Boltzmann Equation Research Articles

On a pointwise inequality for even Legendre polynomials in high dimensional spheres

We present a pointwise inequality for adjacent even Legendre polynomials in high dimensional spheres featuring the effect of spectral gaps. This improves a recent result of Imbert, Silvestre and Villani that is crucially used in their study of the Fisher information for the Boltzmann equation.

Research on propagation characteristics of an electromagnetic wave in magnetized fully ionized dusty plasma

When some spacecraft fly at high speed in near-earth space, the dusty plasma is attached to the surface of the spacecraft due to the intense friction between the spacecraft and the space. The dusty plasma attached to the aircraft hinders the propagation of electromagnetic (EM) waves and affects the flight safety of the pilot. In this paper, we replace the collision term of the general Boltzmann equation with the Fokker-Planck-Landau collision term and obtain the general correlation expression of the electron distribution function in the fully ionized dusty plasma. The dielectric relation of EM waves in magnetized fully ionized dusty plasma is calculated considering the impact effect and charging effect of dust particles in the case of an external magnetic field. The reflection coefficient (R) and transmission coefficient (T) of the EM wave in magnetized fully ionized dusty plasma are calculated by a scattering matrix method. The effects of applied magnetic field intensity and dust particle parameters on the propagation characteristics of EM waves in magnetized fully ionized dusty plasma are discussed by simulation. In addition, the propagation characteristics of EM waves in ordinary plasma and magnetized fully ionized dusty plasma are compared. The results show that with the increase in the additional magnetic field, the reflection of the EM wave in the fully ionized dusty plasma decreases and the transmission increases. In the 0–100 GHz band, dust particles block EM waves from penetrating fully ionized dusty plasma. Other dusty plasma parameters also affect the propagation characteristics of EM waves to varying degrees. These results provide a theoretical reference for solving the communication related problems of high-speed aircraft.

An investigation of firm size distributions involving the growth functions

Following the ideas of prospect theory, a class of growth functions is used to characterize deterministic variations of firm size, which portrays the asymmetric efforts of the firm to achieve the desired size. Considering the differences in the ability of different firm size to cope with uncertainties, the Boltzmann equation for the evolution of firm size is constructed. Utilizing a suitable scaling limit, the Fokker–Planck equation is acquired and its explicit steady-state solution is derived. Our results illustrate that different choices of parameters in the growth function lead to various statistical laws for firm size, such as the Amoroso distribution, the lognormal distribution and Zipf’s law. Under certain conditions, inequality for the distribution of firm size decreases as firm size increases. The numerical analyses are presented to illustrate our results.

Thermal conductivity predictions in monolayer MoSi2N4: Integrating neural network potentials with phonon scattering analysis

Two-dimensional (2D) materials, known for their exceptional thermal conductivity and mechanical flexibility, have emerged as promising candidates for thermal management applications. Recently, increasing attention has been given to investigating the lattice thermal conductivity of these materials. While traditional methods combining density functional theory (DFT) with the Boltzmann transport equation (BTE) can produce accurate results, these approaches are computationally expensive and demand substantial resources. To address this challenge, we employed machine learning to successfully model the interatomic potential of monolayer MoSi2N4. This neural network potential (NNP), combined with BTE, facilitated the theoretical calculation of MoSi2N4′s thermal conductivity. Using NNP, we efficiently and accurately calculated the lattice thermal conductivity of MoSi2N4, highlighting the importance of selecting an appropriate interaction cutoff distance to ensure calculation accuracy. Furthermore, using this NNP, we investigated how four-phonon scattering influences the heat conduction properties of MoSi2N4, thereby strengthening our comprehension of phonon scattering dynamics. This study not only optimized computational efficiency but also provided fresh perspectives on the heat transfer mechanisms in complex 2D materials.

