Article
Physics, Mathematical
Bernt Wennberg
Summary: We study the Lorentz gas in a distribution of scatterers and find that it satisfies a linear Boltzmann equation in the low density limit, while the periodic Lorentz gas does not satisfy the Boltzmann equation in the limit.
JOURNAL OF STATISTICAL PHYSICS
(2023)
Article
Physics, Multidisciplinary
Alan Rodrigo Mendoza Sosa, Atahualpa S. Kraemer
Summary: We introduce an algorithm based on generalized dual method (GDM) for efficient study of particle dynamics in quasiperiodic environments. The algorithm eliminates the need for periodic approximations or saving information of the quasiperiodic lattice vertices, allowing for realistic simulations with low computational resources. The algorithm can be applied to study any quasiperiodic lattice produced by the cut-and-project method. The study reveals that the distribution of free paths in quasiperiodic systems depends on the rank of the system rather than its symmetry.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics, Applied
Francois Golse
Summary: This note proposes a slightly different proof of Gallavotti's theorem in deriving the linear Boltzmann equation for the Lorentz gas with obstacles following a Poisson distribution in the Boltzmann-Grad limit.
KINETIC AND RELATED MODELS
(2022)
Article
Statistics & Probability
Peter Balint, Henk Bruin, Dalia Terhesiu
Summary: We prove limit laws for infinite horizon planar periodic Lorentz gases when both the time n and scatterer size rho tend to infinity and zero simultaneously at a sufficiently slow pace. Our results include a non-standard Central Limit Theorem and a Local Limit Theorem for the displacement function. These results are the first in an intermediate case between two well-studied regimes: (i) fixed infinite horizon configurations with superdiffusive root n log n scaling, and (ii) Boltzmann-Grad type situations.
PROBABILITY THEORY AND RELATED FIELDS
(2023)
Article
Mathematics, Applied
Corentin Le Bihan
Summary: This paper rigorously derives the Boltzmann equation in a compact domain with isotropic boundary conditions, showing that the dynamics of a system of hard spheres converge to the solution of the Boltzmann equation at a specific scale.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Physics, Multidisciplinary
Robertus Potting
Summary: In this study, we applied the principles of relativistic kinetic theory and the Boltzmann equation to particles with Lorentz-violating dispersion relations. We considered both classical (Maxwell-Boltzmann) and quantum (Fermi-Dirac and Bose-Einstein) statistics. Our results showed that Boltzmann's H-theorem still holds when the entropy current is appropriately defined. We derived equilibrium solutions and identified the effects of Lorentz violation on various thermodynamic variables and Bose-Einstein condensation. Additionally, we investigated a scenario involving nonelastic collisions between multiple species of particles, which corresponds to chemical or nuclear reactions.
EUROPEAN PHYSICAL JOURNAL PLUS
(2023)
Article
Statistics & Probability
Songzi Li
Summary: This article investigates the convergence rates for superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas. By applying Stein's method, convergence rates in Wasserstein distance for discrete-time displacement and a result for the Berry-Essen type bound for continuous-time displacement are obtained.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2022)
Article
Mathematics
Tianfang Wu, Xiongfeng Yang
Summary: In this paper, the hydrodynamic limit of Boltzmann equations for gas mixture is studied using the Hilbert expansion method. The terms in the Hilbert expansion are derived formally according to different orders of the Knudsen number. The truncation of the expansion and the justification of the hydrodynamic limit are done by establishing uniform estimates of the remainder term. The approach is based on the L2 - L infinity framework, which is motivated by the study of the single Boltzmann equation in a previous work [22].
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Evaldo M. F. Curado, Carlos E. E. Cedeno, Ivano Damiao Soares, Constantino Tsallis
Summary: In this paper, the Juttner probability density function (PDF) is studied for relativistic gases. A new Lorentz-invariant PDF is obtained by introducing the rapidity variable, and its validity is confirmed through computational dynamics simulations.
Article
Physics, Mathematical
Andrea Bondesan, Marc Briant
Summary: This study investigates the diffusion asymptotics of the Boltzmann equation for gaseous mixtures in a perturbative regime around a local Maxwellian vector. By introducing a suitable modified Sobolev norm and utilizing a hypocoercive formalism, a Cauchy theory that is uniform with respect to the Knudsen number epsilon is established. It is proven that the Maxwell-Stefan system is stable for the Boltzmann multi-species equation, ensuring a rigorous derivation in the vanishing limit epsilon -> 0.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Physics, Mathematical
Jory Griffin, Jens Marklof
Summary: The study reveals that in a crystal with short-range potentials, the macroscopic transport properties of the quantum Lorentz gas converge to a random flight process in the Boltzmann-Grad limit, which is not compatible with the linear Boltzmann equation. This derivation is based on a hypothesis about the statistical distribution of lattice points in thin domains, closely related to the Berry-Tabor conjecture in quantum chaos.
