Boltzmann Equation Research Articles

Systematic moment expansion for electroweak baryogenesis

We present a systematic moment expansion for solving the semiclassical Boltzmann equations for electroweak baryogenesis. The expansion is developed in powers of adiabatic coordinate velocity, and it is used for computing the CP-violating seed asymmetry at the front of the phase transition wall, that sources the eventual baryon asymmetry of the universe (BAU). We implement the method in a benchmark model, with a CP-violating mass arising from a dimension-5 operator coupling the fermion to a singlet scalar field. We find that the higher moment calculations yield a BAU that can significantly differ from the commonly used two-moment approximation. We discuss in detail the underlying approximations in the moment method and propose a new truncation scheme for the expansion, that appears to give more numerically robust results than the previous schemes.

Resonant reheating

We investigate a novel reheating scenario proceeding through s-channel inflaton annihilation, mediated by a massive scalar. If the inflaton ϕ oscillates around the minimum of a monomial potential ∝ ϕ n, we reveal the emergence of resonance phenomena originating from the dynamic evolution of the inflaton mass for n>2. Consequently, a resonance appears in both the radiation and the temperature evolution during the reheating process. By solving the coupled Boltzmann equations, we present solutions for radiation and temperature. We find non-trivial temperature characteristics during reheating, depending on the value of n and the masses of the inflaton and mediator. Some phenomenological aspects of the model are explored. As a concrete example, we show that the same mediator participates in the genesis of dark matter, modifying the standard freeze-in dynamics. In addition, we demonstrate that the resonant reheating scenario could be tested by next-generation low- and high-frequency gravitational wave detectors.

Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

Nano Thermal Simulation of Graphene Field Effect Transistor Based on Ballistic Diffusive Model

This paper investigates temperature evolution in graphene field-effect transistors. We present a refined Ballistic-diffusive equation model (BDE) tailored for scrutinizing the phonon transport within GNRFET transistors. Furthermore, we examine the influence of the channel length on thermal characteristics. COMSOL Multiphysics is employed to compute the phonon transport and temperature distribution in nano-GNRFET. Our results reveal that the phonon transport anticipated by the suggested BDE model closely corresponds to that derived from the Boltzmann transport equation. The proposed model proficiently gauges the thermal performance of GNRFET, with the outcomes suggesting that the characteristic length of the GNR has a negligible impact on the temperature increase.

First principle investigation of structural, optoelectronic and thermoelectric properties of lead-free Cs2AgAuX6 (X= Br, I) for energy harvesting applications

In this research work, the properties of double perovskites (DPs) are investigated by using the full potential linear augmented plane-wave (FP-LAPW) technique. To compute structural characteristics, we used the generalized gradient approximation (GGA) functional. On the other hand, the optoelectronic and thermoelectric properties of Cs2AgAuX6 (X = Br, I), were calculated using the modified Becke and Johnson (mBJ) potential functional. According to our work, these materials have band gaps of 1.13 eV for X = I and 1.24 eV for X = Br. These materials' optical properties have been investigated through the use of reflectivity, absorption, refractive index, and dielectric constants. With the highest transition values versus photon energy in the visible spectrum, the optical properties point to the potential application of these double perovskites in solar cells. In addition, our studies of the transport properties using the Boltzmann transport equation suggest that our understudy DPs is optimal for thermoelectric uses.

Inverse Modeling of Heterogeneous Structures in Electron Probe Microanalysis.

Electron probe microanalysis (EPMA) is a powerful tool for chemical characterization of materials on a microscopic scale. However, EPMA has the drawback that its information volume has a spatial extent of some 100 nm to a few µm. With the introduction of new electron sources, i.e., Schottky Thermal Field and Cold Field Emitter, where the electron beam is focused down to a few nm, measurements can be nowadays performed on the sub-micrometer scale. The goal of the work is to reveal the chemical composition of structures smaller than the excitation volume. New strategies are presented where the acquisition is performed at different positions on the sample and as a scan across a fine structure by using one or more single beam energies. Besides the well-known Monte-Carlo simulation, a deterministic model is also used. The deterministic model is based on moment equations of the Boltzmann equation. Inverse modeling is presented for several case studies. Due to the highly complex nonlinearity of the inverse model, an ill-posed and well-posed problem is shown as well. Finally, the method is extended to reconstruct 2D structures, i.e., rectangular shaped particles, with heterogeneous composition on lateral and depth scale.

