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Fluid dynamical simulations are often performed using cheap macroscopic models like the Euler equations. For rarefied gases under near-equilibrium conditions, however, macroscopic models are not sufficiently accurate and a simulation using more accurate microscopic models is often expensive. In this paper, we introduce a hierarchical micro-macro acceleration based on moment models that combines the speed of macroscopic models and the accuracy of microscopic models. The hierarchical micro-macro acceleration is based on a flexible four step procedure including a micro step, restriction step, macro step, and matching step. We derive several new micro-macro methods from that and compare to existing methods. In 1D and 2D test cases, the new methods achieve high accuracy and a large speedup.

An approximation by diffusion of a nonlinear Boltzmann equation modeling a Fermi-Dirac statistics is analyzed for an unbounded velocity space and Poisson coupling. A careful analysis of the entropy and entropy-dissipation allows to control the distribution function and to pass to the limit using duality method.

A generalized physics-based expression for the drag coefficient of spherical particles moving in a fluid is developed. The proposed correlation incorporates essential rarefied effects, low-speed hy...

In this paper, a high-order implicit-explicit flux reconstruction lattice Boltzmann method (FRLBM) is proposed for simulating viscous incompressible flows. The discrete velocity Boltzmann equation (DVBE) is solved by using a high-order flux reconstruction scheme (FR) for spatial discretization and a high-order implicit-explicit Runge-Kutta scheme (IMEX RK) for time discretization in the generalized curvilinear coordinates with uniform and non-uniform grids to obtain high simulation efficiency. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) doubly periodic shear layer flow, (c) lid-driven cavity flow, (d) cylindrical Couette flow and (e) flow over a circular cylinder. Numerical results demonstrate that the present method is stable and accurate even at high Reynolds number. Compared with the existing high-order off-lattice Boltzmann methods, the present method is more efficient. In addition, the present method holds the compactness of the standard LBM for parallel computing, but greatly improves the adaptability of complicated geometries.

In this paper, we extend a recently introduced multi-fidelity control variate for the uncertainty quantification of the Boltzmann equation to the case of kinetic models arising in the study of multiagent systems. For these phenomena, where the effect of uncertainties is particularly evident, several models have been developed whose equilibrium states are typically unknown. In particular, we aim to develop efficient numerical methods based on solving the kinetic equations in the phase space by Direct Simulation Monte Carlo (DSMC) coupled to a Monte Carlo sampling in the random space. To this end, exploiting the knowledge of the corresponding mean-field approximation we develop novel mean-field Control Variate (MFCV) methods that are able to strongly reduce the variance of the standard Monte Carlo sampling method in the random space. We verify these observations with several numerical examples based on classical models , including wealth exchanges and opinion formation model for collective phenomena.

We develop the idea that a natural link between Boltzmann schemes and finite volumes exists naturally: the conserved mass and momentum during the collision phase of the Boltzmann scheme induces general expressions for mass and momentum fluxes. We treat a unidimensional case and focus our development in two dimensions on possible flux boundary conditions. Several test cases show that a high level of accuracy can be achieved with this scheme.

In this paper we study the unidimensional Stationary Boltzmann Equation by an approach via Morse theory. We define a functional J whose critical points coincide with the solutions of the Stationary Boltzmann Equation. By the calculation of Morse index of J’’0(0)h and the critical groups C2(J, 0) and C2(J, ∞) we prove that J has two different critical points u1 and u2 different from 0, that is, solutions of Boltzmann Equation.

It has generally been believed that the lattice thermal conductivity (κL) is mainly contributed by acoustic phonons and the contribution of optical phonons can be neglected in crystalline materials. However, in this work, we demonstrate that the contribution of optical phonons play an important role in the complex structural low- κ L oxides. The mode-wised κL of three representative oxides, MgO, BaHfO3 and La2Zr2O7, with different structural complexity was calculated using first-principles phonon Boltzmann transport equation (BTE) method. It is found that the contribution of optical phonons become significant as the structural complexity increases. Especially, for La2Zr2O7, the κL is dominated by optical phonons due to the large density of states. Optical phonons exhibit amorphous-like phonon-phonon scattering characteristics that are different from those of acoustic phonons. Furthermore, calculations for materials other than the three representative oxides were conducted and the results confirmed the validity of our conclusions.

