Landau Kinetic Equation Research Articles

Radially Symmetric Models of the Landau Kinetic Equation and High Energy Tails

Radially Symmetric Models of the Landau Kinetic Equation and High Energy Tails

Collective effects in an incompressible electronic liquid.

Starting from Landau's kinetic equation, we show that an electronic liquid in d = 2, 3 spatial dimensions depicted by a Landau-type effective theory will become incompressible on condition that the Landau parameters satisfy either (i) [Formula: see text] or (ii) [Formula: see text]. Condition (i) is the Pomeranchuk instability in the current channel and suggests a quantum spin liquid (QSL) state with a spinon Fermi surface; while condition (ii) means that the strong repulsion in the charge channel leads to a conventional charge and thermal insulator. In the collisionless regime (ωτ ≫ 1) and the hydrodynamic regime (ωτ ≪ 1), the zero and first sound modes have been studied and classified by symmetries, including the longitudinal and transverse modes in d = 2, 3 and the higher angular momentum modes in d = 3. The sufficient (and/or necessary) conditions of these collective modes have been revealed. It has been demonstrated that some of these collective modes will behave in quite different manners under incompressibility condition (i) or (ii). Possible nematic QSL states and a hierarchy structure for gapless QSL states have been proposed in d = 3.

Fluctuating quantum kinetic theory

We consider a quantum Langevin kinetic equation for a system of fermions. We first derive the Langevin force noise correlation functions in Landau's Fermi-liquid kinetic theory from general considerations. We then use the resulting equation to calculate the equilibrium dynamic structure factor in the collisionless regime at low temperatures. The result is in agreement with the conventional many-body result. We then use the theory to derive both the fluctuating Navier-Stokes equations for a quantum fluid and the fluctuating hydrodynamic equations for fermions in the presence of quenched disorder. We also discuss the modifications needed for the fluctuating hydrodynamic equations to describe an electron fluid with long-ranged interactions, and we prove an $H$ theorem for the nonlinear Landau kinetic equation.

Pairing Fluctuation Corrections to the Kinetic Theory of Liquid $$^3\mathrm {He}$$

Liquid $$^3\mathrm {He}$$ is a Fermi liquid that undergoes a BCS-type phase transition to a spin-triplet superfluid, making it valuable for understanding interacting fermions. When the temperature approaches the transition temperature $$T_{\mathrm {c}}$$ from above, physical properties can be modified by Cooper pair fluctuations, leading to deviations from the predictions of Fermi liquid theory. We use nonequilibrium Green’s function theory to study the role of pair fluctuations on quasiparticle transport. The Boltzmann–Landau kinetic equation, which describes the transport properties of Fermi liquids, acquires corrections from the interaction of quasiparticles with pair fluctuations. As an application, we study the effects of pair fluctuations on the propagation of zero sound in liquid $$^3\mathrm {He}$$ . We calculate the temperature-dependent correction to the zero sound velocity due to pair fluctuations and compare the result with existing experimental data.

Vortex domain configuration for energy-storage ferroelectric ceramics design: A phase-field simulation

The utilization of ferroelectrics in forms of ceramics, films, and composites toward energy-storage applications is of great interest recent years. However, the simultaneous achievement of high polarization, high breakdown strength, low energy loss, and weakly nonlinear polarization–electric field (P–E) correlation has been a huge challenge, which impedes progress in energy storage performance. In this work, a vortex domain engineering constructed via the core–shell structure in ferroelectric ceramics is proposed. The formation and the switching characteristics of vortex domains (VDs) were validated through a phase-field simulation based on the time-dependent Ginzburg–Landau kinetic equation. Benefiting from the smaller depth of a potential well in the energy profiles, the switching of VDs was much easier than that of conventional large-sized domains, which was found to be the origin of the lower coercive field, lower remanent polarization, and weaker nonlinear P–E correlation. Choosing BaTiO3 (BT) as a representative of ferroelectric ceramics, the shell fractions and permittivity values were varied in our phase-field simulation to optimize the energy storage performance. As a result, a large discharge energy of 6.5 J/cm3 was obtained in BT ferroelectric ceramics with a shell fraction of 5% and a shell permittivity of 20 under the applied electric field of 100 kV/mm, which is almost 140% higher than that with no shell structure. In general, the vortex domain engineering proposed in this work can serve as a universal method in designing high-performance ferroelectrics with simultaneous high breakdown strength, high discharge energy density, and high energy efficiency.

