Towards a microscopic derivation of the large phonon Boltzmann equation

Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling

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…

The Derivation of the Linear Boltzmann Equation from a Rayleigh Gas Particle Model

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…

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 estimatin…

Some Considerations on the derivation of the nonlinear Quantum Boltzmann Equation

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…

Rigorous derivation of a binary-ternary Boltzmann equation for a dense gas of hard spheres

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…

Microscopic solutions of the Boltzmann-Enskog equation in the series representation

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.

希尔伯特第六问题：通过玻尔兹曼动理论推导流体方程

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。

Long-time derivation at equilibrium of the fluctuating Boltzmann equation

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].

Derivation of the Boltzmann equation and entropy production in functional mechanics

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…

Monitoring derivation of the quantum linear Boltzmann equation

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.

Rigorous derivation of a ternary Boltzmann equation for a classical system of particles

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.

从非理想 Rayleigh 气体推导线性 Boltzmann 方程

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。

Approaches to Derivation of the Boltzmann Equation with Hard Sphere Collisions

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…

从多组分 Boltzmann 方程形式化推导等熵两相流模型

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。

Derivation of the Boltzmann equation with moderately soft potentials from a perturbed Nanbu particles system

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…

A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction

We study the dynamics of a Fermi gas with a Coulomb interaction potential, and show that, in a mean-field limiting regime, the dynamics is described by the Hartree-Fock equation. This extends previous work of Bardos et al. to the case of unbounded interaction potentials. We also express the mean-field limit as a "superhamiltonian" system, and state our main result in terms of a Heisenberg-picture dynamics of observables. This is a Egorov-type theorem.

硬球高阶 Boltzmann 方程的推导

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。

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 intermediate regime between the weak-coupling and the low-density. 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.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 limi…

从Enskog-Boltzmann系统形式推导出厚喷雾模型

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。

非弹性 Lorentz 气体的 Boltzmann-Grad 极限：第一部分。存在性、唯一性及基于弱收敛的严格推导

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。

Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics

In these lecture notes we discuss recent progress in the rigorous derivation of effective evolution equations for the description of the dynamics of quantum mechanical many-body systems.

Microscopic and soliton-like solutions of the Boltzmann-Enskog and generalized Enskog equations for elastic and inelastic hard spheres

N.N. Bogolyubov discovered that the Boltzmann-Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann-Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if th…

Derivation of a Boltzmann equation with higher-order collisions from a generalized Kac model

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 …

A microscopic derivation of the quantum mechanical formal scattering cross section

We prove that the empirical distribution of crossings of a "detector'' surface by scattered particles converges in appropriate limits to the scattering cross section computed by stationary scattering theory. Our result, which is based on Bohmian mechanics and the flux-across-surfaces theorem, is the first derivation of the cross section starting from first microscopic principles.

Boltzmann's Entropy and Large Deviation Lyapunov Functionals for Closed and Open Macroscopic Systems

I give a brief overview of the resolution of the apparent problem of reconciling time symmetric microscopic dynamic with time asymmetric equations describing the evolution of macroscopic variables. I then show how the large deviation function of the stationary state of the microscopic system can be used as a Lyapunov function for the macroscopic evolution equations.

玻尔兹曼方程从量子多体动力学的推导

The Derivation of the Boltzmann Equation from Quantum Many-body Dynamics

本文从量子多体动力学严格推导出量子玻尔兹曼方程，通过双重临界正则性界和BBGKY层级分析，证明平均场极限下时间不可逆性。

一般外场中 Ginzburg-Landau 理论的微观推导与 BCS 临界温度偏移

Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in General External Fields

本文在弱宏观外电场和外磁场下从 BCS 自由能微观推导 Ginzburg-Landau 泛函，并给出临界温度随外场变化的渐近公式，将既有恒定磁场结果推广至非零磁通的一般磁场。

Scattering theory from microscopic first principles

We sketch a derivation of abstract scattering theory from the microscopic first principles defined by Bohmian mechanics. We emphasize the importance of the flux-across-surfaces theorem for the derivation, and of randomness in the impact parameter of the initial wave function---even for an, inevitably inadequate, orthodox derivation.

