Transition state theory

Reactive Flux for a double well system

In the open quantum system context, like a double well system, the dynamics of the system are typically described by the density matrix rather than a wavefunction. The density matrix provides a complete description of the system's state, including all quantum correlations, and evolves according to the Lindblad master equation if the system-environment interaction is Markovian.

To define a reactive flux, we would need to specify a dividing surface or a transition state that separates the two wells. This could be the barrier top, or some other region of the phase space.

The reactive flux, in this case, would correspond to the rate at which the system transitions from being localized in one well to being localized in the other well, as evidenced by the time evolution of the density matrix.

Typically, the population of a state can be calculated as the expectation value of the projection operator onto that state. For a two-level system, if we denote the projection operator onto the first state as P1 and the density matrix as ρ(t), the population of the first state at time t would be Tr[P1 ρ(t)], where Tr denotes the trace.

The rate of change of this population then gives a measure of the reactive flux, under the assumption that changes in the population are primarily due to transitions between the two states (wells). However, keep in mind that this is a simplified picture and the actual calculation may involve more complex factors, especially when the system-environment interaction and quantum coherence effects are strong.