JOURNAL OF STATISTICAL PHYSICS
(2021)
Article
Statistics & Probability
Alessia Nota, Chiara Saffirio, Sergio Simonella
Summary: In this study, we investigated a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the influence of a perpendicular magnetic field. We proved that, in the low-density limit, the particle distribution follows a generalized linear Boltzmann equation, which includes non-Markovian terms. By adapting the ideas from (Phys. Rev. 185 (1969) 308-322), we were able to demonstrate the convergence of the process with memory.
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES
(2022)
Article
Physics, Mathematical
Francoise Pene, Dalia Terhesiu
Summary: The article discusses the local limit theorem for the Sinai billiard map, obtaining sharp error rates and mixing rates in the process. It also provides an asymptotic estimate for the tail probability of the first return time to the initial cell, while studying transfer operators and higher order expansions for eigenvalues and eigenprojectors.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mechanics
Heiko Pleskun, Tobias Bode, Andreas Bruemmer
Summary: The mass flow rate of Couette flow in a long rectangular channel is calculated for various gas rarefaction levels and width-to-height ratios. Analytical solutions are derived for different width-to-height ratios in the hydrodynamic, slip, and free molecular regimes. Simulations using the direct simulation Monte Carlo method are performed in the transitional regime. The results, presented as tabulated data, can be applied to cases with constant or linear increasing wall velocity.
Article
Physics, Multidisciplinary
Gilberto M. Kremer
Summary: The post-Newtonian hydrodynamic equations for a non-perfect fluid are derived from a post-Newtonian Boltzmann equation. The energy-momentum tensor components are determined using the relativistic Eckart decomposition for a viscous and heat conducting fluid. The post-Newtonian expression of the relativistic Grad distribution function is obtained. The hydrodynamic equations for mass density, mass-energy density, and momentum density are derived using a post-Newtonian transfer equation and Grad's distribution function. In the non-relativistic limit, the Newtonian hydrodynamic equations are recovered.
Article
Physics, Mathematical
Jens Marklof, Balint Toth
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2016)
Article
Mathematics
Jens Marklof, Ilya Vinogradov
GEOMETRIAE DEDICATA
(2017)
Article
Physics, Mathematical
Carl P. Dettmann, Jens Marklof, Andreas Strombergsson
JOURNAL OF STATISTICAL PHYSICS
(2017)
Article
Mathematics
Min Lee, Jens Marklof
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2018)
Article
Mathematics, Applied
Jens Marklof
Article
Mathematics
Jens Marklof, Andreas Strombergsson
AMERICAN MATHEMATICAL MONTHLY
(2017)
Article
Mathematics
Jens Marklof, Nadav Yesha
COMPOSITIO MATHEMATICA
(2018)
Article
Mathematics
Jens Marklof, Ilya Vinogradov
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2018)
Article
Mathematics
Jens Marklof
MONATSHEFTE FUR MATHEMATIK
(2020)
Correction
Physics, Mathematical
Jens Marklof, Andreas Strombergsson
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2020)
Article
Mathematics
Jens Marklof
AMERICAN MATHEMATICAL MONTHLY
(2020)
Article
Mathematics
Alan Haynes, Jens Marklof
Summary: This paper extends the three-distance theorem and provides a generalization for higher-dimensional Kronecker sequences. It proves that in 2D, there can be at most five possible distances between nearest neighbors, and conjectures similar upper bounds for higher dimensions. Additionally, it studies the number of possible distances from a point to its nearest neighbor in a restricted cone of directions and provides results under different conditions.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Physics, Mathematical
Jory Griffin, Jens Marklof
Summary: The study reveals that in a crystal with short-range potentials, the macroscopic transport properties of the quantum Lorentz gas converge to a random flight process in the Boltzmann-Grad limit, which is not compatible with the linear Boltzmann equation. This derivation is based on a hypothesis about the statistical distribution of lattice points in thin domains, closely related to the Berry-Tabor conjecture in quantum chaos.
JOURNAL OF STATISTICAL PHYSICS
(2021)
Article
Mathematics
Jens Marklof, Matthew Welsh
Summary: We establish limit laws for the distribution of roots in small intervals of a quadratic congruence. By translating the problem into convergence of random line processes in the hyperbolic plane, we obtain an explicit expression for the pair correlation density of the roots.
DUKE MATHEMATICAL JOURNAL
(2023)
Article
Mathematics
Alan Haynes, Jens Marklof
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2020)
