Linearized Landau Equation Research Articles

A Remark about Time-Analyticity of the Linear Landau Equation with Soft Potential

A Remark about Time-Analyticity of the Linear Landau Equation with Soft Potential

The analytic smoothing effect of linear Landau equation with soft potentials

The analytic smoothing effect of linear Landau equation with soft potentials

Variance reduced particle solution of the Fokker-Planck equation with application to rarefied gas and plasma dynamics

We develop an importance-weight-based variance reduction method for direct Monte Carlo simulations of the Fokker-Planck equation with applications to rarefied gas and plasma dynamics. We show that there exists a class of Fokker-Planck equations that admit stable weight evolution processes along particle trajectories, including those with drift and diffusion coefficients independent of moments higher than the kinetic energy, such as the linearized Landau equation. When drift or diffusion coefficients depend on higher-order moments, such as in the cubic Fokker-Planck equation, maintaining stability and accuracy of the weight evolution becomes challenging. In this work, stability and conservation for the cubic FP model are achieved using the maximum cross-entropy formulation introduced in recent work Sadr and Hadjiconstantinou (2023) [30]. Several test cases show that significant speed-up is obtained using the proposed variance reduced method compared to the standard Monte Carlo solution in the low-signal limit.

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.

Well-posedness of Cauchy problem for Landau equation in critical Besov space

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.

The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit

We consider a system of N particles interacting via a short-range smooth potential, in a weak-coupling regime. This means that the number of particles \begin{document}$N$\end{document} goes to infinity and the range of the potential \begin{document}$e$\end{document} goes to zero in such a way that \begin{document}$Ne^{2} = α$\end{document} , with \begin{document}$α$\end{document} diverging in a suitable way. We provide a rigorous derivation of the Linear Landau equation from this particle system. The strategy of the proof consists in showing the asymptotic equivalence between the one-particle marginal and the solution of the linear Boltzmann equation with vanishing mean free path. This point follows [ 3 ] and makes use of technicalities developed in [ 16 ]. Then, following the ideas of Landau, we prove the asympotic equivalence between the solutions of the Boltzmann and Landau linear equation in the grazing collision limit.

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.

Derivation of the Linear Landau Equation and Linear Boltzmann Equation from the Lorentz Model with Magnetic Field

We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in \cite{BMHH}, which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.

A Diffusion Limit for a Test Particle in a Random Distribution of Scatterers

We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.

Relativistic hydrodynamics from the projection operator method

We study relativistic hydrodynamics in the linear regime, based on Mori's projection operator method. In relativistic hydrodynamics, it is considered that an ambiguity about the fluid velocity occurs from the choice of a local rest frame: the Landau and Eckart frames. We find that the difference of the frames is not the choice of the local rest frame, but rather that of dynamic variables in the linear regime. We derive hydrodynamic equations in both frames by the projection operator method. We show that the natural derivation gives the linearized Landau equation. Also we find that, even for the Eckart frame, the slow dynamics is actually described by the dynamic variables for the Landau frame.

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker–Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.

Rigorous derivation of the Landau equation in the weak coupling limit

We examine a family of microscopic models of plasmas, with a parameter $\alpha$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $\alpha$ to the whole range $(0, 1/2]$, by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusion coefficient of the Landau equation and show it to be independent of the parameter.

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inRv is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith→0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.

Kinetic theory in the weak-coupling approximation: III. Classical systems: extension of the theory and further applications

The kinetic theory of open classical systems in the weak-coupling approximation presented in a previous paper (paper I), is extended here to a wider class of systems so that it is applicable when the Liouville operator of the open system has a continuous spectrum. It is applied (i) to an anharmonic-oscillator and (ii) to a free test-particle model, thus leading to a generalization of the conventionally used linearized Landau equation. Problems related to the divergences that appear when the kinetic equations, obtained by the formalism of paper I, are considered in the limit of a continuous spectrum of the open system's Liouville operator are also discussed.

Kinetic theory in the weak-coupling approximation: II. A classical harmonic oscillator model

The kinetic theory of open systems presented in a previous paper is applied to a classical harmonic oscillator model. It leads to a kinetic equation, very different from the conventionally used linearized Landau equation, that shows explicitly the influence of the field on the interaction of the harmonic particle with its surroundings, contrary to what is often assumed. A self-consistent approach based on our equation, suggests a one-dimensional generalization of Kramers' model equation. The limit of our equation when the harmonic frequency vanishes contains a divergent term whose nature is briefly discussed.

The Cauchy problem for the linearized Landau equation

A theorem of the existence as a whole and the uniqueness of the classical solution of the Cauchy problem for the linerized Landau equation is proved.