The class of Divergence-free symmetric tensors is ubiquitous in Continuum Mechanics. We show its invariance under projective transformations of the independent variables. This action, which preserves the positiveness, extends Sophus Lie's group analysis of Newtonian dynamics. When applied to models of gas dynamics --~such as Euler system or Boltzmann equation,~-- in combination with Compensated Integrability, this yields new dispersive estimates. The most accurate one is obtained for mono-atomic gases. Then the space-time integral of $t\rho^\frac1d p$ is bounded in terms of the total mass and moment of inertia alone.

We discuss the theoretical basis of the Boltzmann transport equation (BTE) and the Monte Carlo (MC) simulation scheme to take into account the discrete nature of impurities in semiconductor substrates. This is achieved by explicitly considering the self-consistent coupling between the BTE and Poisson’s equation and by separating the impurity potential into long- and short-range parts in those equations. We show that the Hartree potential is represented by the long-range part of the potential, whereas the scattering potential for the BTE is naturally derived within the framework, rather than an ad hoc assumption. The proposed scheme is validated by showing that MC simulations with discrete impurities in bulk can reproduce the conventional simulation results with the “jellium” impurity. Also, by including both static and dynamic potential fluctuations, the MC simulation can predict the correct mobility even under discrete impurities.

A stabilization method is presented for the Boltzmann equation (BE) that is based on Godunov’s approach applied directly in the phase space. This makes it possible to perform transient simulations under quasiballistic conditions and to include the Pauli principle in the scattering integral without further approximations. The method ensures positive distribution functions by construction. The BE is solved self-consistently together with the Poisson and Schrodinger equations by the Newton–Raphson method for a nanowire silicon nMOSFET, for which the stationary ${I}$ – ${V}$ characteristics, the small-signal admittance parameters, and the switching behavior are calculated. The large-signal behavior of the MOSFET as an amplifier is evaluated by cyclostationary simulations including power gain and the output power at higher harmonics. The current responsivity of the MOSFET operated as a passive mixer is simulated up to terahertz (THz) frequencies including the quasiballistic case with negligible scattering.

The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. The analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof., Comment: 51 pages

We present theoretical framework investigations of the influence of temperature variation on the electrical conductivity of zigzag carbon nanotubes (CNTs) under the applied homogeneous axial dc field. This study was done semiclassically by solving Boltzmann transport equation to derive the current density of zigzag CNT as a function of homogenous axial dc field and temperature. Plots of the normalized current density versus homogeneous dc field applied along the axis of semiconducting zigzag CNTs as room temperature increases from 293 to 299 K revealed a significant increase in electrical conductivity, whereas in metallic zigzag CNTs, almost constant or a negligible decrease in electrical conductivity is observed. The study predicts semiconducting zigzag CNT as a potential material for temperature sensors since it exhibits a faster response and a substantially higher sensitivity to room temperature changes than the metallic counterpart. The electrical conductivity of metallic zigzag CNTs increases immensely as the temperature is reduced to a very low value which could probably lead to a potential superconductivity property that usually occurs at very low temperatures. These potential temperature sensors and superconductors of nanomaterial have vast applications in current-day science and technology.

The Seebeck effect measures the electric potential built up in materials under a temperature gradient. For organic thermoelectric materials, the Seebeck coefficient shows more complicated temperatu...

The thermal transport properties of biphenylene network (BPN), a novel sp2-hybridized two-dimensional allotrope of carbon atoms recently realized in experiments [Fan et al., Science, 372 852–856 (2021)], are studied using the density functional theory-driven solution of the Boltzmann transport equation. The thermal transport in BPN is anisotropic and the obtained thermal conductivities are more than an order of magnitude lower than that in graphene, despite similar sp2-hybridized planar-structure of both allotropes. The lower thermal conductivity in BPN is found to originate from enhanced anharmonicity which in turn is a result of reduced crystal symmetry of BPN.

