Grazing Collision Limit Research Articles

The grazing collisions limit from the linearized Boltzmann equation to the Landau equation for short-range potentials

The Landau equation and the Boltzmann equation are connected through the limit of grazing collisions. This has been proved rigorously for certain families of Boltzmann operators concentrating on grazing collisions. In this contribution, we study the collision kernels associated to the two-particle scattering via a finite range potential $ \Phi(x) $ in three dimensions. We then consider the limit of weak interaction given by $ \Phi_ \epsilon(x) = \epsilon \Phi(x) $. Here $ \epsilon\rightarrow 0 $ is the grazing parameter, and the rate of collisions is rescaled to obtain a non-trivial limit. The grazing collisions limit is of particular interest for potentials with a singularity of order $ s\geq 0 $ at the origin, i.e. $ \phi(x) \sim |x|^{-s} $ as $ |x|\rightarrow 0 $. For $ s\in [0,1] $, we prove the convergence to the Landau equation with diffusion coefficient given by the Born approximation, as predicted in the works of Landau and Balescu. On the other hand, for potentials with $ s>1 $ we obtain the non-cutoff Boltzmann equation in the limit. The Coulomb singularity $ s = 1 $ appears as a threshold value with a logarithmic correction to the diffusive timescale, the so-called Coulomb logarithm.

Boltzmann to Landau from the gradient flow perspective

We revisit the grazing collision limit connecting the Boltzmann equation to the Landau(-Fokker–Planck) equation from their recent reinterpretations as gradient flows. Our results are in the same spirit as the Γ-convergence of gradient flows technique introduced by Sandier and Serfaty [39]; Serfaty [41]. In this setting, the grazing collision limit reduces to showing the lower semi-continuous convergence of the Boltzmann entropy-dissipation to the Landau entropy-dissipation.

Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

We consider a homogeneous plasma composed of $N$ particles of the same electric charge which interact through a Coulomb potential. In the large plasma parameter limit, classical kinetic theories justify that the empirical density is the solution of the Balescu-Guernsey-Lenard equation, at leading order. This is a law of large numbers. The Balescu-Guernsey-Lenard equation is approximated by the Landau equation for scales much smaller than the Debye length. In order to describe typical and rare fluctuations, we compute for the first time a large deviation principle for dynamical paths of the empirical density, within the Landau approximation. We obtain a large deviation Hamiltonian that describes fluctuations and rare excursions of the empirical density, in the large plasma parameter limit. We obtain this large deviation Hamiltonian either from the Boltzmann large deviation Hamiltonian in the grazing collision limit, or directly from the dynamics, extending the classical kinetic theory for plasmas within the Landau approximation. We also derive the large deviation Hamiltonian for the empirical density of $N$ particles, each of which is governed by a Markov process, and coupled in a mean field way. We explain that the plasma large deviation Hamiltonian is not the one of $N$ particles coupled in a mean-field way.

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.

Kinetic description of collision avoidance in pedestrian crowds by sidestepping

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

Sharp bounds for Boltzmann and Landau collision operators

The aim of the work is to provide a stable method to get sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. All the sharp bounds are given for the original Boltzmann and Landau operators. The sharpness means the lower and upper bounds for the operators are consistent with the behavior of the linearized operators. Moreover, we make clear the difference between the bounds for the original operators and those for the linearized ones. According to the Bobylev's formula, we introduce two types of dyadic decompositions performed in both phase and frequency spaces to make full use of the interaction and the cancellation. It allows us to see clearly which part of the operator behaves like a Laplace type operator and which part is dominated by the anisotropic structure. It is the key point to get the sharp bounds in weighted Sobolev spaces and in anisotropic spaces. Based on the geometric structure of the elastic collision, we make a geometric decomposition to capture the anisotropic structure of the collision operator. More precisely, we make it explicit that the fractional Laplace-Beltrami operator really exists in the structure of the collision operator. It enables us to derive the sharp bounds in anisotropic spaces and then complete the entropy dissipation estimates. The structures mentioned above are so stable that we can apply them to the rescaled Boltzmann collision operator in the process of the grazing collisions limit. Then we get the sharp bounds for the Landau collision operator by passing to the limit. We remark that our analysis used here will shed light on the investigation of the asymptotics from Boltzmann equation to Landau equation.

A kinetic equation for economic value estimation with irrationality and herding

A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.

Entropy dissipation estimates for the linear Boltzmann operator

We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani's conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to [27] , [32] . For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.

