Generalized Boltzmann Equation Research Articles

An efficient solution of the multi-term multi-harmonic electron Boltzmann equation for use in global models

Solving the electron Boltzmann equation is an essential but costly step in simulating low-temperature plasma kinetics. This work addresses the problem by introducing a solution of the general multi-term multi-harmonic Boltzmann equation (MTMH-BE) optimized for electrons in time-dependent non-equilibrium gases and electric fields. This is accomplished by configuring the numerical Jacobian as the product of a time-independent sparse matrix and an array of time-dependent coefficients. As this approach requires a fixed energy grid, the MTMH-BE is discretized using a finite volume scheme with support for non-uniform cell size, enabling robust solutions across a wide range of reduced field strengths. The new MTMH-BE solver is tested against a range of benchmarks, demonstrating excellent agreement with existing methods. Performance with time-dependent electron energy is evaluated using LoKI-B's pulsed electric field model, with the new solver demonstrating improved wall time scaling and an improvement in grid convergence of over three orders of magnitude. Alternatively, performance with quasi-stationary electrons in an evolving non-equilibrium gas is tested using a nitrogen cross-section set with up to 58 vibrational states and 3,570 processes. Here, the MTMH-BE solver achieves a wall time reduction of up to 99.6% over MultiBolt for error-controlled simulations in the two-term limit, reducing the average solution time from 1-2 seconds to less than 10 ms. These results indicate that the new MTMH-BE solver can improve accuracy and performance in state-to-state global models without the use of grid refinement, cross-section pruning, and other time-saving procedures.

Attenuation characteristics analysis of the THz circularly polarized waves in inhomogeneous fully ionized dusty plasma using FPL model

When the hypersonic vehicle is flying, the plasma in the area near the stagnation point of the front end of the vehicle can be approximately seen as the fully ionized dusty plasma. Due to the existence of dust particles, dusty plasma affects the communication quality of the hypersonic vehicle. In this paper, the general Boltzmann equation applicable to dusty plasmas containing electrons and the Fokker–Planck–Landau collision model are combined to derive a general formula for the electron distribution function of fully ionized dusty plasmas. Considering the contribution of the collision effect and charging effect to the dispersion relationship of fully ionized dusty plasma, the dielectric constant of fully ionized dusty plasma under an external magnetic field is solved. The Wentzel–Kramers–Brillouin method is used to calculate the attenuation coefficient (α) of the THz wave in fully ionized dusty plasma, and the influence of the external magnetic field strength and other dusty plasma parameters on the attenuation characteristics of the THz circularly polarized wave is analyzed. The research results show that the α of the THz left-hand circularly polarized wave decreases with the increase in the external magnetic field strength, while the α of the THz right-hand circularly polarized wave increases. In addition, increasing the dust particle radius, dust particle density, and electron density in a certain frequency range can increase the α of the THz circularly polarized waves. These research results provide theoretical guidance for the exploration of the interaction mechanism between the THz waves and fully ionized dusty plasma.

Perturbative Page curve induced by external impulse

In this work, we extend the recent study of entropy dynamics induced by an external impulse in open quantum systems, where the entropy response follows the Page curve. For small system-bath coupling κ, we expect that the entropy first increases exponentially κ2eϰt in the early-time regime t ≲ |log κ| due to quantum many-body chaos, and then decreases as e−λ0t with λ0 ∝ κ2 due to the energy relaxation. These results are confirmed through explicit calculations using two methods: (1) generalized Boltzmann equation for systems with quasi-particles; (2) scramblon effective theory in the early-time regime and perturbation theory in the late-time regime for 0+1-d systems. We also prove that in the second stage, the entropy of the system is equal to the coarse-grained entropy.

