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 Schrodinger 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.
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