Comparison of typicality in quantum and classical many-body systems

Quantum typicality refers to the phenomenon that the expectation values of any given observable are nearly identical for the overwhelming majority of all normalized vectors in a sufficiently high-dimensional Hilbert (sub-)space. As a consequence, we show that the thermal equilibrium fluctuations in many-body quantum systems can be very closely imitated by the purely quantum mechanical uncertainties (quantum fluctuations) of suitably chosen pure states. On the other hand, we find that analogous typicality effects of similar generality are not encountered in classical systems. The reason is that the basic mathematical structure, in particular the description of pure states, is fundamentally different in quantum and classical mechanics.