Efficient spectrally-resolved electron transport for thermal property prediction

Predicting thermal conductivity from the transport of electrons is a complex challenge which must be addressed for the design and production of nano- and microscale devices. In this work, we describe our progress in simulating thermal electron transport in silicon for the purposes of thermal conductivity prediction. We describe a new approach to solve the Boltzmann transport equation for electrons, using the self-adjoint angular flux formulation coupled with a novel approach in computing electron temperature and Fermi energy which does not require the combination of linear “inner” and nonlinear “outer” iterations. The electron Boltzmann transport equation is discretized in space by the continuous finite element method, and in angle by discrete ordinates, using the Multiphysics Object Oriented Simulation Environment, implemented in the radiation transport code Griffin. We rely on density functional theory calculations to provide material properties such as electron density of states, energy, wavevector, mean free path, and more. Our method couples the discrete electron groups through temperature and Fermi energy, to simulate thermal electron transport (in the absence of electric fields) for the prediction of thermal conductivity, thermal and electrical flux, and heat capacity. We show effective thermal conductivity and temperature results for different sized 2D slabs of Si at 300 K, which agree with other studies in the literature. Additionally, we report Fermi energy and efficiency of this method using the generalized minimal residual method.

Unconventional Thermal Rectification in Assemblies of Supertetrahedral T3-Type Sn4In6Se20 Clusters.

Going beyond the conventional design paradigm with atoms as building blocks, we propose the concept of cluster-assembled thermal rectifiers comprising metal chalcogenide supertetrahedral clusters. Different from the experimentally reported T4,∞ and T5,∞, for the first time we assemble T3-Sn4In6Se20 clusters into a stable T3,∞ framework without needing extra ions, based on which the thermal rectification (TR) effect is explored using machine-learning molecular dynamics and the mode-resolved phonon Boltzmann transport equation. The tetrahedron-shaped cluster assembly serves as a novel TR switch, where the open state shows an outstanding TR efficiency (∼40%) arising from the asymmetric lateral confinement due to not only the phonon particle behavior but also the phonon wave nature. The prism-shaped assembly has a symmetric skeleton but with asymmetric surface roughness induced by clusters, thus exhibiting an unusual TR effect, distinguished from atom-based symmetric systems with atomically flat surfaces. Our findings demonstrate the unique potential of cluster assemblies for unconventional thermal rectification.

Inspecting the structural stability, magneto-opto-electronic, and transport characteristics of half-metallic ferromagnets double perovskite oxide (Sr2MoSbO6): A DFT study

The structural, elastic, mechanical, electronic, thermoelectric, and magnetic properties of the double perovskite Sr2MoSbO6 have investigated in this manuscript using Perdew-Burke-Ernzerhof Generalized Gradient Approximation (PBE-GGA) with an enhanced Trans Blaha modified Becke Johnson potential (TB-mBJ) approach. Through electro-magnetic and elastic exploration, we have determined that this compound is semiconductor, ferromagnetic, and brittle. Strong hybridisation between the Mo and Sr-d orbitals was seen in the Density of states (DOS) results, which, according to their relative quantities, supports the two states' ionic nature. The Mo atoms contribute significantly to the overall magnetic moment, which is 3.0 μB in total. The semiconducting nature of Sr2MoSbO6 is confirmed by the calculation of the overall electronic parameters. Calculations of thermodynamic parameters for temperature ranges of 0–1200 K and pressure ranges of roughly 0–30 GPa show good agreement between theoretical and experimental data. The DFT Boltzmann transport equations have been used to compute thermoelectric properties in relation to temperature and chemical potential. The p-type character of Sr2MoSbO6 is identified by positive values of the Seebeck coefficient. The power factor (PF), Seebeck coefficient (S), figure of merit (ZT), electrical conductivity, and lattice thermal conductivity were also calculated. It was discovered that this perovskite had a merit figure that was almost equal to one, a very high Seebeck coefficient, and strong electrical conductivity—all of which are consistent with its semiconductor nature. These findings suggest a substance with a great deal of promise for thermoelectrical uses. The results are taken into consideration for future experiments and may be future candidates for spintronics applications.