Information scrambling of the dilute Bose gas at low temperature

We calculate the quantum Lyapunov exponent $\lambda_L$ and butterfly velocity $v_B$ in the dilute Bose gas at temperature $T$ deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time ordered correlators, from which $\lambda_L$ and $v_B$ are extracted. At very low temperature where elementary excitations are phonon-like, we find $\lambda_L\propto T^5$ and $v_B\sim c$, the sound velocity. At relatively high temperature, we have $\lambda_L\propto T$ and $v_B\sim c(T/T_*)^{0.23}$. We find $\lambda_L$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on $T$. The chaos diffusion constant $D_L=v_B^2/\lambda_L$, on the other hand, differs from the energy diffusion constant $D_E$. We find $D_E\ll D_L$ at very low temperature and $D_E\gg D_L$ otherwise.

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

Introducing a lattice Boltzman time-domain method: A thermodynamics-based approach for simulating quantum effects.

In this work, using the generalized Boltzmann equationof Zadehgol [Phys. Rev. E 94, 023316 (2016)2470-004510.1103/PhysRevE.94.023316] a lattice Boltzmann time-domain (LBTD) method is proposed. The time-domain methods, such as the finite-difference time-domain method (FDTD), have been proposed by researchers, as tools in the study and design of semiconductor and optoelectronic devices. The LBTD inherits the main advantages of the lattice Boltzmann methods over the conventional methods, i.e., simplicity of the implementation, easy handling of complex geometries, and explicit algorithms which make the method highly suitable for efficient parallel processing. The theoretical findings have been verified by performing LBTD analysis on some benchmark structures.

Stability of measure solutions to a generalized Boltzmann equation with collisions of a random number of particles

In the paper we study a measure version of the evolutionary nonlinear Boltzmann-type equation in which we admit a random number of collisions of particles. We consider first a stationary model and use two methods to find its fixed points: the first based on Zolotarev seminorm and the second on Kantorovich-Rubinstein maximum principle. Then a dynamic version of Boltzmann-type equation is considered and its asymptotical stability is shown.

Hydrodynamics of weak integrability breaking

We review recent progress in understanding nearly integrable models within the framework of generalized hydrodynamics (GHD). Integrable systems have infinitely many conserved quantities and stable quasiparticle excitations: when integrability is broken, only a few residual conserved quantities survive, eventually leading to thermalization, chaotic dynamics and conventional hydrodynamics. In this review, we summarize recent efforts to take into account small integrability breaking terms, and describe the transition from GHD to standard hydrodynamics. We discuss the current state of the art, with emphasis on weakly inhomogeneous potentials, generalized Boltzmann equations and collision integrals, as well as bound-state recombination effects. We also identify important open questions for future works.

Wigner function and kinetic theory for massive spin-1/2 particles

We calculate the Wigner function for massive spin-1/2 particles in an inhomogeneous electromagnetic field to leading order in the Planck constant ħ. Going beyond leading order in ħ we then derive a generalized Boltzmann equation in which the force exerted by an inhomogeneous electromagnetic field on the particle dipole moment arises naturally. Furthermore, a kinetic equation for this dipole moment is derived. Carefully taking the massless limit we find agreement with previous results. The case of global equilibrium with rotation is also studied. Our framework can be used to study polarization effects induced by vorticity and magnetic field in relativistic heavy-ion collisions.

Subdynamics of fluctuations in an equilibrium classical many-particle system and generalized linear Boltzmann and Landau equations.

Exact completely closed homogeneous generalized master equations (GMEs) governing the evolution in time of equilibrium two-time correlation functions for dynamic variables of a subsystem of s particles (s<N) selected from N≫1 particles of a classical many-body system are obtained. These time-convolution and time-convolutionless GMEs differ from the known GMEs (e.g., Nakajima-Zwanzig GME) by the absence of inhomogeneous terms containing correlations between all N particles at the initial moment of time and preventing the closed description of s-particle subsystem evolution. Closed homogeneous GMEs describing the subdynamics of fluctuations are obtained by applying a special projection operator to the Liouville-type equation governing the dynamics of the correlation function with the related to the Gibbs distribution initial state, which is more natural than the conventional factorized initial state. No common approximation, like the "molecular chaos," is needed. In the linear approximation in the particles' density, the linear generalized Boltzmann equation accounting for initial correlations and valid at all timescales is obtained. This equation for a weak interparticle interaction converts into the generalized linear Landau equation in which the initial correlations are also accounted for. The connection of these equations to the nonlinear Boltzmann and Landau equations is discussed. The same approach is applicable to studying the kinetics of the conventional reduced s-particle distribution functions for a classical N-particle system driven from an equilibrium state by an external force.