Landau kinetic equation for dry aligning active models

The Landau equation is a kinetic equation based on the weak coupling approximation of the interaction between the particles. In the framework of dry active matter this new kinetic equation relies on the weak coupling approximation of both the alignment strength and the magnitude of the angular noise, instead of the hypothesis of diluteness. Therefore, it is a kinetic equation bridging between the Boltzmann (Bertin et al 2006 Phys. Rev. E 74 022101), and the Smoluchowski (Baskaran et al 2010 J. Stat. Mech. P04019) approximations, and allowing analytical descriptions at moderate densities. The form of the equation presents non-linear and density dependent diffusions and advections fully derived by the microscopic equations of motions. Finally, implementing the BGL procedure (Peshkov et al 2014 Eur. Phys. J. Spec. Top. 223 1315–44), the parameters of the Toner–Tu equations are derived showing the appearance of linearly stable homogeneous ordered solutions and mimicking the results obtained from the Boltzmann approach.

Atypical behavior of collective modes in two-dimensional Fermi liquids

Using the Landau kinetic equation to study the non-equilibrium behavior of interacting Fermi systems is one of the crowning achievements of Landau’s Fermi liquid theory. While thorough study of transport modes has been done for standard three-dimensional Fermi liquids, an equally in-depth analysis for two dimensional Fermi liquids is lacking. In applying the Landau kinetic equation (LKE) to a two-dimensional Fermi liquid, we obtain unconventional behavior of the zero sound mode c0. As a function of the usual dimensionless parameter s = ω/qvF, we find two peculiar results: first, for |s| > 1 we see the propagation of an undamped mode for weakly interacting systems. This differs from the three dimensional case where an undamped mode only propagates for repulsive interactions and the mode experiences Landau damping for any arbitrary attractive interaction. Second, we find that regardless of interaction strength, a propagating mode is forbidden for |s| < 1. This is profoundly different from the three-dimensional case where a mode can propagate, albeit damped. In addition, we present a revised Pomeranchuk instability condition for a two-dimensional Fermi liquid as well as equations of motion for the fluid that follow directly from the LKE. In two dimensions, we find a constant minimum for all Landau parameters for ℓ ⩾ 1 which differs from the three dimensional case. Finally we discuss the effect of a Coulomb interaction on the system resulting in the plasmon frequency ωp exhibiting a crossover to the zero sound mode.

On relaxation processes in a completely ionized plasma

Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma. The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in one- and two-polynomial approximation.

Corrections to the Landau kinetic equation for a weakly dissipative randomly driven system and the fluctuation-dissipation theorem

This paper is devoted to the derivation of a kinetic equation for many-body one-component dissipative systems in an external random field. The system under consideration was discussed in the recent paper by Sliusarenko et al. [J. Math. Phys. 56, 043302 (2015)], where a generalization of the Vlasov kinetic equation was obtained. The potential interaction is assumed to be small (the corresponding small parameter λ), the dissipation interactions and the correlation functions of the external random field are considered as small quantities estimated by one small parameter μ for the simplicity. The kinetic equation is obtained up to the terms of the orders λ2μ0, λ1μ1, λ0μ2 inclusive. In weakly spatially nonuniform states of the system, this gives corrections to the Landau–Vlasov kinetic equation caused by the dissipation and external random field. In the case λ ≫ μ after the mean free time, the system reaches a state approximately described by the Maxwell distribution. The dissipation and random field will lead to the evolution of the system temperature. A time evolution equation for the temperature in spatially uniform states is derived on the basis of the obtained kinetic equation by a generalized Chapman–Enskog method which takes into account that the kinetic equation is an approximate one. This temperature time evolution equation is investigated up to the terms of the order μ3 inclusive. It is shown that under some conditions the final stage of the system evolution is a steady state. A fluctuation-dissipation theorem for this state is discussed. In this steady state, the system is described by a distribution function that contains corrections to the Maxwell distribution.

