Boltzmann Kinetic Theory Research Articles

Applications of the kinetic theory for a model of a confined quasi-two dimensional granular mixture: Stability analysis and thermal diffusion segregation

The Boltzmann kinetic theory for a model of a confined quasi-two dimensional granular mixture derived previously [Garzó et al., “Navier–Stokes transport coefficients for a model of a confined quasi-two dimensional granular binary mixture,” Phys. Fluids 33, 023310 (2021)] is considered further to analyze two different problems. First, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state (HSS) is carried out to identify the conditions for stability as functions of the wave vector, the coefficients of restitution, and the parameters of the mixture. The analysis, which is based on the results obtained by solving the Boltzmann equation by means of the Chapman–Enskog method to first order in spatial gradients, takes into account the (nonlinear) dependence of the transport coefficients and the cooling rate on the coefficients of restitution and applies in principle to arbitrary values of the concentration, and the mass and diameter ratios. In contrast to the results obtained in the conventional inelastic hard sphere (IHS) model, the results show that all the hydrodynamic modes are stable so that the HSS is linearly stable with respect to long enough wavelength excitations. On the other hand, this conclusion agrees with previous stability analysis performed in earlier studies for monocomponent granular gases. As a second application, segregation induced by both a thermal gradient and gravity is studied. A segregation criterion based on the dependence of the thermal diffusion factor Λ on the parameter space of the mixture is derived. In the absence of gravity, the results indicate that Λ is always positive, and hence, the larger particles tend to accumulate near the cold plate. However, when gravity is present, our results show the transition between Λ>0 (larger particles tend to move toward the cold plate) to Λ<0 (larger particles tend to move toward the hot plate) by varying the parameters of the system (masses, sizes, composition, and coefficients of restitution). Comparison with previous results derived from the IHS model is carried out.

On the interpretation of single and bilayer graphene minimal conductivity

Within the Boltzmann kinetic theory minimal dc conductivity of single and bilayer graphene is studied. It is shown that the uncertainty principle plays a key role in graphene band structure. In the paper a new alternative interpretation for minimal universal conductivity of single and bilayer graphene is proposed. Minimal conductivity is determined by degenerate electrons and holes of the overlap range of electron and hole bands. It is established that the minimal conductivity of single and bilayer graphene can be explained in the framework of one definite physical approach based on the uncertainty relations. Within the proposed theoretical approach minimal conductivity of both single-layer and bilayer graphene equals 4e2/h. It is found that in crystals with parabolic dispersion quantum expansion of an energy level E is asymmetric with respect to E; in a crystal with linear dispersion an energy level quantum expansion is symmetric.

Hydrodynamic approach to many-body systems: Exact conservation laws

In this paper I present a pedagogical derivation of continuity equations manifesting exact conservation laws in an interacting electronic system based on the nonequilibrium Keldysh technique. The purpose of this exercise is to lay the groundwork for extending the hydrodynamic approach to electronic transport to strongly correlated systems where the quasiparticle approximation and Boltzmann kinetic theory fail.

Two-dimensional stable lattice Boltzmann simulation of turbulent flow in wavy walled channel

In recent years, the lattice Boltzmann equation has developed into a promising technique for computational fluid dynamics (CFD). The lattice Boltzmann model (LBM) approach is derived from the Boltzmann equation and kinetic theory, as opposed to the standard CFD methods that are based on direct discretization of the Navier–Stokes equations. In this paper, Newtonian flow passing through a wavy walled channel has been examined for laminar to turbulent transition by using the LBM. The simple LBM for this problem becomes unstable as the Reynolds number increases and the laminar to turbulent transition begins. When Ehrenfest’s limiters are introduced in the LBM, the simulation becomes stable for higher Reynolds numbers. Two types of channel geometries are studied here, the channel walls of relatively small amplitude and channel walls of large amplitude. Our findings are that for large amplitude channel walls, flow becomes unsteady for lower Reynolds numbers as compared to that for small amplitude channel walls. For large amplitude walls, the vortices formed exhibit periodic shedding inside the channel furrows and remain there. For small amplitude walls, the vortex shedding starts downstream of the channel. The present LBM variant is able to simulate small amplitude channel flow for a Reynolds number Re = 800 and large amplitude channel flow for a Reynolds number R = 570.

