  This paper address the question of thermodynamic entropy production in the
context of the dynamical Casimir effect. Specifically, we study a scalar
quantum field confined within a one-dimensional ideal cavity subject to
time-varying boundary conditions dictated by an externally prescribed
trajectory of one of the cavity mirrors. The central question is how the
thermodynamic entropy of the field evolves over time. Utilizing an effective
Hamiltonian approach, we compute the entropy production and reveal that it
exhibits scaling behavior concerning the number of particles created in the
short-time limit. Furthermore, this approach elucidates the direct connection
between this entropy and the emergence of quantum coherence within the mode
basis of the field. In addition, by considering a distinct approach based on
the time evolution of Gaussian states we examine the long-time limit of entropy
production within a single mode of the field. This approach results in
establishing a connection between the thermodynamic entropy production in a
single field mode and the entanglement between that particular mode and all
other modes. Consequently, by employing two distinct approaches, we
comprehensively address both the short-term and long-term dynamics of the
system. Our results thus link the irreversible dynamics of the field, as
measured by entropy production and induced by the dynamical Casimir effect, to
two fundamental aspects of quantum mechanics: coherence and entanglement.
