  Non-equilibrium thermodynamics can provide strong advantages when compared to
more standard equilibrium situations. Here, we present a general framework to
study its application to concrete problems, which is valid also beyond the
assumption of a Gaussian dynamics. We consider two different problems: 1) the
dynamics of a levitated nanoparticle undergoing the transition from an harmonic
to a double-well potential; 2) the transfer of a quantum state across a
double-well potential through classical and quantum protocols. In both cases,
we assume that the system undergoes to decoherence and thermalisation. In case
1), we construct a numerical approach to the problem and study the
non-equilibrium thermodynamics of the system. In case 2), we introduce a new
figure of merit to quantify the efficiency of a state-transfer protocol and
apply it to quantum and classical versions of such protocols.
