Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T07:28:06.145Z Has data issue: false hasContentIssue false
  • English
  • Français

Classical motion in random potentials

Published online by Cambridge University Press:  05 December 2012

ANDREAS KNAUF  and
CHRISTOPH SCHUMACHER
ANDREAS KNAUF
Affiliation:
Department Mathematik, Universität Erlangen-Nürnberg, Cauerstr. 11, D-91058 Erlangen, Germany (email: knauf@mi.uni-erlangen.de)
CHRISTOPH SCHUMACHER
Affiliation:
Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainerstr. 41, D-09126 Chemnitz, Germany (email: christoph.schumacher@mathematik.tu-chemnitz.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the motion of a classical particle under the influence of a random potential on ℝd, in particular the distribution of asymptotic velocities and the question of ergodicity of time evolution.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 557 - 593
Copyright
©2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Aar97]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.Google Scholar
[AK98]Asch, J. and Knauf, A.. Motion in periodic potentials. Nonlinearity 11(1) (1998), 175–200.Google Scholar
[AM78]Abraham, R. and Marsden, J. E.. Foundations of Mechanics, 2nd edn. Benjamin/Cummings, Reading, MA, 1978.Google Scholar
[Arn78]Arnol’d, V. I.. Mathematical Methods of Classical Mechanics. Springer, Berlin, 1978.Google Scholar
[BBI01]Burago, D., Burago, Y. and Ivanov, S.. A Course in Metric Geometry (Graduate Studies in Mathematics, 33). American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
[Bow75]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer, Berlin, 1975.Google Scholar
[BS81]Bunimovich, L. A. and Sinai, Y. G.. Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 (1981), 479–497.CrossRefGoogle Scholar
[CLS10]Cristadoro, G., Lenci, M. and Seri, M.. Recurrence for quenched random Lorentz tubes. Chaos 20 (2010), 023115.Google Scholar
[DL91]Donnay, V. and Liverani, C.. Potentials on the two-torus for which the Hamiltonian flow is ergodic. Comm. Math. Phys. 135 (1991), 267–302.Google Scholar
[EO73]Eberlein, P. and O’Neill, B.. Visibility manifolds. Pacific J. Math. 46 (1973), 45–109.Google Scholar
[GHL48]Gallot, S., Hulin, D. and Lafontaine, J.. Riemannian Geometry. Springer, Berlin, 1948.Google Scholar
[Hir76]Hirsch, M. W.. Differential Topology (Graduate Texts in Mathematics, 33). Springer, Berlin, 1976.Google Scholar
[Hop39]Hopf, E.. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.Google Scholar
[Hop41]Hopf, E.. Statistik der Lösungen geodätischer Probleme vom unstabilen Typus II. Math. Ann. 117 (1941), 590–608.Google Scholar
[Kel98]Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Student Texts, 42). Cambridge University Press, Cambridge, 1998.Google Scholar
[KK92]Klein, M. and Knauf, A.. Classical Planar Scattering by Coulombic Potentials (Lecture Notes in Physics, 13). Springer, Berlin, 1992.Google Scholar
[KK08]Knauf, A. and Krapf, M.. The non-trapping degree of scattering. Nonlinearity 21 (2008), 2023–2041.Google Scholar
[Kli95]Klingenberg, W. P. A.. Riemannian Geometry, 2nd edn(de Gruyter Studies in Mathematics, 1). de Gruyter, Berlin, 1995.Google Scholar
[Kna87]Knauf, A.. Ergodic and topological properties of Coulombic periodic potentials. Comm. Math. Phys. 110 (1987), 89–112.Google Scholar
[Kna90]Knauf, A.. Closed orbits and converse KAM theory. Nonlinearity 3 (1990), 961–973.Google Scholar
[Kna11]Knauf, A.. Mathematische Physik: Klassische Mechanik (Master Class). Springer, Berlin, 2011.Google Scholar
[Len03]Lenci, M.. Aperiodic Lorentz gas: recurrence and ergodicity. Ergod. Th. & Dynam. Sys. 23 (2003), 869–883.Google Scholar
[LMW03]Leschke, H., Müller, P. and Warzel, S.. A survey of rigorous results on random Schrödinger operators for amorphous solids. Markov Process. Related Fields 9 (2003), 729–760.Google Scholar
[LW95]Liverani, C. and Wojtkowski, M. P.. Ergodicity in Hamiltonian systems. Dynamics Reported (Dynam. Report. Expositions Dynam. Systems (N.S.), 4). Springer, Berlin, 1995, pp. 130–202.Google Scholar
[Nat01]Natterer, F.. The Mathematics of Computerized Tomography (Classics in Applied Mathematics, 32). Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.Google Scholar
[PF92]Pastur, L. A. and Figotin, A.. Spectra of Random and Almost-Periodic Operators (Grundlehren der mathematischen Wissenschaften, 297). Springer, Berlin, 1992.Google Scholar
[P{ö}s82]Pöschel, J.. Integrability of Hamiltonian systems on cantor sets. Commun. Pure Appl. Math. 35 (1982), 653–695.CrossRefGoogle Scholar
[PP94]Paternain, G. P. and Paternain, M.. On Anosov energy levels of convex Hamiltonian systems. Math. Z. 217(1) (1994), 367–376.Google Scholar
[RF11]Roeck, W. D. and Fröhlich, J.. Diffusion of a massive quantum particle coupled to a quasi-free thermal medium. Comm. Math. Phys. 303 (2011), 613–707.Google Scholar
[Rue87]Ruelle, D.. A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 (1987), 225–239.CrossRefGoogle Scholar
[Sch98]Schmidt, K.. On joint recurrence. C. R. Acad. Sci. Paris 327 (1998), 837–842.Google Scholar
[Sch04]Schumacher, C.. Klassische Bewegung in zufälligen Potenzialen. Diplomarbeit, University Erlangen–Nuremberg, 2004.Google Scholar
[Sch10]Schumacher, C.. Klassische Bewegung in zufälligen Potenzialen mit Coulomb–Singularitäten. PhD Thesis, University Erlangen–Nuremberg, 2010.Google Scholar
[SKM87]Stoyan, D., Kendall, W. S. and Mecke, J.. Stochastic Geometry and its Applications. John Wiley & Sons, Chichester, 1987.Google Scholar
[Ves08]Veselić, I.. Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators (Lecture Notes in Mathematics, 1917). Springer, Berlin, 2008.Google Scholar
[Wal82]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1982.Google Scholar