Asymptotic Analysis of the Spatially Homogeneous Boltzmann Equation: Grazing Collisions Limit

In the present work, we consider the asymptotic problem of the spatially homogeneous Boltzmann equation when almost all collisions are grazing, that is, the deviation angle \(\theta \) of the collision is limited near zero (i.e., \(\theta \le \epsilon \)). We show that by taking the proper scaling to the cross-section which was used in [37], that is, assuming $$\begin{aligned} B^\epsilon ( v-v_{*},\sigma )=2(1-s)|v-v_*|^{\gamma }\epsilon ^{-3}\sin ^{-1}\theta \left( \frac{\theta }{\epsilon }\right) ^{-1-2s}\mathrm {1}_{\theta \le \epsilon }, \end{aligned}$$ where \(\theta = \langle \theta ={\frac{\upsilon -\upsilon _*}{|\upsilon -\upsilon _*|}}.\sigma \rangle , \) the solution \(f^\epsilon \) of the Boltzmann equation with initial data \(f_0\) can be globally or locally expanded in some weighted Sobolev space as $$\begin{aligned} f^\epsilon = f+ O(\epsilon ), \end{aligned}$$ where the function \(f\) is the solution of Landau equation, which is associated with the grazing collisions limit of Boltzmann equation, with the same initial data \(f_0\). This gives the rigorous justification of the Landau approximation in the spatially homogeneous case. In particular, if taking \(\gamma =-3\) and \(s=1-\epsilon \) in the cross-section \(B^\epsilon \), we show that the above asymptotic formula still holds and in this case \(f\) is the solution of Landau equation with the Coulomb potential. Going further, we revisit the well-posedness problem of the Boltzmann equation in the limiting process. We show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution \(f^\epsilon \). Thanks to the weak convergence results on the grazing collisions limit in [37], in other words, we establish a unified framework to establish the well-posedness results for both Boltzmann and Landau equations.

Well-Posedness and Asymptotics of Grazing Collisions Limit of Boltzmann Equation with Coulomb Interaction

In the present work, we consider the well-posedness and asymptotics of grazing collisions limit of the spatially inhomogeneous Boltzmann equation with Coulomb interaction. Under the screening hypothesis on the cross-section $B^\epsilon(z,\sigma)$ of the Boltzmann collision operator, that is, $ B^\epsilon(z, \sigma)=|\log \epsilon|^{-1}|z|^{-3}\theta^{-3}(\sin\theta)^{-1}{1}_{\{\theta\ge \epsilon\}}$, where $\cos\theta= \frac{z}{|z|}\cdot\sigma$, we prove that there exists a common lifespan $T$ such that for any $\epsilon$, the Boltzmann equation admits a unique nonnegative solution $f^\epsilon(t,x,v)$ in the function space $C([0,T]; L^2_x L^2_{2N(4N+5/2)+l}\cap \mathcal{H}^N_l)$ when the initial data belongs to the weighted Sobolev space $L^2_xL^2_{2N(4N+5/2)+l}\cap \mathcal{H}^N_l $ with $N\in \mathbb{N}$ and $l\in \mathbb{R}^+$. Moreover, it is proved that the solution $f^\epsilon$ is uniformly bounded with respect to the parameter $\epsilon$ in the above function spaces. As a consequence, by using the equation and taking the grazing collisions limit $\epsilon\rightarrow0$, we derive the local existence and uniqueness of the nonnegative solution $f$ of the Landau equation with a Coulomb potential corresponding to the same initial data as the one to the Boltzmann equation. In other words, we establish the well-posedness for both Boltzmann and Landau equations with Coulomb interaction in a unified framework. Finally, we show that the limiting process ($\epsilon\rightarrow0$) can be explicitly described by the asymptotic formula $f^\epsilon= f+ O(|\log \epsilon|^{-1})$, which holds in some weighted Sobolev spaces. It gives the first rigorous result on the justification of the Landau approximation for the Coulomb potential in the general spatially inhomogeneous setting.

Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials

We give an explicit bound for the Wasserstein distance with quadratic cost between the solutions of the Boltzmann and Landau equations in the case of soft and Coulomb potentials. This gives an explicit rate of convergence for the grazing collisions limit. Our result is local in time for very soft and Coulomb potentials and global in time for moderately soft potentials.

The Grazing Collision Limit of the Inelastic Kac Model around a Lévy-type Equilibrium

This paper is devoted to the grazing collision limit of the inelastic Kac model introduced in [A. Pulvirenti and G. Toscani, J. Statist. Phys., 14 (2004), pp. 1453–1480], when the equilibrium distribution function is a heavy-tailed Levy-type distribution with infinite variance. We prove that solutions in an appropriate domain of attraction of the equilibrium distribution converge to solutions of a Fokker–Planck equation with a fractional diffusion operator.

The grazing collision limit of Kac caricature of Bose–Einstein particles

We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modification of the one- dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in [13 ]. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.

A Finite Element Approximation of Grazing Collisions

In this article, we propose a finite element discretization of the Boltzmann-Lorentz operator for which it is possible to define a grazing collision limit. We illustrate this discretization by considering the evolution of a system of particles in a slab, subject to collisions both with the boundaries and with themselves. A comparison is made with isotropic collisions or with the Laplace-Beltrami operator. Moreover, in the case of multiplying boundary conditions, it is proven by numerical simulations the existence of a critical value for the absorption coefficient, which is independent on the grazing collision parameter. Finally, the focalization of a beam is studied and the numerical results are compared with previous simulations.

Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit

In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of [21,23]. We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved.