  Driven-dissipative phase transitions are currently a topic of intense
research due to the prospect of experimental realizations in quantum optical
setups. The most paradigmatic model presenting such a transition is the Kerr
model, which predicts the phenomenon of optical bistability, where the system
may relax to two different steady-states for the same driving condition. These
states, however, are inherently out-of-equilibrium and are thus characterized
by the continuous production of irreversible entropy, a key quantifier in
thermodynamics. In this paper we study the dynamics of the entropy production
rate in a quench scenario of the Kerr model, where the external pump is
abruptly changed. This is accomplished using a recently developed formalism,
based on the Husimi $Q$-function, which is particularly tailored for
driven-dissipative and non-Gaussian bosonic systems [Phys. Rev. Res. 2, 013136
(2020)]. Within this framework the entropy production can be split into two
contributions, one being extensive with the drive and describing classical
irreversibility, and the other being intensive and directly related to quantum
fluctuations. The latter, in particular, is found to reveal the high degree of
non-adiabaticity, for quenches between different metastable states.
