We calculate the structure function and intermittent exponent of the 1.) Keck data, which consists of 29 high resolution, high signal to noise ratio (S/N) QSO Ly$alpha$ absorption spectra, and 2.)the Ly$alpha$ forest simulation samples produced via the pseudo hydro scheme for the low density cold dark matter (LCDM) model and warm dark matter (WDM) model with particle mass $m_W=300, 600, 800$ and 1000 eV. These two measures detect not only non-gaussianities, but also the type of non-gaussianty in the the field. We find that, 1.) the structure functions of the simulation samples are significantly larger than that of Keck data on scales less than about 100 h$^{-1}$ kpc, 2.) the intermittent exponent of the simulation samples is more negative than that of Keck data on all redshifts considered, 3.) the order-dependence of the structure functions of simulation samples are closer to the intermittency of hierarchical clustering on all scales, while the Keck data are closer to a lognormal field on small scales. These differences are independent of noise and show that the intermittent evolution modeled by the pseudo-hydro simulation is substantially different from observations, even though they are in good agreement in terms of second and lower order statistics. (Abridged)
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