  Flip-flop processes due to magnetic dipole-dipole interaction between
neighbouring ions in rare-earth-ion-doped crystals is one of the mechanisms of
relaxation between hyperfine levels. Modeling of this mechanism has so far been
macroscopic, characterized by an average rate describing the relaxation of all
ions. Here however, we present a microscopic model of flip-flop interactions
between individual nuclear spins of dopant ions. Every ion is situated in a
unique local environment in the crystal, where each ion has different distances
and a unique orientation relative to its nearest neighbors, as determined by
the lattice structure. Thus, each ion has a unique flip-flop rate and the
collective relaxation dynamics of all ions in a bulk crystal is a sum of many
exponential decays, giving rise to a distribution of rates rather than a single
average decay rate. We employ this model to calculate flip-flop rates in
Pr$^{3+}$:Y$_2$SiO$_5$ and show experimental measurements of population decay
of the ground state hyperfine levels at $\sim$2 K. We also present a new method
to measure rates of individual transitions from hole burning spectra that
requires significantly fewer fitting parameters in theoretical rate equations
compared to earlier work. Furthermore, we measure the effect of external
magnetic field on the flip-flop rates and observe that the rates slow down by
two orders of magnitude in a field of 5 - 10 mT.