We report a phenomenon occurring in some chalcopyrite structures with space group $$R\overline{3 }m$$ (space group number: 166) where heavier mean atomic mass does not necessarily lead to lower lattice thermal conductivity, in contradiction to the conventional Keyes’ theory of an inverse relation between lattice thermal conductivity and heavier elements. The lattice thermal conductivity is calculated for three chalcopyrite structures, namely TlBiS2, TlBiSe2, and TlBiTe2, at room temperature using phonon Boltzmann transport equation with interatomic force constants evaluated by density functional theory (DFT). According to Keyes’ theory, the decedent order of lattice thermal conductivity should follow: TlBiS2 > TlBiSe2 > TlBiTe2. However, our first-principles calculations demonstrate that TlBiSe2 has higher lattice thermal conductivity than TlBiS2, although the Se atoms are heavier than S atoms. Resorting to full first-principles calculations, we reveal the relatively lower lattice thermal conductivity of TlBiS2 compared to TlBiSe2 originates from the largely suppressed phonon relaxation time, in particular for optical phonons, which negatively affects thermal transport, although both heat capacity and phonon group velocity of TlBiS2 are higher than that of TlBiSe2, which should have led to higher lattice thermal conductivity in TlBiS2. Detailed analysis of phonon partial density of states (PDOS) demonstrates that the short phonon relaxation time in TlBiS2 is due to the extensive strongly anharmonic phonon interactions solely associated with the vibrations of S atoms as shown in PDOS, which reduces the life of the phonon transport in TlBiS2, while in TlBiSe2 and TlBiTe2 the thermally conductive collective modes are present. The higher anharmonic phonon–phonon interactions are also attributed to the higher transferred charges present in TlBiS2. We also provide a microscopic picture connecting the strong phonon anharmonicity to the stereo-chemically lone-pair electrons present in the chalcopyrite structures. Moreover by analyzing the atomic charges, our calculations show that these materials possess a recently discovered new type of bonding named metavalent bonding, which is featured by giant phonon anharmonicity, high transverse optical Gruneisen parameters, high Born effective charges, and high dielectric constants. These results provide deep understanding of metavalent bonding induced unusual heat transfer mechanism and offer a new route to designing novel materials with ultralow lattice thermal conductivity, which is extremely attractive for waste-heat recovery (thermoelectrics) and thermal insulation applications.

Recent proceedings in radiation belt studies have proposed new requirements for numerical methods to solve the kinetic equations involved. In this article, we present a numerical solver that can solve the general form of the radiation belt Fokker–Planck equation and Boltzmann equation in arbitrarily provided coordinate systems and with user-specified boundary geometry, boundary conditions, and equation terms. The solver is based upon the mathematical theory of stochastic differential equations, whose computational accuracy and efficiency are greatly enhanced by specially designed adaptive algorithms and a variance reduction technique. The versatility and robustness of the solver are exhibited in four example problems. The solver applies to a wide spectrum of radiation belt modeling problems, including the ones featuring non-diffusive particle transport such as that arising from nonlinear wave–particle interactions.

Understanding and controlling the phonon, the dominant heat carrier of semiconductor materials, is essential to developing a wide variety of applications. This article studies the theoretical and computational approach of the calculation of lattice thermal conductivity of semiconducting materials. Despite having different methods to calculate the lattice thermal conductivity, first-principle estimates predict more accurately in most applications. This motivates to present the descriptive explanation on first-principle calculation with the combination of lattice dynamics and Boltzmann transport equation. Finally, we summarized an overview of the recent achievements and opportunities.

In this paper we study the regularity of the non-cutoff Vlasov–Poisson–Boltzmann system for plasma particles of two species in the whole space $$\mathbb {R}^3$$ with hard potential. The existence of global-in-time nearby Maxwellian solutions is known for soft potential from Duan and Liu (Commun Math Phys 324(1):1–45, 2013). However the smoothing effect of these solutions has been a challenging open problem. We establish the global existence and regularizing effect to the Cauchy problem for hard potential with large time decay. Hence, the solutions are smooth with respect to (t, x, v) for any positive time $$t>0$$ . This gives regularity to the Vlasov–Poisson–Boltzmann system, which enjoys a similar smoothing effect as the Boltzmann equation. The proof is based on the time-weighted energy method building also upon the pseudo-differential calculus.

