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\def\lesssim{\mathrel{\hbox{\rlap{\hbox{\lower4pt \hbox{$\sim$}}} \hbox{$<$}}}} The hydrodynamical evolution of spherical perturbations consisting of dark matter and a hydrogen/helium baryonic component is computed to model the evolution of Ly$\alpha$ forest systems. If the IGM is photoionized for $z\lesssim6$, the baryons in perturbations with circular velocities less than the sound speed will expand into the IGM driven by their thermal pressure. The hydrodynamical motion of these clouds will preserve to high accuracy a Voigt line profile, with a Doppler $b$-parameter, measuring the width of the absorption lines, in the range $25\lesssim b\lesssim 70{\rm\,km\,s^{-1}}$. For low column density systems ($N_{HI} \lesssim 10^{14}{\rm\,cm^{-2}}$), $b$ is found to decrease for decreasing column density. These lines arise in the interface between the cloud and the IGM. Expansion cooling is found to cool the clouds effectively, but the required velocities are too large to account for the very low $b$-values ($b<20{\rm\,km\,s^{-1}}$), reported in the literature. The clouds have diameters at $z=2.5$ as large as $100{\rm\,kpc}$ for $q_0=0.5$, and $400{\rm\,kpc}$ for $q_0=0.1$, both for $H_0=50{\rm\,km\,s^{-1} \,Mpc^{-1}}$.
