  We unveil the role of the long time average of Loschmidt echo in the
characterization of nonequilibrium quantum phase transitions by studying sudden
quench processes across quantum phase transitions in various quantum systems.
While the dynamical quantum phase transitions are characterized by the
emergence of a series of zero points at critical times during time evolution,
we demonstrate that nonequilibrium quantum phase transitions can be identified
by nonanalyticities in the long time average of Loschmidt echo. The nonanalytic
behaviours are illustrated by a sharp change in the long time average of
Loschmidt echo or the corresponding rate function or the emergence of
divergence in the second derivative of rate function when the driving quench
parameter crosses the phase transition points. The connection between the
second derivative of rate function and fidelity susceptibility is also
discussed.