On the role of geometry in statistical mechanics and thermodynamics. II. Thermodynamic perspective

The General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) provides the structure of mesoscopic multiscale dynamics that guarantees the emergence of equilibrium states. Similarly, a lift of the GENERIC structure to iterated cotangent bundles, called a rate GENERIC, guarantees the emergence of the vector fields that generate the approach to equilibrium. Moreover, the rate GENERIC structure also extends Onsager’s variational principle. The maximum entropy principle in the GENERIC structure becomes the Onsager variational principle in the rate GENERIC structure. In the absence of external forces, the rate entropy is a potential that is closely related to the entropy production. In the presence of external forces when the entropy does not exist, the rate entropy still exists. While the entropy at the conclusion of the GENERIC time evolution gives rise to equilibrium thermodynamics, the rate entropy at the conclusion of the rate GENERIC time evolution gives rise to rate thermodynamics. Both GENERIC and rate GENERIC structures are put into the geometrical framework in the first paper of this series. The rate GENERIC is also shown to be related to Grad’s hierarchy analysis of reductions of the Boltzmann equation. Chemical kinetics and kinetic theory provide illustrative examples. We introduce rate GENERIC extensions (and thus also Onsager-variational-principle formulations) of both chemical kinetics and the Boltzmann kinetic theory.

Barkas effect in strongly magnetized plasmas

Strongly magnetized plasmas, which are characterized by the particle gyrofrequency exceeding the plasma frequency, exhibit novel transport properties. For example, recent work showed that the friction force on a test charge moving through a strongly magnetized plasma not only consists of the typical stopping power component but also includes components perpendicular to the test charge's velocity. However, these studies only considered test charges that have the same sign as the charge of the plasma particles. Here, we extend these calculations to the case of charges with opposite signs (such as an ion interacting with strongly magnetized electrons). This is done with both a novel generalized Boltzmann kinetic theory and molecular dynamics simulations. It is found that the friction force changes dramatically depending on the sign of the interacting charges. Likewise, the stopping power component for oppositely charged particles decreases in magnitude compared with like-charged particles, and the perpendicular components increase in magnitude. Moreover, the difference between the two cases increases as the gyrofrequency becomes larger compared with the plasma frequency. The electrical resistivity is calculated from the friction force, where it is found that strong magnetization in conjunction with oppositely charged interactions significantly decreases the parallel resistivity and increases the perpendicular resistivity.

Diffusion matrix associated with the diffusion processes of multiple conserved charges in a hot and dense hadronic matter

Bulk matter produced in heavy ion collisions has multiple conserved quantum numbers like baryon number, strangeness and electric charge. The diffusion process of these charges can be described by a diffusion matrix describing the interdependence of diffusion of different charges. The diffusion coefficient matrix is estimated here from the Boltzmann kinetic theory for the hadronic phase within relaxation time approximation. In the derivation for the same, we impose the Landau-Lifshitz conditions of fit. This leads to e.g. the diagonal diffusion coefficients to be manifestly positive definite. The explicit calculations are performed within the ambit of hadron resonance gas model with and without excluded volume corrections. It is seen that the off-diagonal components can be significant to affect the charge diffusion in a fluid with multiple conserved charges. The excluded volume correction effects is seen to be not significant in the estimation of the elements of the diffusion matrix.

Thermal transport in two-dimensional nematic superconductors

We study the thermal transport in a two-dimensional system with coexisting superconducting (SC) and nematic orders. We analyze the nature of the coexistence phase in a tight-binding square lattice where the nematic state is modelled as a $d$-wave Pomeranchuk-type instability and the feedback of the symmetry breaking nematic state on the SC order is accounted for by mixing of the $s, d$ paring interaction. The electronic thermal conductivity is computed within the framework of Boltzmann kinetic theory where the impurity scattering collision integral is treated in the Born and unitary limits. We present qualitative, analytical, and numerical results that show that the heat transport properties of SC states emerging from a nematic background are quite distinct and depend on the degree of anisotropy of the SC gap induced by nematicity. We describe the influence of the Fermi surface topology, the van Hove singularities, and the presence or absence of zero-energy excitations in the coexistence phase on the the low-temperature behavior of the thermal conductivity. Our main conclusion is that the interplay of nematic and SC orders has visible signatures in the thermal transport, which can be used to infer SC gap structure in the coexistence phase.

