiparticle anisotropic hydrodynamics with a Boltzmann distribution function (q rightarrow 1). We show that the evolution of the temperature, pressure ratio, and scaled energy density, are quite insensitive to which distribution function is assumed. However, we find significant differences in the early-time evolution of the bulk pressure which are observed for even small deviations from the Boltzmann distribution function. Finally, we discuss the existence of non-conformal hydrodynamic attractors for the longitudinal and transverse pressures, the bulk and shear viscous corrections, and the nonextensivity parameter q.

Far-from-equilibrium kinetic dynamics of λϕ4 theory in an expanding universe

We investigate the far-from-equilibrium behavior of the Boltzmann equation for a gas of massless scalar field particles with quartic (tree level) self-interactions ($\lambda \phi^4$) in Friedmann-Lemaitre-Robertson-Walker spacetime. Using a new covariant generating function for the moments of the Boltzmann distribution function, we analytically determine a subset of the spectrum and the corresponding eigenfunctions of the linearized Boltzmann collision operator. We show how the covariant generating function can be also used to find the exact equations for the moments in the full nonlinear regime. Different than the case of a ultrarelativistic gas of hard spheres (where the total cross section is constant), for $\lambda \phi^4$ the fact that the cross section decreases with energy implies that moments of arbitrarily high order directly couple to low order moments. Numerical solutions for the scalar field case are presented and compared to those found for a gas of hard spheres.

Propagation pattern for moment representation of the lattice Boltzmann method.

A propagation pattern for the moment representation of the regularized lattice Boltzmann method (LBM) in three dimensions is presented. Using effectively lossless compression, the simulation state is stored as a set of moments of the lattice Boltzmann distribution function, instead of the distribution function itself. An efficient cache-aware propagation pattern for this moment representation has the effect of substantially reducing both the storage and memory bandwidth required for LBM simulations. This paper extends recent work with the moment representation by expanding the performance analysis on central processing unit (CPU) architectures, considering how boundary conditions are implemented, and demonstrating the effectiveness of the moment representation on a graphics processing unit (GPU) architecture.

Semiclassical response of disordered conductors: Extrinsic carrier velocity and spin and field-corrected collision integral

The semiclassical equations of motion are widely used to describe carrier transport in conducting materials. Nevertheless, the substantial challenge of incorporating disorder systematically into the semiclassical model persists, leading to quantitative inaccuracies and occasionally erroneous predictions for the expectation values of physical observables. In the present work we provide a general prescription for reformulating the semiclassical equations of motion for carriers in disordered conductors by taking the quantum mechanical density matrix as the starting point. We focus on external electric fields, without magnetic fields, and spin-independent disorder. The density matrix approach allows averaging over impurity configurations, and the trace of the velocity operator with the disorder-averaged density matrix can be reinterpreted as the semiclassical velocity weighted by the Boltzmann distribution function. Through this rationale the well-known intrinsic group and anomalous velocities are trivially recovered, while we demonstrate the existence of an extrinsic interband velocity, namely a disorder correction to the semiclassical velocity of Bloch electrons, mediated by the interband matrix elements of the Berry connection. A similar correction is present in the non-equilibrium expectation value of the spin operator, contributing to spin-orbit torques. To obtain agreement with diagrammatic approaches the scattering term in the Boltzmann equation is corrected to first order in the electric field, and the Boltzmann equation is solved up to sub-leading order in the disorder potential. Our prescription ensures all vertex corrections present in diagrammatic treatments are taken into account, and to illustrate this we discuss model cases in topological insulators, including the anomalous Hall effect as well as spin-orbit torques.

