  The quench dynamics of many-body quantum systems may exhibit
non-analyticities in the Loschmidt echo, a phenomenon known as dynamical phase
transition (DPT). Despite considerable research into the underlying mechanisms
behind this phenomenon, several open questions still remain. Motivated by this,
we put forth a detailed study of DPTs from the perspective of quantum phase
space and entropy production, a key concept in thermodynamics. We focus on the
Lipkin-Meshkov-Glick model and use spin coherent states to construct the
corresponding Husimi-$Q$ quasi-probability distribution. The entropy of the
$Q$-function, known as Wehrl entropy, provides a measure of the coarse-grained
dynamics of the system and, therefore, evolves non-trivially even for closed
systems. We show that critical quenches lead to a quasi-monotonic growth of the
Wehrl entropy in time, combined with small oscillations. The former reflects
the information scrambling characteristic of these transitions and serves as a
measure of entropy production. On the other hand, the small oscillations imply
negative entropy production rates and, therefore, signal the recurrences of the
Loschmidt echo. Finally, we also study a Gaussification of the model based on a
modified Holstein-Primakoff approximation. This allows us to identify the
relative contribution of the low energy sector to the emergence of DPTs. The
results presented in this article are relevant not only from the dynamical
quantum phase transition perspective, but also for the field of quantum
thermodynamics, since they point out that the Wehrl entropy can be used as a
viable measure of entropy production.
