  The time dynamics of the energy flow from electronic to lattice degrees of
freedom in pump-probe setups could be strongly affected by the presence of a
hot-phonon bottleneck, which can sustain longer coherence of the optically
excited electronic states. Recently, hot-phonon physics has been experimentally
observed and theoretically described in MgB$_2$, the electron-phonon based
superconductor with $T_{\rm c}\approx 39$ K. By employing a combined ab-initio
and quantum-field-theory approach and by taking MgB$_2$ as an example, here we
propose a novel path for revealing the presence and characterizing the
properties of hot phonons through a direct analysis of the information encoded
in the lattice inter-atomic correlations. Such method exploits the underlying
symmetry of the $E_{2g}$ hot modes characterized by a out-of-phase in-plane
motion of the two boron atoms. Since hot phonons occur typically at
high-symmetry points of the Brillouin zone, with specific symmetries of the
lattice displacements, the present analysis is quite general and it could aid
in revealing the hot-phonon physics in other promising materials, such as
graphene, boron nitride, or black phosphorus.