Snow melting water infiltration mechanism of farmland freezing-thawing soil and determination of meltwater infiltration parameter in seasonal frozen soil areas

In seasonal frozen soil areas, much meltwater is formed during the thawing period, which easily causes snowmelt floods and waterlogging disasters. The meltwater simultaneously replenishes groundwater and produces surface runoff to provide water for crop roots. However, there are insufficient researches on the infiltration process of meltwater under day thaw and night freezing conditions. Meltwater infiltration process and estimation could provide an important reference for spring irrigation, meltwater disaster early warning systems and soil pollution research. In this study, the law of mass conservation was adopted for meltwater infiltration estimation under 15 and 30 cm snow covers. According to the hypothetical curve trend characteristics of meltwater infiltration rate, it is proposed to adopt Boltzmann distribution function and determine the curve parameters through experimental results, and the average daily maximum infiltration rate of frozen soil is 0.0622 cm/hour. During the thawing period, the soil moisture content changes periodically, and the period lasts approximately 10 h. We also put forward the concept of the maximum meltwater storage volume (Ϝ) of farmland meltwater, which provides a new analysis structure for meltwater runoff research. During the thawing periods, when the snow cover thickness in the early stage of thawing period exceeds Z, the amount of meltwater infiltration does not greatly change, and the maximum infiltration volume is between 3.15 and 5.25 cm. The average daily meltwater infiltration amounts to 0.15 cm.

Molecular dynamics lattice gas equilibrium distribution function for Lennard-Jones particles.

The molecular dynamics lattice gas (MDLG) method maps a molecular dynamics (MD) simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method (LBM) by taking a Boltzmann average over the MDLG. A key property of the LBM is the equilibrium distribution function, which was originally derived by assuming that the particle displacements in the MD simulation are Boltzmann distributed. However, we recently discovered that a single Gaussian distribution function is not sufficient to describe the particle displacements in a broad transition regime between free particles and particles undergoing many collisions in one time step. In a recent publication, we proposed a Poisson weighted sum of Gaussians which shows better agreement with the MD data. We derive a lattice Boltzmann equilibrium distribution function from the Poisson weighted sum of Gaussians model and compare it to a measured equilibrium distribution function from MD data and to an analytical approximation of the equilibrium distribution function from a single Gaussian probability distribution function. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Analysis of Agglomeration Kinetics of Magnetic Nanoparticles With Boltzmann Distribution of Energy Barrier

The agglomeration kinetics of magnetic nanoparticles under the magnetic field is studied by measuring the temporal change of the magnetic weight. The kinetics of stretched exponential is observed for the growth of the magnetic weight, which results from the distribution of the energy barrier. The agglomeration kinetics is simulated with the Boltzmann distribution function of the energy barrier. The Boltzmann distribution seems plausible for the agglomeration of nanoparticles where the diffusion of low energy barrier is dominant. The comparison with the experimental results indicates that the Boltzmann distribution is inappropriate for energy barrier distribution function even though the dynamics close to the stretched exponential is reproduced. The kinetic parameters deduced from the simulation results suggest that the complex structural relaxation of the agglomerate with higher energy barrier is more dominant than the diffusion in the growth of the magnetic weight.

Characteristics of negative pions produced in dC collisions at 4.2 GeV/c per nucleon

Temperature is considerably an important and commonly used parameter to study characteristics of matter formed during high-energy nuclear collisions. Experimental data from JINR and UrQMD (latest code 3.3p2) models’ simulations have been used to estimate the temperature and other properties of negative pions in collisions of deuteron with carbon nuclei at an incident momentum of 4.2 GeV/c. Transverse mass and transverse momentum spectra have been used to get the temperature of said particles, with the help of some fittings. These fittings are referred to as Hagedorn Thermodynamic and Boltzmann Distribution functions. Such functions or equations are used to describe the particles spectra. Temperature of negative pions has been found to be equal to 98 ± 2 and 114 ± 2 MeV in experimental and model, respectively, using Hagedorn function. Results from both experimental and model calculations have also been compared with each other and thus most reliable fitting function has been suggested. It is found that Hagedorn Thermodynamic function is the most reliable function to get pions’ temperature in said collision system at given incident momentum. Similarly temperature obtained in this research has been compared with results from other experiments in the world and worthy conclusions have been reached and reported.