Dynamical evolution of gravitational leptogenesis

Radiatively-induced gravitational leptogenesis is a potential mechanism to ex- plain the observed matter-antimatter asymmetry of the universe. Gravitational tidal effects at the quantum loop level modify the dynamics of the leptons in curved spacetime and may be encoded in a low-energy effective action Seff . It has been shown in previous work how in a high-scale BSM theory the CP odd curvature-induced interactions in Seff modify the dispersion relations of leptons and antileptons differently in an expanding universe, giving rise to an effective chemical potential and a non-vanishing equilibrium lepton-antilepton asymmetry. In this paper, the CP even curvature interactions are shown to break lepton number current conservation and modify the evolution of the lepton number density as the universe expands. These effects are implemented in a generalised Boltzmann equation and used to trace the dynamical evolution of the lepton number density in different cosmological scenarios. The theory predicts a potentially significant gravitationally-induced lepton-antilepton asymmetry at very early times in the evolution of the universe.

Semiclassical theory of the circular photogalvanic effect in gyrotropic systems

We develop a theory of circular photogalvanic effect (CPGE) for classically high photon energies which exceed the electron scattering rate but are small compared to the average electron kinetic energy. In this frequency range one can calculate the CPGE by using two different approaches. In the fully quantum-mechanical approach we find the photocurrent density by applying Fermi's golden rule for indirect intraband optical transitions with virtual intermediate states both in the conduction and valence bands. In the framework of the semiclassical approach, we apply a generalized Boltzmann equation with accounts for the Berry-curvature induced anomalous velocity, side jumps and skew scattering. The calculation is carried out for a wurtzite symmetry crystal. Both methods yield the same results for the CPGE current demonstrating consistency between the two approaches and applicability of the semiclassical theory for the description of nonlinear high-frequency transport.

Transport theory within a generalized Boltzmann equation for multiband wave packets

The authors present a multiband transport theory of thermoelectric currents that relies on a density operator for multiband wave packets, whose dynamics is captured in a generalized equation of motion.

On the phase space in Double Field Theory

We present a model of (double) kinetic theory which paves the way to describe matter in a Double Field Theory background. Generalized diffeomorphisms acting on double phase space tensors are introduced. The generalized covariant derivative is replaced by a generalized Liouville operator as it happens in relativistic kinetic theory. The section condition is consistently extended and the closure of the generalized transformations is still given by the C-bracket. In this context we propose a generalized Boltzmann equation and compute the moments of the latter, obtaining an expression for the generalized energy-momentum tensor and its conservation law.

Heat conductivity and Dufour effect in the weakly ionized, magnetized, and kappa‐distributed plasma

AbstractBy using the generalized Boltzmann equation of transport and the first‐order approximation of Chapman‐Enskog expansion on the κ‐distribution function, we study the thermal conductivity and Dufour effect in the weakly ionized and magnetized plasma. We show that the thermal conductivity and Dufour coefficient in the κ‐distributed plasma are significantly different from those in the Maxwell‐distributed plasma, and the transverse thermal conductivity and Dufour coefficient in the κ‐distributed plasma are generally greater than those in the Maxwell‐distributed plasma, and the Righi‐Leduc coefficient and Hall Dufour coefficient in the κ‐distributed plasma are also generally greater than those in the Maxwell‐distributed plasma.

