The Master Density Function

We shall deal with a system of particles distributed according to a vector particle state as considered in Section 2.1. However, we shall not seek to distinguish between internal coordinates x and external coordinates r, and instead regard the joint particle state as z. 

It will ease our discussion considerably to deal with the scalar case first, so that we denote the particle state by z. Further, we neglect any dependence on the continuous phase variables and consider this in Section 7.1.3. 

The actual total number of particles in the system is obtained by integrating the number density over all possible states of all particles. Thus, 

PrjThere are at time t, v particles in the system with one each in 

The probability density function is symmetric with respect to all the particle states in the sense that its value is unaltered by the permutations of the z/s. This follows from the particles being indistinguishable except through their chosen states. The proper normalization condition for the density function may be arrived at by ordering the particle states as Zi Z2 z in order to avoid redundancies so that 

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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. 

Since the master density function is insensitive to the permutation of the particle state arguments, the integrals in the inner sum are all the same so that... 

Notice in particular the first term on the right-hand side where a Dirac delta function appears because of integration with respect to z. The second term on the right-hand side has been identified as such based on the assumption that z z since Zj z while integrating the previous equation. We now make use of the symmetry properties of the master density function. In both terms on the right-hand side of the preceding equation, the summands within the inner sums are independent of the index of summation so that we may write... 

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. 

The integral U((7m) represents the area under the engineering stress-strain master curve up to the maximum stress (7. The static stress is a function of the energy density function U([Pg.18]

The master density should be concerned not only with the population of particles and their states, but also with the state of the environment. Thus, we define the function 2 0 such that... 

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. 

We now define a more general master density function than that defined in Section 7.1.1. Again, we confine the treatment to the scalar particle state, denoted x, since the extension to the vector case, as seen in Section 7.1.2, is straightforward. 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

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). 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

In these review we consider different methods. The density matrix can be determined from the master equation. For Green functions the EOM method or Keldysh method can be applied. Traditionally, the density matrix is used in the case of very weak system-to-lead coupling, while the NGF methods are more successful in the description of strong and intermediate coupling to the leads. The convenience of one or other method is determined essentially by the type of interaction. Our aim is to combine the advantages of both methods. 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

The functions g(9) and h 6) represent two separate effects of disentanglement g(6) is the fractional reduction of entanglement density due to steady shear flow and h(ff) is the fractional reduction in energy dissipation rate per molecule due to dis-entanglement in steady shear flow (for details, see Graessley, 1974, Chap. 8). Eqs. (16.52)-(16.55) define an implicit expression for the master curve rj/rj0 vs. qzn, the reduced viscosity also being present in the arguments of the functions g(6) and h(6). 

Figure 9.15 Lambda as function of the foam density line, master curve points, measured values with particle foam on basis of Sconapor1 (trade name of EPS by BSL/Dow Chemical)...

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