Long Wave Asymptotics for the Vlasov–Poisson–Landau Kinetic Equation

The work is devoted to some mathematical problems of dynamics of collisional plasma. The difficulty is that in plasma case we have at least three different length scales: Debye radius $$ r_D $$ , mean free path l and macroscopic length L. This is true even for the simplest model (plasma of electrons with a neutralizing background of infinitely heavy ions), considered in the paper. We study (at the formal level of mathematical rigour) solutions of the VLPE, having the typical length of the order $$ l>> r_D $$ , and try to clarify some mathematical questions related to corresponding limit. In particular, we study the existence of the limit for electric field and show that, generally speaking, it does not exist because of rapidly oscillating terms. An approximate asymptotic formula for the oscillating electric field near this limit is derived from VLPE. Still the limiting equations, which are used in many publications by physicists, can lead in some cases to correct results for the distribution function. Both these conclusions are confirmed by more explicit analysis of the linearized Vlasov–Poisson equation. We also study the well-posedness of limiting kinetic equations and the corresponding criterion in the class of weakly inhomogeneous initial data. It is shown that the collisional effects do not play an important role in this problem. In particular the equations are well-posed in the case of small deviations from equilibrium, as it was already known for related collisionless models.

Three dimensional phase-field simulations on the frequency dependence of polarization vectors and hysteresis loops in ferroelectric crystals

The phase field approach has been widely used to study the domain structure of ferroelectric crystals in both two and three dimensions (2D and 3D), but in the 3D case, little has been done to address the frequency dependence of ferroelectric characteristics. In this work, we adopt the 3D time-dependent Ginzburg–Landau kinetic equation to calculate the evolution of local polarization vectors and the overall hysteresis loops of ferroelectric crystals under the frequencies from 0.4 kHz to 120 kHz, and then use the fast Fourier transform to analyze the frequency characteristics of the polarizations. It shows the phenomenon of multiple frequencies at low field frequency but not at high one. The distribution and evolution of polarization vectors in x, y, and z directions are obtained, and various forms of electrical hysteresis loops are found from the average of local polarization vectors. The results indicate that, as the frequency increases, the hysteresis loops of Pz versus Ez change from the standard shape to the oval shape, but the loops for Px and Py change from the dumbbell shape to an oblique ellipse, and then to figure-eight curve and eventually to the superparaelectric one. The detailed distribution and evolution of the polarization vectors in the crystal are also vividly displayed. Finally, the effects of lattice size, amplitude of the applied field, depolarization energy, and the initial state of polarizations in the crystal are investigated. It shows that the nature of polarization evolution in a 3D crystal is highly complex and that each of these factors can have a significant effect.

Collisional transport coefficients of dense high-temperature plasmas within the quantum Landau-Fokker-Planck framework

We extend the long-established formulas for the transport coefficients of classical plasmas inside the dense plasma regime for temperatures and densities where the classical Landau equation breaks down but its quantum extension that includes quantum degeneracy effects is valid. To this end, the quantum Landau kinetic equation is solved by the Chapman-Enskog method. The mathematical derivation is done in full generality, i.e., for multicomponent systems and to all orders of the polynomials expansion used to approximate the distribution functions. We apply the general results to two important examples, the electron gas model and an electron-ion plasma model consisting of one type of ions of any charge. We discuss the combined effects of the Pauli exclusion principle, of the electron-electron, and of the electron-ion collisions on the transport coefficients and on the convergence of the Chapman-Enskog method. For the electron gas model, the effect of the Pauli exclusion principle on the transport coefficients rapidly becomes non-negligible outside the domain of validity of the classical Landau equation. For the electron-ion plasmas, the effect of the Pauli exclusion principle depends sensitively on the ion charge Z and varies non-monotonically with Θ. For instance, for ion charge Z = 1, the electrical conductivity is increased by up to ∼30% compared to its classical value over the range of degeneracy parameters studied, the thermal conductivity is reduced by up to ∼9%, and the shear viscosity coefficient is increased by up to ∼13%. In the Lorentz gas (Z→∞) limit, the electrical conductivity is reduced by up to ∼14% compared to its classical value over the range of degeneracy parameters studied, the thermal conductivity is reduced by up to ∼39%, and the shear viscosity coefficient is not affected.

