We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.
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
Abstract. We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n = 1 there are no bounded orbits, and for n = 2 there is just one closed orbit, for n ≥ 3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates. The theory includes the n-centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres. Finally we show that different, non-universal, phenomena occur for collinear configurations.
The quotient ζ(s -1) /ζ(s) of Riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperature s. Contents 1. Introduction 2. The Zeta Function and the Spin Chain 78 3. General Framework 82 4. The Grand Canonical Ensemble 5. Ferromagnetism 6. Upper Bounds for the Interaction Coefficients 7. Asymptotic Translation Invariance 8. Decay Properties of the Potential A. Numerical Calculations 112 References 115
We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.
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