On the dynamics of reaction coordinates in classical, time-dependent, many-body processes

Hugues Meyer, Thomas Voigtmann, Tanja Schilling

Published 2019-01-28Version 1

Complex microscopic many-body processes are often interpreted in terms of so-called "reaction coordinates", i.e. in terms of the evolution of a small set of coarse-grained observables. A rigorous method to produce the equation of motion of such observables is to use projection operator techniques, which split the dynamics of the observables into a main contribution and a marginal one. The basis of any derivation in this framework is the classical (or quantum) Heisenberg equation for an observable. If the Hamiltonian of the underlying microscopic dynamics does not explicitly depend on time, this equation is obtained by a straight-forward derivation. However, the problem is more complicated if one considers Hamiltonians which depend on time explicitly as e.g.~in systems under external driving. We use an analogy to fluid dynamics to derive the classical Heisenberg picture and then apply a projection operator formalism to derive the non-stationary generalized Langevin equation for a coarse-grained variable. We show, in particular, that the results presented for a time-independent Hamiltonian in J. Chem. Phys. 147, 214110 (2017) can be generalized to the time-dependent case.