Three alternative explanations for the thermalization of isolated quantum systems are often proposed:
# For initial states of physical interest, the coefficients  exhibit large fluctuations from eigenstate to eigenstate, in a fashion which is completely uncorrelated with the fluctuations of  from eigenstate to eigenstate. Because the coefficients and matrix elements are uncorrelated, the summation in the diagonal ensemble is effectively performing an unbiased sampling of the values of  over the appropriate energy window. For a sufficiently large system, this unbiased sampling should result in a value which is close to the true mean of the values of  over this window, and will effectively reproduce the prediction of the microcanonical ensemble. However, this mechanism may be disfavored for the following heuristic reason. Typically, one is interested in physical situations in which the initial expectation value of  is far from its equilibrium value. For this to be true, the initial state must contain some sort of specific information about , and so it becomes suspect whether or not the initial state truly represents an unbiased sampling of the values of  over the appropriate energy window. Furthermore, whether or not this were to be true, it still does not provide an answer to the question of when arbitrary initial states will come to equilibrium, if they ever do.
# For initial states of physical interest, the coefficients  are effectively constant, and do not fluctuate at all. In this case, the diagonal ensemble is precisely the same as the microcanonical ensemble, and there is no mystery as to why their predictions are identical. However, this explanation is disfavored for much the same reasons as the first.
# Integrable quantum systems are proved to thermalize under condition of simple regular time-dependence of parameters, suggesting that cosmological expansion of the Universe and integrability of the most fundamental equations of motion are ultimately responsible for thermalization.