**Boris Altshuler**

Columbia University

*Between Localization and Ergodicity in Classical and Quantum Systems*

Can we describe conventional Statistical Physics isolated systems with large number of degrees of freedom? Can such systems reach a thermal state being decoupled from the thermostat? One can think about two limiting cases: (*) integrable systems which are characterized by a conservation law per each degree of freedom and (*) completely chaotic ergodic systems, which in contrast with the integrable ones do thermalize. According to the Kolmogorov-Arnold-Moser (KAM) theorem weak enough violation of the integrability does not lead to the qualitative change of the system behavior. However several examples, such as e.g. Solar System, suggest that thermalization (ergodicity) is not restored even far from the KAM region. Quantum analog of the KAM behavior is Many Body Localization of the eigenstates of the system in the Hilbert space. The laws of the Statistical Physics based on the Gibbs equipartition postulate are obviously inapplicable in the MBL state. However it turns out that delocalization does not immediately produce the ergodic state of the system. It looks like there exists the intermediate regime with the quantum states that are neither localized nor extended and ergodic.

I will present the arguments in favor of the existence of this extended non-ergodic phase with multifractal eigenfunctions and discuss several examples of quantum and classical non-ergodic behavior.