Rigorous hydrodynamics from linear Boltzmann equations and viscosity-capillarity balance

Exact closure for hydrodynamic variables is rigorously derived from the linear Boltzmann kinetic equation. Our approach, based on spectral theory, structural properties of eigenvectors, and the theory of slow manifolds, allows us to define a unique, optimal reduction in phase space close to equilibrium. The hydrodynamically constrained system induces a modification of entropy that ensures pure dissipation on the hydrodynamic manifold, which is interpreted as a nonlocal variant of Korteweg's theory of viscosity-capillarity balance. The rigorous hydrodynamic equations are exemplified on the Knudsen minimum paradox in a channel flow. Published by the American Physical Society 2024

The Harnack inequality fails for nonlocal kinetic equations

We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.

Influence of polymorphism on the lattice thermal conductivity of Ga2O3

In this paper, the lattice thermal conductivity of Ga2O3 in its β, α, ɛ(κ), and γ phase is systematically investigated based on the first principles calculation and iterative approaches to solve the phonon Boltzmann equation. The results indicate that the crystal microstructure of Ga2O3 has a significant effect on the lattice thermal conductivity. In addition, the results also find that γ-Ga2O3 has an ultralow lattice thermal conductivity within the temperature range from 50 to 700 K. As for γ-Ga2O3, the obtained lattice thermal conductivity at room temperature (300 K) is 0.1189 W/(m K) along the [100] and [010] directions, and 0.1159 W/(m K) along the [001] direction. The lattice thermal conductivity exhibits the following order: γ-Ga2O3 ≪ ɛ(κ)-Ga2O3 < α-Ga2O3 < β-Ga2O3. The disruptive effect of Ga3+ cation vacancies on the spinel structure's symmetry is responsible for the ultralow lattice thermal conductivity observed in γ-Ga2O3. This disruption increases the complexity of the lattice and hampers the propagation and scattering of phonons. Another contributing factor is the presence of weak chemical bonding, which intensifies the oscillation of Ga atoms. The results of this study have significant implications for further investigating the factors influencing the thermal conductivity of Ga2O3 and developing thermoelectric materials.

Effective phonon dispersion and low field transport in AlxGa1−xN alloys using supercells: An ab initio approach

To investigate the transport properties in random alloys, it is important to model the alloy disorder using supercells. Although computationally expensive, the local disorder in the system is accurately captured as translational symmetry that is imposed on the system over larger length scales. Additionally, in supercells, the error introduced by self-image interaction between the impurities is reduced. In this work, we have investigated the Effective Phonon Dispersion (EPD) and transport properties, from first principle calculations using supercells in AlxGa1−xN alloy systems. Using an in-house developed code for phonon-band unfolding, the EPD of AlGaN is obtained and the individual phonon modes are identified with good agreement with experimental values. Moreover, we report an in-house developed method to calculate low-field transport properties directly from supercells without phonon band unfolding. First, to validate our methods, we have solved the Boltzmann transport equation using Rode’s method to compare the phonon limited mobility in the 4 atom GaN primitive cell and 12 atom GaN supercell. Using the same technique, we have investigated the low field transport in random AlxGa1−xN alloy systems. The quadrupole interaction is included for transport properties of GaN and AlN to accurately capture the physics in these materials. Our calculations show that along with alloy scattering, electron–phonon scattering may also play an important role at room temperature and high-temperature device operation. This technique opens up the path for calculating phonon-limited transport properties in random alloy systems.

Revealing the correlation between asymmetric structure and low thermal conductivity in Janus-graphene via machine learning force constant potential

Understanding the fundamental link between structure and functionalization is crucial for designing and optimizing functional materials, since different structural configurations could trigger materials to demonstrate diverse physical and chemical properties. However, the correlation between crystal structure and thermal conductivity (κ) remains unclear. In this study, taking two-dimensional (2D) carbon allotropes Janus-graphene and graphene as study cases, we utilize phonon Boltzmann transport equation combined with machine learning potential to thoroughly investigate the complex folding structure of pure sp2 hybridized Janus-graphene from the perspective of crystal structure, phonon modal resolved thermal transport, and atomic interactions, with the goal of identifying the underlying relationship between 2D geometry and κ. The results reveal that the folded structure in Janus-graphene causes strong symmetry breaking, significantly reduces phonon group velocities, increases phonon–phonon scattering, and ultimately leads to low κ. These findings enhance our understanding of how atomic structure folding affects thermal transport and the relationship between structure and functionalization.