Dielectric Boundary Force for the Minimization of the Mean-Field Electrostatic Free-Energy Functional Including the Ionic Size Effect

The generalized Poisson–Boltzmann equation with the uniform ionic size is investigated. In mean-field theory, the ionic size effect is added by modifying the entropy effect of solvent molecules in the electrostatic free-energy functional of ion concentrations. Taking minimization of the mean-field electrostatic free-energy functional including the ionic size effect with respect to the ion concentrations, the minimum free-energy is obtained (see (4.6) in Li [(2009a) “Continuum electrostatics for ionic solutions with non-uniform ionic sizes,” Nonlinearity 22(4), 811–833]. Maximizing the minimum free-energy with respect to the electrostatic potential determines the generalized Poisson–Boltzmann equation. By the definition of shape derivative, the first variation of this minimum free-energy with respect to variation of the dielectric boundary is derived. Thus, the dielectric boundary force of an ionic solution with multiple ionic species with the uniform ionic size is obtained.

Spectral asymptotics and metastability for the linear relaxation Boltzmann equation

We consider the linear relaxation Boltzmann equation in a semiclassical framework. We construct a family of sharp quasimodes for the associated operator which yields sharp spectral asymptotics for its small spectrum in the low temperature regime. We deduce some information on the long time behavior of the solutions with a sharp estimate on the return to equilibrium as well as a quantitative metastability result. The main novelty is that the collision operator is a pseudo-differential operator in the critical class S^{1/2} and that its action on the Gaussian quasimodes yields a superposition of exponentials.

Evolution of the Secondary Electron Spectrum during Cosmic-Ray Discharge in the Universe

We recently found that streaming cosmic rays (CRs) induce a resistive electric field that can accelerate secondary electrons produced by CR ionization. In this work, we study the evolution of the energy spectrum of secondary electrons by numerically solving the one-dimensional Boltzmann equation and Ohm’s law. We show that the accelerated secondary electrons further ionize a gas, that is, an electron avalanche occurs, resulting in increased ionization and excitation of the gas. Although the resistive electric field becomes weaker than the one before the CR discharge, the weak resistive electric field weakly accelerates the secondary electrons. The quasi-steady state is almost independent of the initial resistive electric field but depends on the electron fraction in the gas. The resistive electric field in the quasi-steady state is larger for the higher electron fraction, which makes the number of secondary electrons that can ionize the gas larger, resulting in a higher ionization rate. The CR discharge could explain the high ionization rate that is observed in some molecular clouds.

Derivation of a Boltzmann Equation with Higher-Order Collisions from a Generalized Kac Model

Derivation of a Boltzmann Equation with Higher-Order Collisions from a Generalized Kac Model

Transient absorption of warm dense matter created by an X-ray free-electron laser

Warm dense matter is at the boundary between a plasma and a condensed phase and plays a role in astrophysics, planetary science and inertial confinement fusion research. However, its electronic structure and ionic structure upon irradiation with strong laser pulses remain poorly understood. Here, we use an intense and ultrafast X-ray free-electron laser pulse to simultaneously create and characterize warm dense copper using L-edge X-ray absorption spectroscopy over a large irradiation intensity range. Below a pulse intensity of 1015 W cm−2, an absorption peak below the L edge appears, originating from transient depletion of the 3d band. This peak shifts to lower energy with increasing intensity, indicating the movement of the 3d band upon strong X-ray excitation. At higher intensities, substantial ionization and collisions lead to the transition from reverse saturable absorption to saturable absorption of the X-ray free-electron laser pulse, two nonlinear effects that hold promise for X-ray pulse-shaping. We employ theoretical calculations that combine a model based on kinetic Boltzmann equations with finite-temperature real-space density-functional theory to interpret these observations. The results can be used to benchmark non-equilibrium models of electronic structure in warm dense matter.

Compressible fluid limit for smooth solutions to the Landau equation

Although the compressible fluid limit of the Boltzmann equation with cutoff has been extensively investigated in Caflisch [Comm. Pure Appl. Math. 33 (1980), 651–666] and Guo, Jang, and Jiang [Comm. Pure Appl. Math. 63 (2010), 337–361], obtaining analogous results in the case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, still remains largely open, essentially due to the velocity diffusion effect of the collision operator such that L^{\infty} estimates are hard to obtain without using Sobolev embeddings. In this paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number \varepsilon>0 , yielding the O(\varepsilon) convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians, together with techniques for viscous compressible fluid and properties of Burnett functions, we design an \varepsilon -dependent energy functional to capture the dissipation in the compressible fluid limit with the feature that only the highest-order derivatives are most singular.

A second-order particle Fokker-Planck model for rarefied gas flows

The direct simulation Monte Carlo (DSMC) method has become a powerful tool for studying rarefied gas flows. However, for the DSMC method to be effective, the cell size must be smaller than the mean free path, and the time step smaller than the mean collision time. These constraints make it difficult to use the DSMC method in multiscale rarefied gas flows. Over the past decade, the particle Fokker-Planck (FP) method has been studied to address computational cost issues in the near-continuum regime. To capture the main features of the Boltzmann equation, various FP models have been proposed, such as the quadratic entropic FP (Quad-EFP) and the ellipsoidal statistical FP (ESFP). Nevertheless, few studies have clearly demonstrated that the FP method offers a computational advantage over the DSMC method without sacrificing accuracy. This is because conventional particle FP methods have employed first-order accuracy schemes. The present study proposes a unified stochastic particle ESFP (USP-ESFP) model. This model improves the accuracy of shear stress and heat flux predictions. Additionally, a spatial interpolation scheme is introduced to the particle FP method. The numerical test cases include relaxation problem, Couette flows, Poiseuille flows, velocity perturbation, and hypersonic flows around a cylinder. The results show that the USP-ESFP model agrees well with both analytical and DSMC results. Furthermore, the USP-ESFP model is found to be less sensitive to cell size and time step than the DSMC method, resulting in a factor of four speed-up for the considered hypersonic flow around a cylinder.

The Normal/Umklapp/Intervally/Intravally transport property of 2D SnSe

Based on the Boltzmann transport equation, both the electrical and thermal transport properties of 2D SnSe have been accurately calculated. The mobility of n-type SnSe is higher than that of p type due to the greater average velocity and smaller electron-phonon coupling matrix. For both n-type and p-type electron-phonon scattering, the LA phonon mode primarily governs the total, Normal, Umklapp, Intervally, and Intravally scattering processes. As the temperature increases from 300K to 800K, the LA phonon remains the dominant scattering mode. Compared to the Umklapp and Intervally scattering process, both Normal and Intravally scattering processes mainly govern electrical transport. In terms of phonon-electron scattering, the Normal/Intravalley processes dominate initially, followed by the Umklapp/Intervalley processes governing thermal transport as the temperature increases, governing thermal transport. The critical carrier concentration changes from 1018 cm−3 to 1019 cm−3, as the temperature rises from 300K to 800K, indicating a shift in the competitive relationship between Normal/Intravalley and Umklapp/Intervalley processes with temperature. Using the iteration method, the optimal power factor for n type@300K, p type@300K, n type@800K, p type@800K are 1.63 μW/cmK, 5.62 μW/cmK, 2.30 μW/cmK, 4.04 μW/cmK, respectively. After considering phonon-electron scattering, the lattice thermal conductivity of n type (p type) is 0.70 W/mK (0.64 W/mK) at 300K and 0.31 W/mK (0.30 W/mK) at 800K. The optimal ZT of n type is 1.17 at 800K, originating from the Normal scattering process, while the optimal ZT of p type is 0.86, stemming from the Intravally scattering process. Our results provide a deeper understanding of the transport processes in SnSe and offer insights for tuning the thermoelectric properties of two-dimensional materials through electron-phonon and phonon-electron scattering processes.

