  Motivated by the experimental observation [1] that driving a non-interacting
Bose gas in a 3D box with weak disorder leads to power-law energy growth, $E
\propto t^{\eta}$ with $\eta=0.46(2)$, and compressed-exponential momentum
distributions that show dynamic scaling, we perform systematic numerical and
analytical studies of this system. Schr\"odinger-equation simulations reveal a
crossover from $\eta \approx 0.5$ to $\eta \approx 0.4$ with increasing
disorder strength, hinting at the existence of two different dynamical regimes.
We present a semi-classical model that captures the simulation results and
allows an understanding of the dynamics in terms of an energy-space random
walk, from which a crossover from $E \propto t^{1/2}$ to $E \propto t^{2/5}$
scaling is analytically obtained. The two limits correspond to the random walk
being limited by the rate of the elastic disorder-induced scattering or the
rate at which the drive can change the system's energy. Our results provide the
theoretical foundation for further experiments.