A physically clear probabilistic model of a gas from hard spheres is considered using the theory of random processes and in terms of the classical kinetic theory for the densities of distribution functions in the phase space: from the system of nonlinear stochastic differential equations (SDEs), first the generalized and then the random and nonrandom Boltzmann integro-differential equations are derived, taking into account the correlations and fluctuations. The main feature of the original model is the random nature of the intensity of the jump measure and its dependence on the process itself. For the sake of completeness, we briefly recall the transition to increasingly rough approximations in accordance with a decrease in the dimensionless parameter, the Knudsen number. As a result, stochastic and nonrandom macroscopic equations are obtained that differ from the system of Navier–Stokes equations or systems of quasi-gas dynamics. The key difference of this derivation is a more accurate averaging over the velocity due to the analytical solution of the SDE with respect to the Wiener measure, in the form of which the intermediate meso-model in the phase space is presented. This approach differs significantly from the traditional one, which uses not the random process itself, but its distribution function. The emphasis is on the transparency of the assumptions when moving from one level of detail to another, rather than on numerical experiments, which contain additional approximation errors.

We complete the investigation of the feasibility in principle to close the moment system for the kinetic equation via invariant relations obtained by group analysis methods. For the one-dimensional equation, we obtain an affirmative answer, find an invariant closure of the moment system, and reveal several new phenomena.

Plasma Modeling: Methods and Applications presents and discusses the different approaches that can be adopted for plasma modeling, giving details about theoretical and numerical methods. The book is intended to assist and direct students and researchers, who want to develop research activity in the field of plasma physics, in the choice of the best model for the problem of interest. The book is organised in three parts. The first describes kinetic models used in plasma investigations, consisting of the solution of the Boltzmann equation using different approaches. The second part develops the theory of fluid equations and of hybrid models, and the third part is devoted to applications, considering some practical problems of interest in different fields.

In the 1860’s emerges a revolutionary idea: many properties of the world around us can be explained by combining the atomistic hypothesis with the statistical theory. Some of the great scientific conquests from this time are the Boltzmann equation, which triggers one of the first qualitative studies of a complicated nonlinear partial differential equation; the notion of statistical entropy, which would later be fundamental in other areas of physics and mathematics, including information theory; and the notion of macroscopic irreversibility emerging from microscopically reversible laws. Thus the basic rules of statistical physics were set until Boltzmann’s irreversibility paradigm was shaken by Landau’s discovery of the Landau damping effect, about 80 years later, which opened the idea that equilibration is compatible with preservation of information, and led to a number of problems concerning the statistical theory of matter.