Information processing and thermodynamic properties of microtubules

The Shannon entropy dependence on temperature, dipole moment and thermodynamic properties of microtubules (MTs) have been investigated using the Landau–Ginzburg phenomenological theory through continuum Boltzmann distribution function. By minimising the loss in energy, we found that there is a possibility that MTs formed from the heterodimers can process information over a long time at higher temperature. We also found that multiple heterodimers under the influence of dipole moment, has the tendency to process information whenever the amount of information stored or transferred decreases with increasing electronegativity of the system. We analyse the dynamic instability phenomenon that infinitely occurs in polymerisation and depolymerisation processes in MTs. Also, under physiological conditions, temperature dependence of thermodynamic properties was investigated and our results exhibited critical behaviour of heat capacity and chemical potential giving room for phase transitions around 302 K.

Characteristics of Positive Pions Produced in p + C Collisions at 4.2 GeV/c

Experimental data from JINR and UrQMD (latest code 3.3p2) model simulations have been used to estimate temperature and other properties of positive pions in collisions of protons with carbon nuclei at incident momentum of 4.2 GeV/c. Transverse mass and transverse momentum spectra have been used to get temperature of said particles, with the help of some fitting equations. These equations are referred as Hagedorn thermodynamic and Boltzmann distribution functions. Such functions or equations are used to describe particles spectra. Temperature of positive pions has been found to be equal to 97 ± 7 and 98 ± 12 MeV in experimental and model, respectively, using Hagedorn function. Results from both experimental and model calculations have also been compared with each other, and thus, most reliable fitting function has been suggested. It is found that Hagedorn thermodynamic function is the best reliable function to get pions’ temperature in said collision system at given momentum. Similarly, temperature obtained in this research has been compared with results from other experiments and worthy conclusions have been reached and reported.

Non-Boltzmann vibrational energy distributions and coupling to dissociation rate.

In this article, we propose a generalized model for nonequilibrium vibrational energy distribution functions. The model can be used, in place of equilibrium (Boltzmann) distribution functions, when deriving reaction rate constants for high-temperature nonequilibrium flows. The distribution model is derived based on the recent ab initio calculations, carried out using potential energy surfaces developed using accurate computational quantum chemistry techniques for the purpose of studying air chemistry at high temperatures. Immediately behind a strong shock wave, the vibrational energy distribution is non-Boltzmann. Specifically, as the gas internal energy rapidly excites to a high temperature, overpopulation of the high-energy tail (relative to a corresponding Boltzmann distribution) is observed in ab initio simulations. As the gas excites further and begins to dissociate, a depletion of the high-energy tail is observed, during a time-invariant quasi-steady state. Since the probability of dissociation is exponentially related to the vibrational energy of the dissociating molecule, the overall dissociation rate is sensitive to the populations of these high vibrational energy states. The non-Boltzmann effects captured by the new model either enhance or reduce the dissociation rate relative to that obtained assuming a Boltzmann distribution. This article proposes a simple model that is demonstrated to reproduce these non-Boltzmann effects quantitatively when compared to ab initio simulations.

Analysis of positive pions obtained in d + C collisions at 4.2AGeV/c

Temperature is an important and commonly used parameter among others to study properties of matter created during high energy collisions of nuclei. Experimental data from JINR and UrQMD (version 3.3p2) model simulations have been used to estimate temperature and other properties of positive pions in collisions of deuteron with carbon nuclei at incident momentum of 4.2[Formula: see text]GeV/[Formula: see text] in this paper. Transverse mass and transverse momentum spectra have been used to get inverse slope parameter/temperature of said particles, with the help of some fitting equations. These equations are referred as Hagedorn Thermodynamic and Boltzmann Distribution functions. Such functions or equations are used to describe particles spectra. Temperature of positive pions has been found to be equal to [Formula: see text] and [Formula: see text][Formula: see text]MeV in experimental and model, respectively, using Hagedorn function. Results from both experimental and model calculations have also been compared with each other and thus most reliable fitting function has been suggested. It is found that Hagedorn Thermodynamic function is the most reliable function to get pions’ temperature in said collision system at given momentum. Similarly, results obtained in this paper have been compared with results from other experiments in the world and worthy conclusions have been reached and reported.

