arXiv subjects

Andreas Strombergsson

Publications and source records attributed to Andreas Strombergsson.

3 recordsLinked to original sources

arXiv2008

Kinetic transport in the two-dimensional periodic Lorentz gas

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation--in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the two-dimensional set-up, and reports an explicit, elementary formula for the collision kernel of the transport equation.

math-ph↗

arXiv2008

The Boltzmann-Grad limit of the periodic Lorentz gas

We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

math.DS↗

arXiv2002

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number $α$ the Kronecker sequence $mα$ ($m=1,...,M$) is equidistributed modulo one in the limit $M\to\infty$, and (ii) closed horocycles of length $\ell$ become equidistributed in the unit tangent bundle $T_1 M$ of a hyperbolic surface $M$ of finite area, as $\ell\to\infty$. In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant $ν> 0$ the Kronecker sequence embedded in $T_1 M$ along a long closed horocycle becomes equidistributed in $T_1 M$ for almost all $α$, provided that $\ell = M^ν \to \infty$. This equidistribution result holds in fact under explicit diophantine conditions on $α$ (e.g., for $α=\sqrt 2$) provided that $ν<1$, or $ν<2$ with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for $ν=2$, our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence $n^2 α$ modulo one.

math.NT↗