The growing demand for miniaturized transistors with increased performance and low power consumption has scaled down the device to the nanoscale regime. The design and modeling of such small devices need a reliable computing technique for simulation. Since the flow of electrons in such short channels is due to differences in electrochemical potential between source and drain which of ballistic transport. The study of novel micro and nano-scaled electronic device using the classical approach is inaccurate whose results are not reliable. The complexity of the current formulation increase as the size of the device is reduced. The general concept for the current in the FET device is considered due to the drift and diffusion of carriers in the device. But in modern concepts, the flow of electrons is explained as a combination of two different processes which are termed elastic transfer and heat generation. The elastic transfer is force driven and the heat generation is entropy-driven. Considering the above-mentioned process the transport of electron is studied by two theories, Semi-classical transport theory, and quantum transport theory. The semi-classical transport theory explains the flow of electrons by a combination of Newtonian mechanics for force-driven and thermodynamics for entropy-driven, called Boltzmann transport equation (BTE). In mesoscopic physics, the quantum transport theory approach is used in which the force-driven transport of electrons is well explained by quantum mechanics. The quantum transport theory is the advanced one that properly describes the current flow in the electronic device and is reliable. In the quantum transport theory, force-driven flow is expressed using the Schrodinger wave equation and thermodynamics for entropy-driven, and the formal integration of these two processes is obtained by the Non-Equilibrium Green’s Function (NEGF) method. The NEGF utilized the process of many-body perturbation theory (MBPT) to explain the dispersed entropy-generating practices, which are dependent upon the second quantization language. In both the equilibrium and non-equilibrium conditions, NEGF has been evolved to examine the many-particle quantization systems. The application of NEGF formalism in the field of quantum optics, quantum correction of BTE, transport in bulk systems corresponding to the high field, quantum electron and holes transport in different materials and devices, resonant tunnel diodes of III-V groups, electron waveguides, quantum cascade lasers, Si nano-pillars, carbon nanotubes, graphene nanoribbons, metal wires, organic molecules, spintronic devices, thermal and thermoelectric devices. A meticulous framework is provided by the NEGF method to deal with whole interaction either elastic (non-dissipative) or inelastic (dissipative) with the help of MBPT which occurs in the channel. This paper will provide a comprehensive understanding of the accurate carrier transport model which can be employed with advanced nanoscale devices where the classical theory fails is. And it is capable to set a strong background for the readers and researchers groups who are actively working in the quantum and ballistic transport regime. It would also be beneficial to graduate/undergraduate students, and industry people who are pursuing the same domain.

The surface current density of charge carriers in two-dimensional doubly periodic n-type semiconductor superlattices in an external magnetic field in the presence of temperature gradient has been calculated in the single-electron approximation. The magnetic field was assumed to be permanent, homogeneous, and oriented perpendicular to the electron-gas plane. A joint solution of the Schrodinger equation and kinetic Boltzmann equation showed that the dependences of the surface density of transverse current on temperature and temperature-gradient magnitude are significantly nonlinear and contain portions with negative transverse differential conductivity. The dependence of the relaxation time on the electron quasi-momentum is phenomenologically taken into account in the model via the dispersion relation for carriers in magnetic subbands.

We consider the evolution of a cascade of atomic collisions in a solid consisting of identical atoms with account for their binding energy ed at lattice sites. Based on the solution to the model Boltzmann kinetic equation, we have derived the expression for the distribution function describing a nonstationary energy distribution of moving atoms. We assume that scattering of moving atoms is elastic and spherically symmetric in the center-of-mass system, and the interaction cross section is a constant quantity. In the particular case when ed = 0, the obtained solution is in good agreement with the exact solution that has been obtained earlier.

International audience; We present and analyze two simple N-particle systems for the spread of an infection, respectively with binary and with multi-body interactions. We establish a convergence result, as N → ∞, to a set of kinetic equations, providing a mathematical justification of related numerical schemes. We analyze rigorously the time asymptotics of these equations, and compare the models numerically.

Low dimensional and 2D materials have been considered as promising candidates for thermoelectric applications due to their intrinsic high carrier mobilities. Amongst them, β-SiC7 has gained interest due to its direct band gap and thermal stability. To further explore the potential of this material, we present a detailed analyses of the lattice thermal conductance ( κ L ) of 2D β-SiC7 through a full iterative solution of the Boltzmann transport equation. We find the thermal transport to be anisotropic with κ yy = 97.62 (10−9 W/K), significantly lower than that of graphene due to the presence of Si atom in the lattice which causes distortion and weakening of the acoustic branches. The TA and LA modes to possess the highest phonon group velocities due to their linear nature around the Γ point in the phonon dispersion, while the ZA mode possesses the lowest group velocity as a result of the quadratic shape of the branch within the vicinity of the Γ point. On the other hand, the flexural ZA branch possesses the longest carrier lifetime that is about three orders of magnitude higher than the LA and TA branches, since the mass disorder in the lattice softens the acoustic modes. The thermodynamic limit to the κ L is attained for a sample domain size of ∼45 μm and nanostructuring emerges as a possibility to reduce the κ L by 50% to enhance the thermoelectric efficiency of the material.

