Master density function product densities

First, the master density function is introduced in Section 7.1. The scalar particle state is discussed in detail in Section 7.1.1 with directions for generalization to the vector case in Section 7.1.2. Coupling with the continuous phase variables is ignored in the foregoing sections, but the necessary modifications for accommodating the environmental effect on the particles are discussed in Section 7.1.3. Thus, from Section 7.1, the basic implements of the stochastic theory of populations along with their probabilistic interpretations become available. These implements are the master density, moment densities that are called product densities, and the resulting mathematical machinery for the calculation of fluctuations. 

A similar derivation is possible for the master density function of an aggregation process but is left to the reader. Instead, we will consider the derivation of equations for an aggregation process in Section 7.3 directly using product densities. 

We now investigate the product densities arising out of this master density function. 

The unfilled parentheses are either for the master density as in (7.3.12) or for the product density functions as in the other equations. The vector case is now fully identified. In Section 7.4.2, we discuss an application of the foregoing development. 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

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