zbMATH — the first resource for mathematics

The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion. (English) Zbl 1233.37025
Results of previous papers by the last author are extended by pursuing the article by W. A. Veech [Invent. Math. 97, No. 3, 553–583 (1989; Zbl 0676.32006)] on compact translation surfaces. Periodic wind-free models are investigated with an emphasis on unbounded planar billiards with periodically located rectangular obstacles. Previous observations by J. Hardy and J. Weber [J. Math. Phys. 21, No. 7, 1802–1808 (1980)] on recurrence and abnormal diffusion of the billiard flow are addressed anew to demonstrate that, depending on the range of rational parameters, the existence of completely periodic directions and recurrences can be proved. In another parameter range, there are escape directions for all trajectories; rates of escape for almost all directions are proved.

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37A60 Dynamical aspects of statistical mechanics
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C27 Periodic orbits of vector fields and flows
37D05 Dynamical systems with hyperbolic orbits and sets
37C75 Stability theory for smooth dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
PDF BibTeX Cite
Full Text: DOI
[1] Ehrenfest P., Encykl. Math. Wissensch. IV 2 pp 6– (1912)
[2] DOI: 10.1088/0951-7715/13/4/316 · Zbl 0983.37051
[3] DOI: 10.1063/1.524633
[4] DOI: 10.1007/BF02777365 · Zbl 1138.37016
[5] Kani E., Collect. Math. 54 pp 1– (2003)
[6] DOI: 10.1007/s00229-006-0012-z · Zbl 1105.14041
[7] DOI: 10.1007/s00208-005-0666-y · Zbl 1086.14024
[8] DOI: 10.1007/BF02392354 · Zbl 0517.58028
[9] DOI: 10.1088/0951-7715/12/3/005 · Zbl 0984.37044
[10] DOI: 10.1070/RD2004v009n01ABEH000259 · Zbl 1049.37024
[11] Troubetzkoy S., Ann. Inst. Fourier 55 pp 29– (2005)
[12] DOI: 10.1007/BF01388890 · Zbl 0676.32006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.