  We theoretically investigate thermalization and spin diffusion driven by a
quantum spin bath for a realistic solid-state NMR experiment. We consider
polycrystalline L-alanine, and investigate how the spin polarization spreads
among several $^{13}$C nuclear spins, which interact via dipole-dipole coupling
with the bath of strongly dipolar-coupled $^1$H nuclear (proton) spins. We do
this by using direct numerical simulation of the many-spin time-dependent
Schr\"odinger equation. We find that, although the proton spins located near
the carbon sites interact most strongly with the $^{13}$C spins, this
interaction alone is not enough to drive spin diffusion and thermalize the
$^{13}$C nuclear spins. We demonstrate that the thermalization within the
$^{13}$C subsystem is driven by the collective many-body dynamics of the proton
spin bath, and specifically, that the onset of thermalization among the
$^{13}$C spins is directly related to the onset of chaotic behavior in the
proton spin bath. Therefore, thermalization and spin diffusion within the
$^{13}$C subsystem is controlled by the proton spins located far from the C
sites. In spite of their weak coupling to the $^{13}$C spins, these far-away
protons help produce a network of strongly coupled proton spins with collective
dynamics, that drives thermalization.