A theoretical derivation of slip boundary conditions based on the Cercignani–Lampis–Lord scattering model

To characterize fluid flow in the slip regime, the use of Navier–Stokes–Fourier (NSF) equations with slip boundary conditions is prevalent. This trend underscores the necessity of developing reliable and accurate slip boundary conditions. According to kinetic theory, slip behaviours are intrinsically linked to the gas scattering processes at the surface. The widely used Maxwell scattering model, which employs a single accommodation coefficient to describe gas scattering processes, reveals its limitations when the difference between accommodation coefficients in the tangential and normal directions becomes significant. In this work, we provide a derivation of velocity slip and temperature jump boundary conditions based on the Cercignani–Lampis–Lord scattering model, which applies two independent accommodation coefficients to describe the gas scattering process. A Knudsen layer correction term is introduced to account for the impact of the surface on the velocity distribution function, which is associated with the scattering model. The governing equation of the correction term is established based on the linearized Boltzmann equation. Additionally, two moments are derived to capture the collision effect in the Knudsen layer: a conserving moment of collision invariants, and an approximate higher-order conserving moment. These moments are then employed to determine the coefficients in the correction term. We demonstrate that the derived slip coefficients align closely with numerical results obtained by solving the Boltzmann equation in the Knudsen layer. Besides, we apply the derived slip boundary conditions within the framework of the NSF equations, yielding numerical results that exhibit excellent consistency with those obtained through molecular-level simulations.

Tl2XY (X, Y = S, Se) monolayers with low lattice thermal conductivity and high thermoelectric performance by full-band approach with four scattering mechanisms

Abstract The low lattice thermal conductivity and high thermoelectric performance of the Janus Tl2SSe monolayer in the temperature region of 300–700 K are identified based on the thermoelectric properties of the Tl2S2, Tl2SSe, and Tl2Se2 monolayers. The transport coefficients for carrier concentrations and temperatures are obtained by solving the linearized Boltzmann transport equation in a full-band electronic structure. Four scattering mechanisms of acoustic deformation potential, optical deformation potential, polar optical phonon, and ionized impurity scatterings are considered. The ionized impurity scattering is recognized as the most important. The lattice thermal conductivity of the Janus Tl2SSe monolayer is substantially smaller than those of the Tl2S2, Tl2Se2 monolayers with higher symmetry. Moreover, we find that the Janus structures of the Tl2SSe monolayer increase the dielectronic constants and enhance the polar optical phonon scattering, then reduce the power factor to some extent. Therefore, the lattice thermal conductivity actually couples with the transport coefficient and cannot be individually regulated as is usually assumed. However, the ZT value of the Tl2SSe monolayer can still reach 1.77 at 700 K even if the intrinsic concentration and the bipolar effect are included. Therefore, the Tl2SSe monolayer is expected to be a promising candidate for thermoelectric materials.

Well/Ill-Posedness of the Boltzmann Equation with Soft Potential

Well/Ill-Posedness of the Boltzmann Equation with Soft Potential

Optimizing thermoelectric properties of CoTiP half-Heusler via doping with Br-, Se- and Ge atoms using first principle study

