Comparison of typicality in quantum and classical many-body systems

TL;DR

The paper investigates whether typicality—the concentration of observable expectations around a common thermal value—arises similarly in quantum and classical many-body systems. It shows that in quantum systems, typicality follows from geometric properties of high-dimensional Hilbert spaces: for a random pure state in a large energy shell, $\\langle\\psi|A|\\psi\\rangle$ deviates from the microcanonical average $A_{mc}$ only with small probability, and quantum fluctuations in pure states can mimic thermal fluctuations. In contrast, classical systems do not exhibit general typicality for microscopic observables; only macroscopic observables tend to display typical behavior as system size grows, with the microscopic case showing non-negligible thermal fluctuations due to equipartition. The results highlight that quantum typicality is a genuinely quantum, geometry-driven phenomenon (not solely entanglement-based) and clarify the limits of classical typicality in reproducing thermal behavior.

Abstract

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.