Ilias, Chenn, Israel Michael Sigal
Starting from the microscopic reduced Hartree-Fock equation, we derive the nanoscopic linearized Poisson-Boltzmann equation for the electrostatic potential associated with the electron density.
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We outline the kinetic limit for weakly nonlinear wave equations
We review some results concerning the derivation of the Boltzmann equation starting from the many-body classical Hamiltonian dynamics. In particular, the celebrated paper by O. E. Lanford III and the more recent papers [13, 23] are discussed.
In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.
Discretization of non-linear Poisson-Boltzmann Equation equations results in a system of non-linear equations with symmetric Jacobian. The Newton algorithm is the most useful tool for solving non-linear equations. It consists of solving a series of linear system of equations (Jacobian system). In this article, we adaptively define the tolerance of the Jacobian systems. Numerical experiment shows that compared to the traditional method our approach can save a substantial amount of computational work. The presented algorithm can be easily incorporated in existing simulators.
It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence -- in the absence of damping the limiting behaviour depends on the exact con…
In this paper, we rigorously derive a Boltzmann equation for mixtures from the many body dynamics of two types of hard sphere gases. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases. As a corollary of the derivation, we prove Boltzmann's propagation of chaos assumption for the case of a mixtures of gases.
Variational derivation of the homogeneous Boltzmann equation
本文通过变分法推导均匀玻尔兹曼方程，证明该方程的唯一能量守恒解源于微观动力学，并建立熵混沌性的传播。
We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally pro…
A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of t…
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 estimatin…
In this paper we analyze a system of N identical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity n…
This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov-Poisson-Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allows that the electric potential may take distinct constant states at both far-fields. The rarefaction wave whose strength is not necessarily small is constructed through the quasineutral Euler equations coming from the zero-order fluid dynamic approximation of the kinetic system. We prove that the local Maxwellian with macroscopic quantities determ…
We establish the time decay rates of the solutions to the Cauchy problem for the two-species Vlasov-Poisson-Boltzmann system near Maxwellians via a refined pure energy method. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov-Poisson-Boltzmann system in whi…
This paper provides the first rigorous derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard-spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this paper introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way…
Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.
We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{ρ_{0}(t,x)}{(2πθ_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2θ_{0}(t,x)},\text{\ }θ_{0}(t,x)=Kρ_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$ where $ρ_{0}(t,x)$ and $ u_{0}(t,x)$ satisfy the Euler-Poisson …
Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory
本文从经历弹性碰撞的硬球粒子系统出发，严格推导了可压缩 Euler 和不可压缩 Navier-Stokes-Fourier 等流体力学基本偏微分方程，通过玻尔兹曼动理论从牛顿定律推导流体方程，解决了希尔伯特第六问题。
We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7].
A derivation of the Boltzmann equation from the Liouville equation by the use of the Grad limiting procedure in a finite volume is proposed. We introduce two scales of space-time: macro- and microscale and use the BBGKY hierarchy and the functional formulation of classical mechanics. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which are rational numbers. Hence, one can speak about the accuracy of the initial probability density function in the Liouville equation. We assume that the initial data for the microscopic density…
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced in [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a non-perturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum.
In this paper, we present a rigorous derivation of a new kinetic equation describing the limiting behavior of a classical system of particles with three particle elastic instantaneous interactions, which are modeled using a non-symmetric version of a ternary distance. The ternary collisional operator we derive can be seen as the first step towards obtaining a toy model for a non-ideal gas where higher order interactions are taken into account.
On the derivation of the linear Boltzmann equation from the nonideal Rayleigh gas
本文改进从非理想 Rayleigh 气体推导线性 Rayleigh–Boltzmann 方程的已有证明，将原收敛速率提升至超过指数级，并支持气体动理学的长时间尺度分析。
In the paper the possible approaches to the rigorous derivation of the Boltzmann kinetic equation with hard sphere collisions from underlying dynamics are considered. In particular, a formalism for the description of the evolution of infinitely many hard spheres within the framework of marginal observables in the Boltzmann--Grad scaling limit is developed. Also we give consideration to one more approach of the description of the kinetic evolution of hard spheres in terms of a one-particle distribution function governed by the non-Markovian generalization of the Enskog equation and the Boltzman…
We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for one-dimensional bipolar Vlasov-Poisson-Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type decomposition around the local Maxwellian related to the bipolar VPB system in our previous work [26], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to 1D bipolar VPB system under some smallness cond…
Formal derivation of an isentropic two-phase flow model from the multi-species Boltzmann equation
本文基于多组分 Boltzmann 方程，在 Knudsen 数趋零且同种组分内共振碰撞主导的区域，形式化推导出含体积分数演化的等熵两相流模型并显式给出其系数。
Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra
本文研究奇导子纳入交换超代数构造转置泊松超代数的方法，提出带两种运算的超向量空间新概念，证明奇导子生成的超向量空间满足其要求，并给出由转置泊松超代数与偶导子构造3-李超代数的方法。
We derive the 3D spatially homogeneous Boltzmann's equation with moderately soft potentials and singular angular interaction, from an interacting particles system. The collision kernel is of the form $B(z,σ)=|z|^γb\left( \frac{z}{|z|}\cdot σ\right)$ and for $K>0$, $\sin(θ)b\left(\cos(θ)\right)\sim Kθ^{-1-ν}$, with $γ\in (-2,-1)$ and $ν\in(1,2)$ satisfying $γ+ν>0$. We use at the particle level the regularizing effects of the grazing collisions, in order to control the singularity of the soft potential. This enables to use a classical compactness argument, and provide a qualitative convergence r…
The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21, 23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave…
The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-tr…
Derivation of the Higher Order Boltzmann Equation for Hard Spheres
该研究从有限 $N$ 个硬球粒子系统出发，识别新的标度区域并推广几何技术，严格推导出包含任意高阶 $M$ 碰撞项的 Boltzmann 型动力学方程。
A Formal Derivation of the Thick Spray Model from the Enskog-Boltzmann System
本研究提出特定标度方法，从Boltzmann-Enskog模型形式推导出厚喷雾Euler-Vlasov模型，验证了其动理学-流体方程，还发现了因体积分数函数引入而被忽略的粘性力高阶修正项，拓展了薄喷雾相关研究结论。
Boltzmann-Grad limit for the inelastic Lorentz gas: Part I. Existence, uniqueness, and rigorous derivation via weak convergence
本文在任意维度下从耗散随机 Lorentz 气体严格推导了非弹性线性 Boltzmann 方程，证明了弱解的存在唯一性及微观解的弱收敛性，并给出了显式收敛速率与强解的存在性证明。
An Efficient Modified "Walk On Spheres" Algorithm for the Linearized Poisson-Boltzmann Equation
提出一种基于改进“球面上行走”算法的无网格随机游走方法，通过生存概率分布函数高效求解线性化泊松-玻尔兹曼方程，并在四个解析解问题中验证了精度。
In this work, we generalize M. Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges …
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
针对泊松-玻尔兹曼方程，提出结合目标导向误差指示与问题特定标记策略的自适应细化算法，并采用多层预处理保持最优计算复杂度，在溶剂化自由能线性泛函上优于基于能量的误差指示方法。
Multiscale Splitting method for Boltzmann-Poisson Equation: Application for Dynamic of Electrons
本文提出一种基于简化Boltzmann方程耦合Poisson方程的电子动力学模型，通过设计分裂方法求解，并利用数值实验讨论方法性能。
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