Going beyond an old shockwave conjecture for improving upon Navier-Stokes.

Nonequilibrium molecular dynamics (NEMD) computer simulations of steady shockwaves in dense fluids and rarefied gases produce detailed shockwave profiles of mechanical and thermal properties. The Boltzmann equation, under the assumption of local thermodynamic equilibrium (LTE), leads to the first-order (linear) continuum theory of hydrodynamic flow: Navier-Stokes-Fourier (NSF). (Expansion of the LTE Boltzmann equationin higher powers of gradients yields so-called Burnett second-order terms, etc.) NEMD simulations of strong shockwaves with high gradients are not well modeled by NSF theory. Many years ago, a conjecture for going "beyond Navier-Stokes" was proposed, applying the empirical observation of anisotropic thermal enhancement in the shock front to the temperature dependence of the NSF transport coefficients, whose dissipation determines the slope at the center of the shock profile: for weak shocks, the actual coefficients in NEMD simulations appear to be smaller than in NSF predictions, leading to steeper gradients being observed, while for strong shocks, the NEMD coefficients appear to be larger, leading to less steep shock rises than predicted by NSF calculations. In this paper, we show that adding significant Burnett nonlinearity into an LTE continuum theory reproduces the early shock rise and slope of NEMD profiles, for both weak and strong shocks in dense fluids, as well as strong shockwaves in the ideal gas. Moreover, we show that "Holian's conjecture" incorporates significant Burnett nonlinearity, but like all the other LTE continuum theories, it fails to describe the slow NEMD return to equilibrium beyond the shock front. We show that Maxwell relaxation has to be applied to the hydrodynamic variables themselves (rather than attempting indirect relaxation of their gradients) in order to more accurately model the entire shockwave profile. Non-LTE Maxwell relaxation is the only way to bring the entire profile into agreement with NEMD, most noticeably for strong shockwaves.

Well/Ill-Posedness Bifurcation for the Boltzmann Equation with Constant Collision Kernel

Well/Ill-Posedness Bifurcation for the Boltzmann Equation with Constant Collision Kernel

Spectral Convergence of a Semi-discretized Numerical System for the Spatially Homogeneous Boltzmann Equation with Uncertainties

Spectral Convergence of a Semi-discretized Numerical System for the Spatially Homogeneous Boltzmann Equation with Uncertainties

Linear operator theory of phase mixing

ABSTRACT We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals.

Atomistic wave packet investigation of phonon scattering at rough surfaces

Investigating the phonon-surface scattering mechanism is essential for the evaluation of thermal transport in nanostructures. The theoretical description to quantify this mechanism remains to be developed at the atomic scale. This work presents a phonon wave packet method to study the phonon-surface scattering behavior at rough surfaces. We obtain the specularity distribution dependent on phonon polarization, wavelength, and surface roughness. The reflection of the diffuse wave packet is primarily attributed to the surface shadowing effect at a higher incident angle, surface disorder, and surface-induced localized modes. Taking the wavevector-dependent specularity data as input, the thermal conductivity of silicon nanowires is calculated based on the Boltzmann Transport Equation. Our specularity model provides an accurate evaluation for predicting thermal conductivity. This work offers an atomic-level analysis for phonon-surface interaction, which is helpful for the understanding of thermal transport in nanostructures.

REV-Scale study of miscible density-driven convection in porous media

Miscible density-driven convection in porous media has important implications for the long-term security of geological CO2 sequestration. In this study, a REV-scale lattice Boltzmann equation method based on the generalized Navier–Stokes equations was used to simulate density-driven convection in porous media, Thus, the effects of the Rayleigh number, the Darcy number, the Schmidt number, and the porosity of porous media can be discussed separately. The results show that density-driven convection only occurs when the Rayleigh–Darcy–Schmidt number RaD-S exceeds 102. The larger the Ra, the more disordered the concentration field, the earlier the convective phenomenon begins, and the more significant the convective mixing; The larger the Da, the finer the generated fingers. These findings provide important insights for the development of geological sequestration technologies.