In this work we solve the nonlinear strong evaporation problem in rarefied gas dynamics. The analysis is based on the BGK model, with three dimensional velocity vector, derived from the Boltzmann equation. We present the complete development of a closed form solution for evaluating density, velocity, temperature perturbations and the heat flux of a gas. Numerical results are presented and discussed.

We consider kinetic theory of granular mixtures in which each solid phase is described by its own granular temperature. The particle velocities are assumed to follow a Mawellian distribution with zero-order approximation of the Boltzmann equation. The literature proposes a number of approximate solutions to the constitutive equations resulting from particle collisions. Here, we derive and show that analytical solutions exist, and reveal that some of the proposed approximate solutions are inaccurate whereas others might give good approximations. We also review the balance laws and constitutive relations resulting from the mono- and poly-disperse kinetic granular flow model frameworks. Such a review has been done here because the model framework on kinetic theory of mono-disperse granular flow systems has sometimes been adopted in models for granular mixtures. The consequence of such a simplification is that the kinetic energy loss in a collision between two particles of different species is equally shared among them.

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.

A disciplinary profile of a country is defined as the versor whose components are the numbers of articles produced in a given discipline divided the overall production of the country. Starting from the Essential Science Indicators (ESI) schema of classification of subject area, we obtained the yearly disciplinary profiles of a worldwide graph, where on each node sits a country, in the two time intervals [1980-1988] and [1992-2017], the fall of the Berlin Wall being the watershed. We analyse the empirical pairwise cross-correlation matrices of the time series of disciplinary profiles. The contrast with random matrix theory proves that, beyond measurement noise, the empirical cross-correlation matrices bring genuine information. Arising from the Shannon theorem as the least-structured model consistent with the measured pairwise correlations, the stationary probability distribution of disciplinary profiles can be described by a Boltzmann distribution related to a generalized $n_d$-dimensional Heisenberg model. The set of network interactions of the Heisenberg model have been inferred and to it they have been applied two clusterization methods, hierarchical clustering and principal component analysis. On a geopolitical plane this allow to obtain a characterization of the worldwide bilateral interactions based on physical modeling. A simple geopolitical analysis reveals the consistency of the results obtained and gives a boost to deeper historical analysis. In order to obtain the optimal set of pairwise interactions we used a Pseudo-Likelihood approach. We analytically computed the Pseudo-Likelihood and its gradient. The analytical computations deserve interest in whatever inference Bayesian problem involving a $n_d$-dimensional Heisenberg model.

By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov–Poisson–Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.

The finite volume method (FVM)-based computational fluid dynamics (CFD) technology has been applied in the non-invasive diagnosis of coronary artery stenosis. Nonetheless, FVM is a time-consuming process. In addition to FVM, the lattice Boltzmann method (LBM) is used in fluid flow simulation. Unlike FVM solving the Navier–Stokes equations, LBM directly solves the simplified Boltzmann equation, thus saving computational time. In this study, 12 patients with left anterior descending (LAD) stenosis, diagnosed by CTA, are analysed using FVM and LBM. The velocities, pressures, and wall shear stress (WSS) predicted using FVM and LBM for each patient is compared. In particular, the ratio of the average and maximum speed at the stenotic part characterising the degree of stenosis is compared. Finally, the golden standard of LAD stenosis, invasive fractional flow reserve (FFR), is applied to justify the simulation results. Our results show that LBM and FVM are consistent in blood flow simulation. In the region with a high degree of stenosis, the local flow patterns in those two solvers are slightly different, resulting in minor differences in local WSS estimation and blood speed ratio estimation. Notably, these differences do not result in an inconsistent estimation. Comparison with invasive FFR shows that, in most cases, the non-invasive diagnosis is consistent with FFR measurements. However, in some cases, the non-invasive diagnosis either underestimates or overestimates the degree of stenosis. This deviation is caused by the difference between physiological and simulation conditions that remains the biggest challenge faced by all CFD-based non-invasive diagnostic methods.

This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular, we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327-355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post-collisional momentum $p'$; specifically we calculate the determinant for $p\mapsto u = \theta p'+\left(1-\theta\right)p$ for $\theta \in [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55-68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649-673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates., Comment: 27 pages, 2 figures

A boundary value problem for the stationary Boltzmann equation with hard sphere in a half-space is considered for a binary mixture of gases, imposing slightly perturbed reflection boundary conditio...

