  The 21-cm hyperfine line of atomic hydrogen (HI) is a promising probe of the
cosmic dark ages. In past treatments of 21-cm radiation it was assumed the
hyperfine level populations of HI could be characterized by a
velocity-independent ``spin temperature'' T_s determined by a competition
between 21-cm radiative transitions, spin-changing collisions, and (at lower
redshifts) Lyman-alpha scattering. However we show here that, if the
collisional time is comparable to the radiative time, the spin temperature will
depend on atomic velocity, T_s=T_s(v), and one must replace the usual hyperfine
level rate equations with a Boltzmann equation describing the spin and velocity
dependence of the HI distribution function. We construct here the Boltzmann
equation relevant to the cosmic dark ages and solve it using a basis-function
method. Accounting for the actual spin-resolved atomic velocity distribution
results in up to a 2 per cent suppression of the 21-cm emissivity, and a
redshift and angular-projection dependent suppression or enhancement of the
linear power spectrum of 21-cm fluctuations of up to 5 per cent. The effect on
the 21-cm line profile is more dramatic -- its full-width at half maximum
(FWHM) can be enhanced by up to 60 per cent relative to the
velocity-independent calculation. We discuss the implications for 21-cm
tomography of the dark ages.