Spectral temperature of π− mesons in proton–carbon interactions at 4.2 GeV/c in the framework of UrQMD model

Transverse momentum [Formula: see text] spectra of [Formula: see text] mesons calculated using ultra-relativistic quantum molecular dynamic (UrQMD) model (Latest version 3.3-p2) simulations have been compared with [Formula: see text] spectra of [Formula: see text] mesons, obtained experimentally in interactions of protons beam with carbon nuclei (propane as target) at momentum of 4.2 GeV/c. Spectral temperatures of negative pions obtained in experimental and UrQMD model simulated interactions of protons beam with carbon nuclei have been calculated by fitting both spectra with four different fitting functions, i.e. Hagedorn thermodynamic, Boltzmann distribution, Gaussian and exponential functions. These functions are used commonly for describing hadron spectra and their spectral temperatures. Hagedorn thermodynamic function has been recommended as the most suitable function to extract the temperature of negative pions at above momentum among these four functions.

Modelling respiration rate of dragon fruit as a function of gas composition and temperature

The application of controlled atmosphere and modified atmosphere packaging to extend the shelf life of dragon fruit requires insight into the respiration behaviour of the fruit under different storage conditions. To this end, this study focused on characterising the respiration kinetics of dragon fruit using a dynamic closed depletion system. An integrated data analysis was performed using a Michaelis-Menten based respiration model extended with an Arrhenius equation to account for the effects of O2, CO2 and temperature. Iterative analysis of the data using different model versions eventually revealed that the classical Arrhenius equation did not suffice to explain the observed respiration behaviour over the whole temperature range (2 °C–35 °C). The experimental data consistently showed lower respiration at 35 °C as compared to what was predicted by the classical Arrhenius equation. To overcome this problem, a modified Arrhenius equation, incorporating the Boltzmann distribution function, was successfully introduced to account for this observed temperature effect, predicting a maximal respiration rate at 33.9 °C. The parameters of the final model were estimated with high precision, explaining 98 % the data variance. Additionally, it was shown that CO2 had an uncompetitive inhibition on oxidative respiration, while O2 concentration affected both oxidative respiration and fermentation. This is the first documented example of postharvest fruit respiration not following the classical Arrhenius’ law at what could still be considered near field temperatures.

Influence of ferroelectricity and dipole turning onelectrocaloric effect

In the three dimensional Landau-Devonshire ferroelectric theory, polarization is a variable of the Gibbs free energy function. The first derivative of the Gibbs free energy to polarization can be used to derive equilibrium values of the polarization and the energy level. When an external electric field ( E ) is applied, the derived polarization and energy level change according to angular degree between the polarization and the E , which can be derived by solving the Gibbs free energy function associated with the electric field. In the E direction the polarization increases with decreased Gibbs free energy level, in the anti- E direction the polarization decreases with increased Gibbs free energy level, and in the directions perpendicular to the E direction, polarization and the Gibbs free energy remain unchanged. If change in a Gibbs free energy level happens, the proportion of the dipole that forms the polarization will also change accordingly, which is derived by solving the Boltzmann distribution function via taking the Gibbs free energy as equilibrium energy level. The result demonstrates that proportion of the dipoles in the E direction increases, and proportions of the dipoles in other directions decrease. The changes in proportions of the dipoles between different directions can be described as turning of dipoles. This turning of dipoles to the E direction causes decrease in the entropy of dipole alignment, and the change in the entropy produces exothermal effect in adiabatic condition. On another aspect, a coupling effect will happen on the parallel aligned neighbor dipoles in the E direction by the action of the electric field to form reversible ferroelectric domains. When the E withdraws, the domains decouple. The coupling or decoupling of the reversible domains corresponds to exothermal effect or endothermal effect, respectively. As the coupling effect of dipoles causes electric hysteresis loop, the coupled thermal effects can be deduced from the polarization of the hysteresis loop which is associated with temperature and the electric field. The result shows that the change in entropy results in an exothermal peak, which shifts to higher temperature with the increase of E , even across the Curie temperature, under higher electric field. The coupling effect of the parallel aligned dipoles produces exothermal effect, decreasing with increase of temperature continuously. The present theoretic results can explain the relevant experimental results by numerical simulation of the derived equations.