Neste artigo, o problema denominado de fonte-fixa (blindagem de nêutrons) em meio não multiplicativo foi estudado. Assim, foi simulada computacionalmente a blindagem da fonte de nêutrons utilizando métodos numéricos determinísticos para estimar os fluxos escalares e angulares de nêutrons em um domínio unidimensional. A equação de transporte linearizada de Boltzmann para partículas neutras foi empregada na modelagem dos problemas estudados. Na resolução numérica da equação de Boltzmann, foram consideradas certas aproximações e empregados métodos numéricos consolidados para gerar soluções precisas. Os métodos numéricos Diamond Difference (DD), Step (ou Degrau) e Step Characteristic (Degrau Característico) foram utilizados nos problemas-modelo abordados. A configuração das blindagens era constituída de três materiais, variando-se a sequência da montagem e o tipo de material em função da interação material-nêutron. Foram simulados e analisados problemas-modelo de modo a identificar blindagens eficientes para o fluxo emergente de nêutrons.

The presence of a nonvanishing thermal gradient and/or a chemical potential gradient in a conducting medium can lead to an electric field—an effect known as thermoelectric effect or Seebeck effect. We discuss here the thermoelectric effects for hot and dense strongly interacting matter within the framework of relativistic Boltzmann equation in the relaxation time approximation. In the context of heavy-ion collisions, the Seebeck coefficients for the quark matter as well as for the hadronic matter are estimated within this approach. The quark matter is modeled by the two flavor Nambu–Jona–Lassinio (NJL) model and the hadronic medium is modeled by the hadron resonance gas (HRG) model with hadrons and their resonances up to a mass cutoff $${\tilde{\varLambda }}\sim 2.6$$ GeV. For the estimation of thermoelectric transport coefficients, for the quark matter, we calculate the relaxation times for the quarks and antiquarks from the quark–quark and quark–antiquark scattering through meson exchange within the NJL model. On the other hand, for the hadronic medium, the relaxation times of hadrons and their resonances are estimated within a hard sphere scattering approximation. We also discuss the formalism of the thermoelectric effect in the presence of a nonvanishing external magnetic field. We give an estimation of the associated magneto-Seebeck coefficient and the Nernst coefficient for the hot and dense QCD matter.

The results of numerical simulation of the effects of H2O and O2 impurities on the Trichel pulses characteristics in the negative point-plane corona discharge in SF6 gas at 0.4 MPa is presented in this paper. The H2O and O2 contents varied from 200 ppm to 10,000 ppm. The continuity equations for three charged species along with Poisson’s equation were solved simultaneously to obtain the Trichel pulses characteristics in SF6/H2O mixtures and SF6/O2 mixtures. The reduced ionization coefficient, the reduced attachment coefficient, the electron mobility and the electron diffusion coefficient for SF6/H2O and SF6/O2 mixtures were determined by solving the two-term Boltzmann equation. The effects of H2O and O2 on the current pulses, the distributions of the charged species and the reduced electric field at the instants of the first and second peak pulse, the time-dependent total number of the three charged species in the gap and the reduced electric field at the discharge tip were investigated to determine the sensitivities of the Trichel pulses characteristics of the SF6 to the H2O and O2 impurities.

The utilization of thermal energy from different sources is an important development direction for conserving energy. With the development of technology, refined and rapid utilization of thermal energy is required. Traditional thermal conductive materials cannot meet the growing needs of human beings. Therefore, people pay attention to two-dimensional graphene film materials for their thermal conductivity. This review collects current modeling group of thermal transport on graphene, including non-equilibrium Green function (NEGF) theory, molecular dynamics (MD) simulations modeling and Boltzmann transport equation method. These models can well explain several phenomena of phonon transport in graphene. Further, structural defects were discussed and expounded the effect for graphene thermal conductivity, including doping, grain boundary and defects. Deeply understanding of defects on graphene, we can better grasp the thermal conductivity of graphene from the microscopic point of view.

