  We study, analytically and numerically, mesoscopic fluctuations of the
off-diagonal matrix elements of the orbital angular momentum between the
nearest energy levels $i=(n_{x},n_{y}) $ and $f=(k_{x},k_{y}) $ in a
rectangular box with incommensurate sides. In the semiclassical regime, where
the level number of $i$ is $\cal{N}\gg 1$, our derivation gives $\left< |
\hat{L}_{if}| ^{2}\right> \sim \sqrt{\cal{N}}$. Numerical simulations, using
simultaneous ensemble averaging (over the aspect ratios of rectangles) and
spectral averaging (over the energy interval), are in excellent agreement with
this analytical prediction. Physically, the mean is dominated by the level
pairs $k_{x}=n_{x}\pm 1$, $k_{y}=n_{y}\mp 1$. Also in a rectangular box, we
investigate the mean orbital susceptibility of a free electron gas and argue
that it reduces, up to a coefficient, to the two-level van Vleck susceptibility
that involves the last occupied (Fermi) level $i$ and the first unoccupied
level $f$. This result is confirmed numerically as well, albeit the effect of
fluctuations is much more pronounced for the susceptibility since it is due
both to large fluctuations in $\left< | \hat{L}_{if}| ^{2}\right>$ and in level
separations $\epsilon_{f}-\epsilon_{i}$ (level bunching).

Mesoscale and Nanoscale Physics