  The spins of gaseous alkali atoms are commonly assumed to oscillate at a
constant hyperfine frequency, which for many years has been used to define the
Second. Indeed, under standard experimental conditions, the spins oscillate
independently, only weakly perturbed and slowly decaying due to random
spin-spin collisions. Here we consider a different, unexplored regime of very
dense gas, where collisions, more frequent than the hyperfine frequency,
dominate the dynamics. Counter-intuitively, we find that the hyperfine
oscillations become significantly longer-lived, and their frequency becomes
dependent on the state of the ensemble, manifesting strong nonlinear dynamics.
We reveal that the nonlinearity originates from a many-body interaction which
synchronizes the electronic spins, driving them into a single collective mode.
The conditions for experimental realizations of this regime are outlined.
