  We investigate the relaxation mechanism of a supercooled tetrahedral liquid
at its limit of stability using isothermal isobaric ($NPT$) Monte Carlo (MC)
simulations. In similarity with systems which are far from equilibrium but near
the onset of jamming [O'Hern et.al., Phys. Rev. Lett. {\bf 93}, 165702 (2004)],
we find that the relaxation is characterized by two time-scales: the decay of
long-wavelength (slow) fluctuations of potential energy is controlled by the
the slope $[\partial (G/N)/\partial \phi]$ of the Gibbs free energy ($G$) at a
unique value of per particle potential energy $\phi = \phi_{mid}$. The
short-wavelength (fast) fluctuations are controlled by the bath temperature
$T$. The relaxation of the supercooled liquid is initiated with a dynamical
crossover after which the potential energy fluctuations are biased towards
values progressively lesser than $\phi_{mid}$. The dynamical crossover leads to
the change of time-scale, i.e., the decay of long-wavelength potential energy
fluctuations (intermediate relaxation). Because of the condition [$\partial^2
(G/N)/\partial \phi^2 = 0$] at $\phi = \phi_{mid}$, the slope $[\partial
(G/N)/\partial \phi]$ has a unique value and governs the intermediate stage of
relaxation, which ends just after the crossover. In the subsequent stage, there
is a relatively rapid crystallization due to lack of long-wavelength
fluctuations and the instability at $\phi_{mid}$, i.e., the condition that $G$
decreases as configurations with potential energies lower than $\phi_{mid}$ are
accessed. The dynamical crossover point and the associated change in the
time-scale of fluctuations is found to be consistent with the previous studies.
