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

## TL;DR

This paper develops a theoretical framework using projection operators to derive equations of motion for reaction coordinates in classical many-body systems with explicitly time-dependent Hamiltonians, extending previous stationary results.

## Contribution

It generalizes the classical Heisenberg equation and projection operator formalism to systems with explicit time dependence in Hamiltonians and observables.

## Key findings

- Derived non-stationary generalized Langevin equation for time-dependent systems
- Extended previous stationary results to explicitly time-dependent Hamiltonians
- Provided a fluid dynamics analogy for the classical Heisenberg picture

## Abstract

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 and the observable under study do 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, or if the observable of interest has an explicit dependence on time. 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 time-independent Hamiltonians and observables in J. Chem. Phys. 147, 214110 (2017) can be generalized to the time-dependent case.