Ergodic and topological properties of coulombic periodic potentials

Abstract

The motion of a classical pointlike particle in a two-dimensional periodic potential with negative coulombic singularities is examined. This motion is shown to be Bernoullian for many potentials and high enough energies. Then the motion on the plane is a diffusion process. All such motions are topologically conjugate and the periodic orbits can be analysed with the help of a group. * Present address: Mathematisch instituut; Budapestlaan 6; NL-3508 TA Utrecht This work is part of a thesis submitted to Freie Universitat Berlin which is conformally equivalent to the Euclidean metric g(x) on M\ (see, e.g. Abraham and Marsden [1, Chap. 3.7]). Now the length L(σ) of a curve σ:]a,b[->M\ is given by L{σ):=]\\σ{t)\\dt

Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

“…What has been invariably discovered, is that, surprisingly, geodesic flows associated with chaotic physical Hamiltonians do not live on everywhere negatively curved manifolds. Few exceptions are known, in particular two low dimensional models [15,16], where chaos is actually associated with negative curvature.…”

Dynamics of Oscillator Chains

“…To give an example, Sinai proved ergodicity and mixing for two hard spheres by just showing that such a system is similar enough to a geodesic flow on a negatively curved compact manifold [33]. Krylov's intuitions have been worked out further by several physicists amongst whom we cite those of [34,35,36,37,38,39,40]. They discovered, much to their surprise, that geodesic flows associated with physical Hamiltonians do not live on negatively curved manifolds, despite their chaoticity.…”

“…They discovered, much to their surprise, that geodesic flows associated with physical Hamiltonians do not live on negatively curved manifolds, despite their chaoticity. Only a few exceptions are known, in particular two low-dimensional models [35,36,41], where chaos is actually associated with hyperbolicity due to everywhere negatively curved manifolds. In fact, for certain models the regions of negative curvature of the mechanical manifolds apparently shrink by increasing the number N of degrees of freedom, thus reducing the frequency of the visits of negatively curved regions.…”

Random paths and current fluctuations in nonequilibrium statistical mechanics

“…Escape occurs as soon as the particle reaches the border Σ of a domain delimited in position space. In Lorentz gases with a regular lattice, the macroscopic diffusion equation applies as proved by Bunimovich and Sinai in hard-disk billiards with a finite horizon and by Knauf in square lattices of Yukawa potentials [52,53]. In these two-dimensional Lorentz gases, there is a single positive Lyapunov exponent λ.…”