  A theory is developed which relates the observed b-parameter of a Lyman-alpha
absorption line to the velocity-curvature of the corresponding peak in the
optical depth fluctuation. Its relation to the traditional interpretation of b
as the thermal broadening width is discussed. It is demonstrated that,
independent of the details of the cosmological model, the differential b
distribution has a high b asymptote of $dN/db \propto b^{-m}$, where $m \geq
5$, when we make the reasonable assumption that low-curvature fluctuations are
statistically favored over high-curvature ones. There in general always exist
absorption lines much broader than the thermal width. We then develop a linear
perturbative analysis of the optical depth fluctuation, which yields a
single-parameter prediction for the full b distribution: in addition to
exhibiting the general high velocity tail, it explains the observed sharp low b
cut-off. The dependence of the b distribution on cosmological parameters, such
as $\Omega$ and the power spectrum, and reionization history as well as
observation/simulation resolution is derived and discussed.
