  The transverse momentum broadening coefficient $\hat{q}$ receives both soft,
classical and radiative, quantum corrections. The former are responsible for a
large O(g) correction, whereas the latter enter at relative order $\alpha_s$,
but are enhanced by a double logarithm of the length of the medium over the
thermal wavelength. We analyze radiative corrections for a weakly-coupled
quark-gluon plasma. We find that a thermal population of dynamical gluons
changes the boundaries and reduces the size of the double-logarithmic phase
space. It also provides new subdominant logarithmic corrections. We also show
how the quantum, double-logarithmic and classical, soft phase spaces are
smoothly connected once the radiated gluon becomes soft enough. Finally, we
discuss a pathway to a determination of radiative corrections beyond the
harmonic-oscillator approximation.