Wick polynomials and time-evolution of cumulants

We show how Wick polynomials of random variables can be defined combinatorially as the unique choice which removes all "internal contractions" from the related cumulant expansions, also in a non-Gaussian case. We discuss how an expansion in terms of the Wick polynomials can be used for derivation of a hierarchy of equations for the time-evolution of cumulants. These methods are then applied to simplify the formal derivation of the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete nonlinear Schrödinger equation (DNLS) with suitable random initial data. We also present a re…

Quantum Transport in a Crystal With Short-Range Interactions: The Boltzmann-Grad Limit

We study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann-Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry-Tabor conjecture in quantum chaos.

Microscopic Derivation of Time-dependent Point Interactions

We study the dynamics of the three-dimensional polaron - a quantum particle coupled to bosonic fields - in the quasi-classical regime. In this case the fields are very intense and the corresponding degrees of freedom can be treated semiclassically. We prove that in such a regime the effective dynamics for the quantum particles is approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point. As a by-product, we also show that the unitary dynamics of a time-dependent point interaction can be approximat…

Microscopic Derivation of the Ginzburg-Landau Model

We present a summary of our recent rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

Derivation of Euler equations from quantum and classical microscopic dynamics

We derive Euler equations from a Hamiltonian microscopic dynamics. The microscopic system is a one-dimensional disordered harmonic chain, and the dynamics is either quantum or classical. This chain is an Anderson insulator with a symmetry protected mode: Thermal fluctuations are frozen while the low modes ensure the transport of elongation, momentum and mechanical energy, that evolve according to Euler equations in an hyperbolic scaling limit. In this paper, we strengthen considerably our previous results, where we established a limit in mean starting from a local Gibbs state: We now control t…

Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit

We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model.

Microscopic Derivation of Ginzburg-Landau Theory

We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

A Microscopic Derivation Of Coupled Spde's With A Kpz Flavor

We consider an interacting particles system composed of a Hamiltonian part and perturbed by a conservative stochastic noise so that the full system conserves two quantities: energy and volume. The Hamiltonian part is regulated by a scaling parameter vanishing in the limit. We study the form of the fluctuations of these quantities at equilibrium and derive coupled stochastic partial differential equations with a KPZ flavor.

A Theorem on Ellipses, an Integrable System and a Theorem of Boltzmann

We study a mechanical system that was considered by Boltzmann in 1868 in the context of the derivation of the canonical and microcanonical ensembles. This system was introduced as an example of ergodic dynamics, which was central to Boltzmann's derivation. It consists of a single particle in two dimensions, which is subjected to a gravitational attraction to a fixed center. In addition, an infinite plane is fixed at some finite distance from the center, which acts as a hard wall on which the particle collides elastically. Finally, an extra centrifugal force is added. We will show that, in the …

On the derivation of the Boltzmann equation in quantum field theory: Flat spacetime

In this paper, we analyze in a mathematically rigorous fashion the validity of the Boltzmann transport equation within quantum field theory. We work within the specific model of a hermitian, scalar field with polynomial self-interaction in two-dimensional Minkowski space. Our main results are as follows: Firstly, that one can obtain a non-perturbative, exact integro-differential equation for the number densities, which we called the pre-Boltzmann equation. We secondly take the long-time-dilute-medium limit of this equation, to obtain a simpler equation. This limiting equation is qualitatively …

Derivation of the particle dynamics from kinetic equations

We consider the microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversi…

Boltzmann方程一般Riemann解的时间渐近稳定性

Time-asymptotic stability of generic Riemann solutions for Boltzmann equation

针对一维Boltzmann方程一般Riemann解的长期稳定性难题，本文首次运用$a$-收缩方法证明含稀疏波、粘性接触波和时移粘性激波的复合波非线性稳定，并克服微观效应耦合困难。

Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystals

A polaron is an electron interacting with a polar crystal, which is able to form a bound state by using the distortions of the crystal induced by its own density of charge. In this paper we derive Pekar's famous continuous model for polarons (in which the crystal is replaced by a simple effective Coulomb self-attraction) by studying the macroscopic limit of the reduced Hartree-Fock theory of the crystal. The macroscopic density of the polaron converges to that of Pekar's nonlinear model, with a possibly anisotropic dielectric matrix. The polaron also exhibits fast microscopic oscillations whic…