Modeling epidemics by means of the stochastic description of complex systems

The aim of this article is to show a way in which the problem of predicting the evolution of an epidemic may be tackled by describing it in the framework of Boltzmann's kinetic theory, as it has been developed and applied in the last years to complex systems by a suitable modification of the Boltzmann equation, via a suitable reinterpretation of state variables and the introduction of the notion of «functional subsystems». Accordingly, in this article we model an arbitrary (national) population S as a complex system, split in two functional subsystems, the first containing all single individuals of S and the second containing the «care tools», that are to be meant as available places in hospitals with a sufficient number of physicians and of equipments for intensive cares. The state variable on the first subsystem will be the «health state», and the state variable on the other will be the «effectiveness». We shall then write a system of nonlinear ordinary differential equations which gives the evolution of the probability distribution on the set of possible values of the health states. By assigning data partly on the basis of plausibility assumptions and partly as estimated from those furnished by institutions of Campania region, the system takes a form allowing the numerical simulation of such evolution, which will be performed and presented in a forthcoming paper.

Quantum diffusion of massive Dirac fermions induced by symmetry breaking

We show that when a two-dimensional $(2\mathrm{D})$ Dirac fermion moves in disordered environments, the weak time-reversal symmetry breaking by a small mass gives rise to the diffusive wave propagation, i.e., that the wave-packet spread obeys the diffusive law of Einstein, up to a---practically inaccessible---exponentially large length. Strikingly, the diffusion constant is larger than that given by the Boltzmann kinetic theory, and grows unboundedly as the energy-to-mass ratio increases. This diffusive phenomenon is of quantum nature and different from weak antilocalization. It implies a new type of transport in topological insulators at zero temperature.

A kinetic model of friction in strongly coupled strongly magnetized plasmas

Plasmas that are strongly magnetized in the sense that the gyrofrequency exceeds the plasma frequency exhibit novel transport properties that are not well understood. As a representative example, we compute the friction force acting on a massive test charge moving through a strongly coupled and strongly magnetized one-component plasma using a generalized Boltzmann kinetic theory. Recent works studying the weakly coupled regime have shown that strong magnetization leads to a transverse component of the friction force that is perpendicular to both the Lorentz force and velocity of the test charge, in addition to the stopping power component aligned antiparallel to the velocity. Recent molecular dynamics simulations have also shown that strong Coulomb coupling in addition to strong magnetization gives rise to a third component of the friction force in the direction of the Lorentz force. Here, we show that the generalized Boltzmann kinetic theory captures these effects and generally agrees well with the molecular dynamics simulations over a broad range of Coulomb coupling and magnetization strength regimes. The theory is also used to show that the “gyro” component of the friction in the direction of the Lorentz force arises due to asymmetries associated with gyromotion during short-range collisions. Computing the average motion of the test charge through the background plasma, the transverse force is found to strongly influence the trajectory by changing the gyroradius and the gyrofriction force is found to slightly change the gyrofrequency of the test charge resulting in a phase shift.

Thermal transport in superconductors with coexisting spin density wave order

We study thermal transport in a two-dimensional system with coexisting $s$- or $d$-wave Superconducting (SC) and Spin Density Wave (SDW) orders. We analyse the nature of coexistence phase in a tight-binding square lattice with $\mathbf{Q}=(\pi,\pi)$ SDW ordering. The electronic thermal conductivity is computed within the framework of the Boltzmann kinetic theory, using Born approximation for the impurity scattering collision integral. We describe the influence of the Fermi surface (FS) topology, the competition between the SC and SDW order parameters, the presence or absence of zero energy excitations in the coexistence phase, on the low temperature behavior of thermal conductivity of the various paring states. We present qualitative analytical, and fully numerical results that show that the heat transport signatures of various SC states emerging from collinear SDW order are quite distinct, and depend on the symmetry properties of the SC order parameter under translation by the SDW nesting vector $\mathbf{Q}$. {A combination of $(\pi,\pi)$-SDW and the $d_{x^2-y^2}$ pairing state results in fully gapped excitations, whereas $(\pi,\pi)$-SDW co-existing with either $d_{xy}$ or $s$-wave pairing states may always have gapless excitations. There appear special stable Dirac nodal points that are not gapped by the SC order in the coexistence phase, resulting in finite residual heat conductivity.

A generalized Boltzmann kinetic theory for strongly magnetized plasmas with application to friction

Coulomb collisions in plasmas are typically modeled using the Boltzmann collision operator, or its variants, which apply to weakly magnetized plasmas in which the typical gyroradius of particles significantly exceeds the Debye length. Conversely, O'Neil has developed a kinetic theory to treat plasmas that are so strongly magnetized that the typical gyroradius of particles is much smaller than the distance of the closest approach in a binary collision. Here, we develop a generalized collision operator that applies across the full range of magnetization strength. Since there is no closed-form solution for the scattering cross section when plasma is strongly magnetized, the input to the collision operator is obtained by numerically calculating particle trajectories. To demonstrate novel physics associated with strong magnetization, it is used to compute the friction force on a massive test charge. In addition to the traditional stopping power component, this is found to exhibit a transverse component that is perpendicular to both the velocity and Lorentz force vectors in the strongly magnetized regime, as was predicted recently using linear response theory. Good agreement is found between the collision theory and linear response theory in the regime in which both apply, but the new collision theory also applies to stronger magnetization strength regimes than the linear response theory is expected to apply in.

Numerical Issues for Solving Eu-type Generalized Hydrodynamic Equations to Investigate Continuum-rarefied Gas Flows

Eu-type generalized hydrodynamic equations have been derived from the Boltzmann kinetic theory and applied to investigate continuum and/or rarefied gas flows. This short communication first reports detailed and important issues in the use of the mixed discontinuous Galerkin method to solve Eu-type generalized hydrodynamic equations in multidimensions. Three major issues are reported. These include the treatment of solid boundary conditions for the nonlinear constitutive equations, a slope limiter to maintain high accuracy and avoid unphysical oscillations, and the computational efficiency compared with that of the particle method. In addition, we implement the present model to a rigid problem, which includes gas flows around the NACA0018 airfoil, a sharp wedge, a sphere and a three-dimensional Apollo configuration.

First-principles thermodynamic theory of Seebeck coefficients

Thermoelectric effects, measured by the Seebeck coefficients, refer to the phenomena in which a temperature difference or gradient imposed across a thermoelectric material induces an electrical potential difference or gradient, and vice versa, enabling the direct conversion of thermal and electric energies. All existing understanding and first-principles calculations of Seebeck coefficients have been based on the Boltzmann kinetic transport theory. Here we demonstrate that the Seebeck coefficient is a well-defined thermodynamic quantity that can be determined from the change in the chemical potential of electrons induced by the temperature change and thus can be efficiently computed solely based on the electronic density of states through first-principles calculations at different temperatures. The proposed approach is demonstrated using the prototype PbTe and SnSe thermoelectric materials. The proposed thermodynamic approach dramatically simplifies the calculations of Seebeck coefficients, making it possible to search for high performance thermoelectric materials using high-throughput first-principles calculations.

A Three-Dimensional Numerical Assessment of Heterogeneity Impact on a Solid Oxide Fuel Cell’s Anode Performance

In this research, a fully three-dimensional, multiphase, microstructure-scale heterogeneous (non-continuous) electrode, Solid Oxide Fuel Cell (SOFC) stack model is implemented in order to assess the impact of homogeneity disturbance in an SOFC anode. The Butler–Volmer model is combined with recent empirical relations for conductivity and aspects of the Maxwell–Boltzmann kinetic theory describing the transport of mass within the porous medium. Methods for the localized quantification of electrode morphology parameters (such as triple phase boundary length) are implemented. The exchange current distribution in the electrode, the partial pressures and the electric potential fields for each phase are computed numerically. In order to simulate heterogeneity, transfer barriers of varying placement and size are added to an otherwise homogeneous, virtual microstructure based on data from FIB-SEM tomography. The results are compared to a model based on the continuous electrode theory, and the points of discrepancy are highlighted.

Collision-dominated conductance in clean two-dimensional metals

We study the temperature-dependent corrections to the conductance due to electron-electron (e-e) interactions in clean two-dimensional conductors, such as lightly doped graphene or other Dirac matter. We use semiclassical Boltzmann kinetic theory to solve the problem of collision-dominated transport between reflection-free contacts. Time-reversal symmetry and the kinematic constraints of scattering in two dimensions (2D) ensure that inversion-odd and inversion-even distortions of the quasiparticle distribution relax with parametrically different rates at low temperature. This entails the surprising result that at lowest temperatures the conductance of very long samples tends to the noninteracting, ballistic conductance, despite the relaxation of the quasiparticle distribution to a drifting equilibrium. The relative correction to the conductance depends on the ratio of relaxation rates of even and odd modes and scales as delta G/G_{ballistic}~(T/E_F)[Log(E_F/T)]^{1/2}, in stark contrast to the behavior in other dimensionalities. This holds generally in 2D systems with simply connected and convex but otherwise arbitrary Fermi surfaces, as long as e-e scattering processes are dominant and umklapp scattering is negligible. These results are especially relevant to the bulk of wide and long suspended high-mobility graphene sheets.

Dispersion Relations for Proton Relaxation in Solid Dielectrics

Frequency-temperature spectra of the complex permittivity are studied for proton semiconductors and dielectrics using the methods of a quasi-classical kinetic theory of dielectric relaxation (the Boltzmann kinetic theory) in the linear approximation with respect to the polarizing field in the radio frequency range at temperatures T = 50-450 K. The effect of the quantum transitions of protons on the Debye dispersion relations is taken into account for crystals with hydrogen bonds (HBC) at low temperatures (50-100 K). The diffusion coefficients and the mobilities under electrical transfer of protons in the HBCs are constructed at high temperatures (100-350 K) in a non-linear approximation with respect to the polarizing field.

Why do Solutions of the Maxwell-Boltzmann Equation Tend to be Gaussians? A Simple Answer

The Maxwell-Boltzmann functional equation has recently attraction renewed interest since besides its importance in Boltzmann's kinetic theory of gases it also characterizes maximizers of certain bilinear estimates for solutions of the free Schrödinger equation. In this note we give a short and simple proof that, under some mild growth restrictions, any measurable complex-valued solution of the Maxwell-Boltzmann equation is a Gaussian. This covers most, if not all, of the applications.

Generalized Boltzmann kinetic theory for EMMS-based two-fluid model

It has long been recognized that the solid particles in circulating fluidized bed risers are distributed heterogeneously in the form of clusters. In response to this fundamental phenomenon, an EMMS-based two-fluid model has been developed recently from the viewpoint of continuum mechanics, however, its microscopic foundation remains unknown. In this study, the statistical mechanics foundation of EMMS-based two-fluid model was presented using generalized Boltzmann kinetic theory. With respect to the gas phase, a new method was developed by considering the fluctuations at different scales simultaneously, with which we can for the first time derive the correct governing equations of gas phase via kinetic theory, in the sense that both the molecular stress and the Reynolds (or pseudo-Reynolds) stress can be obtained simultaneously, whereas all previous kinetic theory analyses failed to predict the appearance of Reynolds (or pseudo-Reynolds) stress in the momentum conservation equation of gas phase due to the assumption of uniform structure, although it is physically always existent no matter how small the Reynolds number is. In case of particle phase, the generalized Boltzmann equation considering the spatio-temporal variation of the volume, density and velocity of clusters was firstly derived, a set of macroscopic transport equations was then derived in different phase spaces. It was shown that the governing equations of dense phase in the EMMS-based two-fluid model derived from continuum mechanics viewpoint corresponds to the macroscopic transport equations at (r,t) space. Therefore, present study launches a solid microscopic foundation of EMMS-based two-fluid model. Finally, CFD simulations have been carried out to validate EMMS-based two-fluid model and to study the effect of gas phase pseudo-turbulence.