@englman] geometric phase [@berry; @qphases; @mead] enters into the propagation of the quantum-classical dynamics. We will then describe the case of a non-Hamiltonian bath, which arises when the bath coordinates coupled to the quantum subsystem are also coupled to a large bath (which does not directly interact with the quantum subsystem and whose detailed dynamics is not of interest). In such cases, the secondary bath acts as a thermal reservoir and can be described either by means of stochastic processes [@gardiner] (e.g., Langevin dynamics [@langevin-qc]), or by means of non-Hamiltonian fictitious coordinates acting as deterministic thermostats (e.g., the Nosé–Hoover thermostat [@nose; @hoover]). Both Langevin and Nosé–Hoover deterministic time evolutions are examples of non-Hamiltonian dynamics. However, only Nosé–Hoover dynamics is defined solely in terms of a quasi-Lie bracket [@b3; @b4]. Instead, explicit dissipative dynamics requires that diffusive terms be added to the bracket.

The quantum-classical equations of motion herein discussed can be implemented [in silico]{} using a variety of simulation algorithms [@sergi-theor; @sstp; @trotter; @asfp; @dauasfp; @dauasfp2; @fmgh; @ddagh; @ddagh2; @ddagh3; @ddagh4; @jlgh]. We will sketch out one such integration algorithm, which unfolds the quantum-classical dynamics of the operator-valued quasi-probability function in terms of piecewise-deterministic trajectories evolving on the adiabatic energy 