  According to the eigenstate thermalization hypothesis (ETH), the
eigenstate-to-eigenstate fluctuations of expectation values of local
observables should decrease with increasing system size. In approaching the
thermodynamic limit - the number of sites and the particle number increasing at
the same rate - the fluctuations should scale as $\sim D^{-1/2}$ with the
Hilbert space dimension $D$. Here, we study a different limit - the classical
or semiclassical limit - by increasing the particle number in fixed lattice
topologies. We focus on the paradigmatic Bose-Hubbard system, which is
quantum-chaotic for large lattices and shows mixed behavior for small lattices.
We derive expressions for the expected scaling, assuming ideal eigenstates
having Gaussian-distributed random components. We show numerically that, for
larger lattices, ETH scaling of physical mid-spectrum eigenstates follows the
ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via
a different exponent. We examine several plausible mechanisms for this
anomalous scaling.