For a nonlinear kinetic Boltzmann equation that describes the model of rough spheres, we construct its approximate solution in the form of a continual distribution with global Maxwellians. We establish sufficient conditions for the coefficient functions and hydrodynamic parameters appearing in the distribution, which enable one to make the analyzed error as small as desired.

This work is devoted to the study of the existence of positive solutions and the hydrodynamic limit of the steady Boltzmann equation with in-flow boundary condition. The proof is based on a L6−L∞ f...

We present a method for predicting thermal conductivity by deterministically solving the Boltzmann transport equation for gray phonons by utilizing arbitrary higher-order continuous finite elements...

The coupling of spin angular momentum and mechanical rotation is utilized for generating a spin current of nuclear spins of the liquid helium-3. A hydrodynamic flow in the liquid helium-3 provides a vorticity gradient which can be regarded as a gradient of effective magnetic field by the spin-rotation coupling. The effective field gradient generates a nuclear spin current owing to the Stern–Gerlach effect. We quantitatively estimate the magnitude of the field gradient based on the microscopic Boltzmann equation. The effective field gradient is large enough to be detected by NMR measurements.

We report a configuration strategy for improving the thermoelectric (TE) performance of two-dimensional (2D) transition metal dichalcogenide (TMDC) WS2 based on the experimentally prepared WS2/WSe2 lateral superlattice (LS) crystal. On the basis of density function theory combined with Boltzmann transport equation, we show that the TE figure of merit zT of monolayer WS2 is remarkably enhanced when forming into a WS2/WSe2 LS crystal. This is primarily ascribed to the almost halved lattice thermal conductivity due to the enhanced anharmonic processes. Electronic transport properties parallel (xx) and perpendicular (yy) to the superlattice period are highly symmetric for both p- and n-doped LS owing to the nearly isotropic lifetime of charger carriers. The spin-orbital effect causes a significant split of conduction band and leads to three-fold degenerate sub-bands and high density of states (DOS), which offers opportunity to obtain the high n-type Seebeck coefficient (S). Interestingly, the separated degenerate sub-bands and upper conduction band in monolayer WS2 form a remarkable stairlike DOS, yielding a higher S. The hole carriers with much higher mobility than electrons reveal the high p-type power factor and the potential to be good p-type TE materials with optimal zT exceeds 1 at 400K in WS2/WSe2 LS.

For continuum mechanics a model is proposed, that is built with consideration outside the integral term when deriving conservation laws using the Ostrogradsky-Gauss theorem. Performed analysis shows discrepancy between accepted classical conservation laws and classical theoretical mechanics and mathematics. As a result, the theory developed for potential flows was extended to flows with significant gradients of physical parameters. We have proposed a model that takes into account the joint implementation of the laws for balance of forces and angular momentums. It does not follow from the Boltzmann equation that the pressure in the Euler and Navier-Stokes equations is equal to one third of the sum the pressures on the corresponding coordinate axes. The vector definition of pressure is substantiated. It is shown that the symmetry condition for the stress tensor is one of the possible conditions for closing the problem. An example of solving the problem of the theory of elasticity is given

The kinetic theory is widely used in the description of thermal transport at the micro- and nanoscale. In the theory, it is assumed that heat is carried by quasi-particles, obeying the Boltzmann transport equation. These quasi-particles are sometimes associated with phonons. However, since phonons are not localized in physical space, they cannot play the role of the quasi-particles used in the kinetic theory. In the present paper, we employ another interpretation of quasi-particles, namely wave packets. Our derivation is carried out for an infinite harmonic chain with a given initial temperature distribution. An exact formula describing the time evolution of the kinetic temperature of the chain is derived. A transition from the exact solution to a continuum limit is performed. It is shown that the resulting continuum solution coincides with the solution of the collisionless Boltzmann equation. The transition yields wave packets, localized in space and moving with group velocities. Therefore, the wave packets can be associated with quasi-particles. It is shown that the quasi-particle has finite lifetime and that its position is determined with some uncertainty.