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We study, analytically and numerically, mesoscopic fluctuations of theoff-diagonal matrix elements of the orbital angular momentum between thenearest energy levels $i=(n_{x},n_{y}) $ and $f=(k_{x},k_{y}) $ in arectangular box with incommensurate sides. In the semiclassical regime, wherethe level number of $i$ is $\cal{N}\gg 1$, our derivation gives $\left< |\hat{L}_{if}| ^{2}\right> \sim \sqrt{\cal{N}}$. Numerical simulations, usingsimultaneous ensemble averaging (over the aspect ratios of rectangles) andspectral averaging (over the energy interval), are in excellent agreement withthis analytical prediction. Physically, the mean is dominated by the levelpairs $k_{x}=n_{x}\pm 1$, $k_{y}=n_{y}\mp 1$. Also in a rectangular box, weinvestigate the mean orbital susceptibility of a free electron gas and arguethat it reduces, up to a coefficient, to the two-level van Vleck susceptibilitythat involves the last occupied (Fermi) level $i$ and the first unoccupiedlevel $f$. This result is confirmed numerically as well, albeit the effect offluctuations is much more pronounced for the susceptibility since it is dueboth to large fluctuations in $\left< | \hat{L}_{if}| ^{2}\right>$ and in levelseparations $\epsilon_{f}-\epsilon_{i}$ (level bunching).

Mesoscopic fluctuations of off-diagonal matrix elements of the angular momentum and orbital magnetism of free electrons in a rectangular box