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We derive an upper bound on the difference between the long-time average andthe microcanonical ensemble average of observables in isolated quantum systems.We propose, numerically verify, and analytically support a new hypothesis,eigenstate randomization hypothesis (ERH), which implies that in the energyeigenbasis the diagonal elements of observables fluctuate randomly. We showthat ERH includes eigenstate thermalization hypothesis (ETH) and makes theaforementioned bound vanishingly small. Moreover, ERH is applicable tointegrable systems for which ETH breaks down. We argue that the range of thevalidity of ERH determines that of the microcanonical description.

Eigenstate Randomization Hypothesis: Why Does the Long-Time Average Equal the Microcanonical Average?