Related papers: De Newton \`a Boltzmann et Einstein: validation de…
These lecture notes provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases…
Consider a microscopic system of $N$ hard spheres that are initially independent (modulo the exclusion condition on particle positions) and identically distributed in $\mathbb{R}^3$. When the number $N$ of particles goes to infinity and the…
The hypothesis of molecular chaos plays the central role in kinetic theory, which provides a closure leading to the Boltzmann equation for quantitative description of classic fluids. Yet how to properly extend it to active systems is still…
The article discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.
A kinetic equation for a dilute gas of hard spheres confined between two parallel plates separated a distance smaller than two particle dimeters is derived. It is a Boltzmann-like equation, which incorporates the effect of the confinement…
We provide a rigorous justification of the linearized Boltzmann- and Landau equations from interacting particle systems with long-range interaction. The result shows that the fluctuations of Hamiltonian $N$- particle systems governed by…
Boltzmann's H-Theorem, formulated 150 years ago in terms of H-function that also bears his name, is one of the most celebrated theorems of science and paved the way for the development of nonequilibrium statistical mechanics. Nevertheless,…
We provide a rigorous derivation of Boltzmann's kinetic equation from the hard sphere system for rarefied gas, which is valid for arbitrarily long times, as long as the solution to the Boltzmann equation exists. This extends Lanford's…
A dilute gas of hard disks confined between two straight parallel lines is considered. The distance between the two boundaries is in between one and two particle diameters, so that the system is quasi-one-dimensional. A Boltzmann-like…
The Choh Uhlenbeck equation was the first successful generalization of the Boltzmann equation to include triple collisions in a systematic manner. Detailed analysis of its triple collision term has led to the important concept of ring…
The paper studies a Milne type problem for a linearized quantum Boltzmann equation. Existence and uniqueness of the solution, together with asymptotic properties are proven for a given energy flow. The energy flow is proportional to the…
We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation.…
In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous…
We propose an extended structural dynamics framework that enriches classical mechanics by treating particle orientation and internal structure as fundamental phase-space coordinates. This extension preserves Hamiltonian structure and…
Stefan-Boltzmann law was empirically deduced by Stefan in 1874 by fitting existing experiments and theoretically validated by Boltzmann in 1884 on the basis of a classical model involving thermodynamics principles and the Maxwell equations.…
We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast…
It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmman equation, fix univocally the entropy of the system, which turn out to be the Boltzmann-Gibbs-Shannon…
The Boltzmann kinetic equation is obtained from an integro-differential master equation that describes a stochastic dynamics in phase space of an isolated thermodynamic system. The stochastic evolution yields a generation of entropy,…
The interplay between electronic interactions and disorder is neglected in the conventional Boltzmann theory of transport, yet can play an essential role in determining the resistivity of unconventional metals. When quasiparticles are…
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…