Splitting master equation

A fundamental property of the master equation is As t -> oo all solutions tend to the stationary solution or - in the case of decomposable or splitting W - to one of the stationary solutions. Again this statement is strictly true only for a finite number of discrete states. For an infinite number of states, and a fortiori for a continuous state space, there are exceptions, e.g., the random walk (2.11). Yet it is a useful rule of thumb for a physicist who knows that many systems tend to equilibrium. We shall therefore not attempt to give a general proof covering all possible cases, but restrict ourselves to a finite state space. There exist several ways of proving the theorem. Of course, they all rely on the property (2.5), which defines the class of W-matrices. 

There may be any number of them. From the fact that the master equation (unless decomposable or splitting) has a single stationary solution Ps one cannot conclude that the macroscopic equation cannot have more than one stationary macroscopic state, as will be seen in XIII. 1. 

The master equation (165) leads to a closed system of 25 equations of motion for the density matrix elements. Since the laser field does not couple to the level d), the system of equations splits into two subsystems a set of 17 equations of motion directly coupled to the driving field and the other of 8 equations of motion not coupled to the driving field. It is not difficult to show that the steady-state solutions for the 8 density matrix elements are zero. Using the trace property, one of the remaining equations can be eliminated, and the system of equations reduces to the 16 coupled linear inhomogeneous equations. 

Blitz and coworkers have also modeled their data with master equation simulations employing empirical high pressure rate values. While these simulations do provide improved agreement with the full set of data, they do so by splitting the difference between theirs and Leone and coworkers experimental results. The remaining discrepancies between the fit and experiment are outside the estimated experimental error bars, suggesting some discrepancy between the various experiments. Thus, it would be useful to have further experimental studies of this reaction. 

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