Velocity and Entropy of Motion in Periodic Potentials
Séminaire Équations aux dérivées partielles (Polytechnique) (1996-1997), Talk no. 17, 11 p.

This is a report on recent joint work with J. Asch, and with T. Hudetz and F. Benatti.

We consider classical, quantum and semiclassical motion in periodic potentials and prove various results on the distribution of asymptotic velocities.

The Kolmogorov-Sinai entropy and its quantum generalization, the Connes-Narnhofer-Thirring entropy, of the single particle and of a gas of noninteracting particles are related.

@article{SEDP_1996-1997____A17_0,
     author = {Knauf, Andreas},
     title = {Velocity and Entropy of Motion in Periodic Potentials},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1996-1997},
     note = {talk:17},
     mrnumber = {1482823},
     zbl = {1055.81526},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1996-1997____A17_0}
}
Knauf, Andreas. Velocity and Entropy of Motion in Periodic Potentials. Séminaire Équations aux dérivées partielles (Polytechnique) (1996-1997), Talk no. 17, 11 p. http://www.numdam.org/item/SEDP_1996-1997____A17_0/

[1] Asch, J., Knauf, A.: Motion in Periodic Potentials. Preprint (1997) | MR 1492956 | Zbl 0896.34027

[2] Benatti, F., Hudetz, T., Knauf, A.: Quantum Chaos and Dynamical Entropy. Preprint (1997) | MR 1670033 | Zbl 0927.37008

[3] Connes, A., Narnhofer, H., Thirring, W.: Dynamical Entropy for C* algebras and von Neumann Algebras. Commun. Math. Phys. 112, 691 (1987). | MR 910587 | Zbl 0637.46073

[4] Gérard, Ch., Nier, F.: Scattering Theory for the Perturbations of Periodic Schrödinger Operators. Preprint Ecole Polytechnique (1997) | MR 1669979 | Zbl 0934.35111

[5] Klein, M., Knauf, A.: Classical Planar Scattering by Coulombic Potentials. Lecture Notes in Physics m 13. Berlin, Heidelberg, New York: Springer; 1993 | Zbl 0783.70001

[6] Knauf, A.: Ergodic and Topological Properties of Coulombic Periodic Potentials. Commun. Math. Phys. 110, 89–112 (1987) | MR 885572 | Zbl 0616.58044

[7] Knauf, A.: Coulombic Periodic Potentials: The Quantum Case. Annals of Physics 191, 205–240 (1989) | MR 1003009

[8] Knauf, A.: Closed orbits and converse KAM theory. Nonlinearity 3, 961–973 (1990) | MR 1067089 | Zbl 0702.70013

[9] Reed, M., Simon, B.: Methods in Mathematical Physics, Vol. IV: Analysis of Operators. New York: Academic Press 1978 | MR 493421 | Zbl 0401.47001

[10] Thomas, L.E.: Time Dependent Approach to Scattering from Impurities in a Crystal. Commun. Math. Phys 33, 335–343 (1973) | MR 334766