弱均匀磁场中 Ginzburg-Landau 理论的微观推导与 BCS 临界温度偏移

Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field

本文从 BCS 自由能泛函微观推导弱均匀磁场下的 Ginzburg-Landau 泛函，并给出临界温度随磁场变化的渐近公式，将既有结果推广至单位晶胞磁通量非零的情形。

一维具有非线性点相互作用的 Schrödinger 方程的微观推导

Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction

本文在一维全同玻色子与微小杂质的三体相互作用体系中，通过平均场与短程联合标度证明混沌传播，首次从微观模型推导出非线性 delta 点相互作用 Schrödinger 方程。

具有紧支撑对势的Vlasov方程的微观推导

Microscopic derivation of Vlasov equation with compactly supported pair potentials

本文通过Gronwall估计，在紧支撑对势条件下证明了三维N粒子系统牛顿轨迹收敛到Vlasov-Dirac-Benney系统特征线，从而从微观动力学导出该方程。

广义谐振子-玻色方程在相互作用玻色气体中的推导

Derivation of renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann equations in an interacting Bose gas

本文通过Bogoliubov旋转和Weyl变换，推导了相互作用玻色气体中修正的HFB和量子玻尔兹曼方程，并改进了误差界。

具有粗糙截断的聚焦\(Φ^6_1\)测度的微扰微观推导

A perturbative microscopic derivation of the focusing $Φ^6_1$ measure with rough cut-off

研究圆上具有粗糙截断的聚焦NLS方程吉布斯测度的推导，基于弗罗利希等人的微扰展开，采用维格纳测度方法和归纳论证克服截断光滑性问题，给出聚焦五次NLS最优截断下吉布斯测度的推导。

On the quantum Boltzmann equation

We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrodinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreeness in the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.

A microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schrödinger equation

We give a microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is our previous work on the time-independent problem and work of the second author on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by the means of complex analytic methods. Our methods…

Microscopic Derivation of an Isothermal Thermodynamic Transformation

We obtain macroscopic isothermal thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics in contact with a heat bath. The microscopic dynamics is given by a chain of anharmonic oscillators subject to a varying tension (external force) and the contact with the heat bath is modeled by independent Langevin dynamics acting on each particle. After a diffusive space-time scaling and cross-graining, the profile of volume converges to the solution of a deterministic diffusive equation with boundary conditions given by the applied tension. This defines an irreversible…

一维聚焦五次非线性薛定谔方程吉布斯测度的微观推导

A microscopic derivation of Gibbs measures for the 1D focusing quintic nonlinear Schrödinger equation

本文从多体量子吉布斯态出发，利用微扰展开方法，在质量与粒子数截断下首次给出了环面上一维聚焦五次非线性薛定谔方程吉布斯测度及其时间相关关联函数的微观推导，填补了三体相互作用情形的空白。

Kinetic Theory of Turbulence Modeling: Smallness Parameter, Scaling and Microscopic Derivation of Smagorinsky Model

A mean-field approach (filtering out subgrid scales) is applied to the Boltzmann equation in order to derive a subgrid turbulence model based on kinetic theory. It is demonstrated that the only Smagorinsky type model which survives in the hydrodynamic limit on the viscosity time scale is the so-called tensor-diffusivity model. Scaling of the filter-width with Reynolds number and Knudsen number is established. This sets the first rigorous step in deriving turbulence models from kinetic theory.

一维聚焦三次非线性薛定谔方程吉布斯测度的微观推导

A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation

本文通过多体量子吉布斯态和微扰展开，在引入质量与粒子数截断的条件下，首次对一维环面上聚焦三次非线性薛定谔方程给出吉布斯测度的微观推导，并研究了时间相关关联函数。

Microscopic derivation of an adiabatic thermodynamic transformation

We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to collisions with external independent particles of "infinite mass". The effect of each collision is to change the sign of the velocity without changing the modulus. This way the energy is conserved by the resulting dynamics. After a diffusive space-time scaling and cross-graining, the profiles of volume and energ…

格子玻尔兹曼方程的极限一致性

Limit Consistency of Lattice Boltzmann Equations

该研究提出极限一致性概念作为LBE与PDE系统一致性证明的模块化部分，以NSE为范例，通过嵌套泰勒展开与有限差分离散BGK玻尔兹曼方程，追踪离散状态与截断误差，证明各方程族的极限一致性，给出保持连续极限的LBM数值格式通用自上而下推导方法。