Effect of magnetic field on transports of charged particles in the weakly ionized plasma with power-law q-distributions in nonextensive statistics

By using the generalized Boltzmann equation of transport in nonextensive statistics, we study transport properties of the diffusion flux and the heat flux of charged particles in the weakly ionized plasma with the power-law q-distributions under the magnetic field. We derive the tensor expressions of diffusion coefficient, thermal diffusion coefficient, mobility and thermal conductivity of electrons and ions in the q-distributed plasma with magnetic field. We show that the tensors of the diffusion coefficient, the thermal diffusion coefficient and the thermal conductivity strongly depend on the q-parameters, and so they are generally not the same as those in the magnetized plasma with a Maxwellian distribution.

Kinetic theory for massive spin−1/2 particles from the Wigner-function formalism

We calculate the Wigner function for massive spin-1/2 particles in an inhomogeneous electromagnetic field to leading order in the Planck constant $\hbar$. Going beyond leading order in $\hbar$ we then derive a generalized Boltzmann equation in which the force exerted by an inhomogeneous electromagnetic field on the particle dipole moment arises naturally. Carefully taking the massless limit we find agreement with previous results. The case of global equilibrium with rotation is also studied. Finally, we outline the derivation of fluid-dynamical equations from the components of the Wigner function. The conservation of total angular momentum is promoted as an additional fluid-dynamical equation of motion. Our framework can be used to study polarization effects induced by vorticity and magnetic field in relativistic heavy-ion collisions.

Quantum chaos for the unitary Fermi gas from the generalized Boltzmann equations

In this paper, we study the chaotic behavior of the unitary Fermi gas in both high- and low-temperature limits by calculating the quantum Lyapunov exponent defined in terms of the out-of-time-order correlator. We take the method of generalized Boltzmann equations, derived from the augmented Keldysh approach (Aleiner et al 2016 Ann. Phys. 375, 378). At high temperatures, the system is described by weakly interacting fermions with two spin components and the Lyapunov exponent is found to be . Here n is the density of fermions for a single-spin component. In the low-temperature limit, the system is a superfluid and can be described by phonon modes. Using the effective action derived in (Son and Wingate 2006 Ann. Phys. 321, 197), we find where TF is the Fermi energy. By comparing these to existing results of heat conductivity, we find that where DE is the energy diffusion constant and v is some typical velocity. We argue that this is related to the conservation law for such systems with quasi-particles.

Collective induced antidiffusion effect and general magnon Boltzmann transport theory

Magnon transport and its efficient manipulation in FIs (ferromagnetic insulators) are important issues in magnon spintronics research, while the relaxation and decay of magnons are still in debates. In this paper, starting from the Hamiltonian of a magnetically ordered system, we derive and solve the general magnon Boltzmann equation. It is found that there exists an ``antidiffusion driving effect'' in magnon transport, which is caused by the collective interaction of a high density (or strong magnetic field) magnon gas in the limited magnetic background. This effect acts as a ``spin pressure'' on magnons, resulting in a decrease in magnon propagation length and an extra accumulation near the magnon generation boundary. By applying the derived magnon Boltzmann equation to several cases and comparing with the experiments, it is shown that our theory cannot only describe the influence of the external magnetic field on magnon transport but also confirm the fast drop regime in magnon transport is attributed to the boundary effect. The magnon Boltzmann method also predicts a new pathway for manipulations of magnon transport in FIs.

The effect of magnetic field on the viscosity in the weakly ionized and magnetized plasma with power-law q-distributions in nonextensive statistics

We study the effect of magnetic field on the viscosity of charged particles in the weakly ionized and magnetized plasma with the power-law q-distributions by using the generalized Boltzmann equation of transport in nonextensive statistics. We derive six viscosity coefficients of electrons and ions in the complex plasma, including the magnetic field introduced four viscosity coefficients, where the q-parameter and magnetic field both play significant roles. By numerical analyses, we show that these viscosity coefficients depend strongly on the magnetic field and q-parameters of the weakly ionized and magnetized plasma, and so have potential applications in low temperature complex plasmas.