Kinetic theory of transport driven current in centrally fuelled plasmas

When a steady-state cylindrical plasma discharge is centrally fuelled, the collisionless radial electron flux is canonically coupled to an axial current. The identification and analysis of this transport driven current, previously reported in collisionless simulations [W. J. Nunan and J. M. Dawson, Phys. Rev. Lett. 73, 1628 (1994)], are addressed analytically and extended to the collisional regime by means of first-principles kinetic models. Collisionless radial transport is described with the standard quasilinear model and collisional velocity anisotropy relaxation with the Landau kinetic equation. When trapped particle corrections are taken into account, the solution of this kinetic model provides the analytical expression for the transport driven current in a centrally fuelled steady-state tokamak as a function of the thermonuclear power and discharge parameters. For ITER type discharges, with central fuelling, a current of about one mega-ampere is predicted by this first-principles analytical kinetic model.

Effects of epitaxial strain, film thickness and electric-field frequency on the ferroelectric behavior of BaTiO3 nano films

Ferroelectric nano films exhibit distinctive properties that are influenced by aspects such as the epitaxial strain, film thickness, and electric-field frequency. In this work, we studied the effects of the above factors on the ferroelectric behavior of BaTiO3 nano films with a phase-field model based on the time-dependent Ginzburg–Landau kinetic equation. For the study of the epitaxial strain, we considered the cases with either compressive or tensile strain, ranging from −3.2% to 1%. Our results showed that the compressive epitaxial strain can generally enhance the remnant polarization and the coercive field, as opposed to the effect of the tensile one. As the film thickness reduces from 100 to 5 nm, the remnant polarization and the coercive field tend to increase, whereas the dielectric constant and the piezoelectric constant tend to decrease. We further investigated the effect of the high-strength electric-field frequency that varies from 0.1 to 100 kHz under two epitaxial-strain values, i.e., −0.1% and −1.74%, respectively. It was found that, at low frequencies, the coercive field increases sharply and the remnant polarization increases slightly with the rise of the applied frequency. Once the frequency gets higher than 20 kHz, the remnant polarization drastically reduces with the increasing frequency. In the meanwhile, the hysteresis loop exhibits an elliptic shape and the sharp tails of the butterfly loop are diminishing. We noticed that the frequency dependency is more sensitive to the epitaxial strain within the low-frequency range. Based on the obtained microstructural evolution process, we found that the 180° polarization reversal is more likely to occur under low-frequency field, while the polarization cannot be fully reversed in high-frequency hysteresis.

Electrostatic fluctuations in collisional plasmas.

We present a theory of electrostatic fluctuations in two-component plasmas where electrons and ions are described by Maxwellian distribution functions at unequal temperatures. Based on the exact solution of the Landau kinetic equation, that includes electron-electron, electron-ion, and ion-ion collision integrals, the dynamic form factor, S(k[over ⃗],ω), is derived for weakly coupled plasmas. The collective plasma responses at ion-acoustic, Langmuir, and entropy mode resonances are described for arbitrary wave numbers and frequencies in the entire range of plasma collisionality. The collisionless limit of S(k[over ⃗],ω) and the strong-collision result based on the fluctuation-dissipation theorem and classical transport at T_{e}=T_{i} are recovered and discussed. Results of several Thomson scattering experiments in the broad range of plasma parameters are described and discussed by means of our theory for S(k[over ⃗],ω).

Generalization of the Grad method in plasma physics

The Grad method is generalized based on the Bogolyubov idea of the functional hypothesis for states at the end of relaxation processes in a system. The Grad problem (i.e., description of the Maxwell relaxation) for a completely ionized spatially uniform two-component electron-ion plasma is investigated using the Landau kinetic equation. The component distribution functions and time evolution equations for parameters describing the state of a system are calculated, and corrections are obtained to the known results in a perturbation theory in a small electron-to-ion mass ratio.