The optimization of thermoelectric materials is crucial for advancing energy conversion technologies. This study explores the electrical and thermoelectric properties of Br-, Ge-, and Se-doped CoTiP half-Heusler compounds using the plane-augmented-wave (PAW) method based on Density Functional Theory (DFT) alongside the semiclassical Boltzmann transport equation (BTE) and Debye-Callaway approximation. While previous research has focused on various doping strategies to enhance thermoelectric performance, specific impacts of Br, Ge, and Se doping on the electronic structure of CoTiP remain unexplored. Our analysis reveals that Ge-doped CoTiP exhibits the largest band gap energy of 1.2597 eV, followed by Se- and Br-doped structures with band gaps of 0.8064 eV and 0.678 eV, respectively. The Fermi level shifts towards the conduction band for both Br- and Se-doped alloys while shifting towards the valence band for Ge-doped alloys. Upon doping, we observe significant enhancements in the Seebeck coefficient and electrical conductivity. Power factor (S2σ) enhancements range from 0.01611 W/m K2 for CoTiP0.875Br0.125, 0.03445 W/m K2 for CoTiP0.875Se0.125 and finally, 0.04191 W/m K2 for CoTiP0.875Ge0.125, surpassing undoped material values by up to 93 %. Finally, the optimal value of figure of merit (ZT) increases to 0.65, 0.57, and 0.2 at 900 K, achieved by doping Ge, Se and Br, respectively, at the P site, with performance gain about 92 %. Hence, doping has optimized the thermoelectric performance of the CoTiP half-Heusler.

Freezing-in cannibal dark sectors

Self-Interacting Dark Matter models can successfully explain dark matter (DM) production through interactions confined within the dark sector. However, they often lack measurable experimental signals due to their secluded nature. Including a feeble interaction with the visible sector through a Higgs portal leads not only to potential detection avenues and richer thermal production dynamics, but also to a possible explanation of the initial dark sector population through the freeze-in mechanism. In this work we study, by solving the full system of coupled Boltzmann equations for the number densities and temperatures of all the involved states, three scenarios of this type where the DM is: a real scalar with broken ℤ2, a complex scalar with unbroken ℤ3, and a ℤ3 scalar with an additional scalar mediator. All of these models have viable dark matter candidates in a cannibal phase while having different detection profiles. We show that cosmological bounds can be either exacerbated or evaded by changing the dark sector interactions, leading to potential signatures in long-lived particle and indirect detection experiments.

The global well-posedness of the relativistic Boltzmann equation with hard potentials and diffuse reflection boundary condition in bounded domains

The global well-posedness of the relativistic Boltzmann equation with hard potentials and diffuse reflection boundary condition in bounded domains

Diffusive Limit of the Boltzmann Equation in Bounded Domains

Diffusive Limit of the Boltzmann Equation in Bounded Domains

Thermoelectric properties of XX- and XY-stacked GeS/GeSe van der Waals heterostructures from DFT and BTP calculations.

This study utilizes density functional theory (DFT) and the Boltzmann transport equation (BTE) to investigate the structural, electronic, and thermoelectric properties of germanium sulfide (GeS) and germanium selenide (GeSe) monolayers, along with their van der Waals (vdW) heterostructures. We analyzed XX-stacked and XY-stacked configurations, where the XX configuration features direct atomic stacking, while the XY configuration exhibits staggered stacking. Our first-principles calculations indicate that the formation of GeS/GeSe heterostructures results in a reduction of bandgaps compared to their bulk and monolayer counterparts, yielding bandgap values of 0.91 eV for the XX configuration and 0.84 eV for the XY configuration. Stability assessments reveal that the XY configuration is more stable, demonstrating a lattice thermal conductivity of 15.21 W/mK compared to 17.95 W/mK for the XX configuration T 300 K. The thermoelectric properties were systematically evaluated across a temperature range of 300-800 K, revealing high Seebeck coefficients of 1.51 mV/K for the XX heterostructure and 1.39 mV/K for the XY heterostructure. reflecting their excellent charge transport capabilities. Notably, the figure of merit (ZT) at 800 K was calculated to be 0.90 for the XX configuration and 1.01 for the XY configuration, underscoring the superior thermoelectric performance of the XY heterostructure. These findings contribute to a comprehensive understanding of 2D GeS/GeSe heterostructures for thermoelectric applications and provide a solid foundation for future research and technological advancements in this domain.