Ergodicity and quantum correlations in irrational triangular billiards

Le résumé fourni par la source

Pseudochaotic properties are systematically investigated in a one-parameter family of irrational triangular billiards (all angles irrational with $\ensuremath{\pi}$). The absolute value of the position correlation function ${C}_{x}(t)$ decays like $\ensuremath{\sim}$${t}^{\ensuremath{-}\ensuremath{\alpha}}$. Fast $(\ensuremath{\alpha}\ensuremath{\approx}1)$ and slow $(0<\ensuremath{\alpha}<1)$ decays are observed, thus indicating that the irrational triangles do not share a unique ergodic dynamics, which, instead, may vary smoothly between the opposite limits of strong mixing $(\ensuremath{\alpha}=1)$ and regular behaviors $(\ensuremath{\alpha}=0)$. Upgrading previous data, spectral statistical properties of the quantized counterparts are computed from $150\phantom{\rule{0.16em}{0ex}}000$ energy eigenvalues numerically calculated for each billiard. Gaussian orthogonal ensemble spectral fluctuations are observed when $\ensuremath{\alpha}\ensuremath{\approx}1$ and intermediate statistics are found otherwise. Our irrational billiards have zero Kolmogorov-Sinai entropy and essentially infinity genus. Thus, differently from previous works on rational (pseudointegrable) enclosures, our results provide a missing classical-quantum correspondence regarding the ergodic hierarchy for a set of nonchaotic systems that might enjoy the strong mixing property.

Ce résumé expose les affirmations des auteurs. BNTIC ne l’interprète pas comme une validation indépendante des résultats.