On the Rate of Relaxation for the Landau Kinetic Equation and Related Models

We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purely exponential relaxation for large values of time \(t>0\). One of the main goals of our work is to numerically identify the large time asymptotics for the relaxation to equilibrium. We recall the work of Strain and Guo (Arch Rat Mech Anal 187:287–339 2008, Commun Partial Differ Equ 31:17–429 2006), who rigorously show that the expected law of relaxation is \(\exp (-ct^{2/3})\) with some \(c > 0\). In this manuscript, we find an heuristic way, performed by asymptotic methods, that finds this “law of two thirds”, and then study this question numerically. More specifically, the linear Landau equation is approximated by a set of ODEs based on expansions in generalized Laguerre polynomials. We analyze the corresponding quadratic form and the solution of these ODEs in detail. It is shown that the solution has two different asymptotic stages for large values of time t and maximal order of polynomials N: the first one focus on intermediate asymptotics which agrees with the “law of two thirds” for moderately large values of time t and then the second one on absolute, purely exponential asymptotics for very large t, as expected for linear ODEs. We believe that appearance of intermediate asymptotics in finite dimensional approximations must be a generic behavior for different classes of equations in functional spaces (some PDEs, Boltzmann equations for soft potentials, etc.) and that our methods can be applied to related problems.

Plasma kinetic coefficients with account for relaxation processes

The hydrodynamics of a completely ionized two-component electron–ion plasma is investigated on the basis of the Landau kinetic equation with account for component temperature and velocity relaxation. The term relaxation is understood here in a narrow sense as a process which can take place in a spatially homogeneous system. The investigation is made on the basis of a generalized Chapman–Enskog method. The relaxation is investigated in the vicinity of the standard hydrodynamic state of plasma: the deviations of the component temperatures and velocities from their standard hydrodynamic values are considered to be small. The integral equations for the component distribution functions are investigated in a perturbation theory in small gradients and small relaxation parameters. The component distribution functions of the first-order both in gradients and in relaxation parameters are calculated. Relaxation corrections to the energy and momentum fluxes and to the kinetic coefficients of the standard hydrodynamic state of the system are obtained.

On Some Properties of the Landau Kinetic Equation

We discuss some general properties of the Landau kinetic equation. In particular, the difference between the “true” Landau equation, which formally follows from classical mechanics, and the “generalized” Landau equation, which is just an interesting mathematical object, is stressed. We show how to approximate solutions to the Landau equation by the Wild sums. It is the so-called quasi-Maxwellian approximation related to Monte Carlo methods. This approximation can be also useful for mathematical problems. A model equation which can be reduced to a local nonlinear parabolic equation is also constructed in connection with existence of the strong solution to the initial value problem. A self-similar asymptotic solution to the Landau equation for large v and t is discussed in detail. The solution, earlier confirmed by numerical experiments, describes a formation of Maxwellian tails for a wide class of initial data concentrated in the thermal domain. It is shown that the corresponding rate of relaxation (fractional exponential function) is in exact agreement with recent mathematically rigorous estimates.

Hydrodynamic, Kinetic Modes of Plasma and Relaxation Damping of Plasma Oscillations

The hydrodynamics of a completely ionized two-component electron-ion plasma is investigated at the end of the component temperature and velocity relaxation. The problem of accounting for the peculiarities of the Coulomb interaction in the plasma kinetics is discussed. The investigation is based on the Landau kinetic equation and the Chapman–Enskog method generalized on the basis of the Bogolyubov idea of the functional hypothesis. Nonlinear hydrodynamic equations are obtained. Linearized hydrodynamic equations are built, and the hydrodynamic and kinetic modes of the Landau kinetic equation are investigated in the hydrodynamic approximation. The effect of the relaxation processes on the evolution of the system is investigated. On the basis of the Vlasov–Landau equation, the plasma modes are investigated in the main hydrodynamic approximation. Some of them describe the relaxation damping of plasma oscillations, which is much more important than the Landau damping at small wave vectors k → 0.