Master density equation solution

Obviously, the preceding hierarchy of master density equations can also be closed at v = N. However, the product density equations may allow closure at a considerably lower value of r, which makes them much more attractive to solve than the master density equations. As pointed out earlier, even an analytical solution to the master density equation is not particularly valuable because of its combinatorial complexity. 

A discrete version of the master density equations (7.3.10), without particle growth, has been solved by Bayewitz et al (1974), and later by Williams (1979), to examine the dynamic average particle size distribution in an aggregating system with a constant kernel. When the population is small EN < 50) their predictions reveal significant variations from those predicted by the population balance equation. However, the solution of such master density equations is extremely difficult even for the small populations of interest for nonconstant kernels. It is from this point of view that a suitably closed set of product density equations presents a much better alternative for analysis of such aggregating systems. We take up this issue of closure again in Section 7.4. 

Second, Section 7.2 provides the derivation of equations in the master density for some of the particulate processes discussed in Chapter 3 and shows the combinatorial complexity of their solution. The objective of this section is to show how Monte Carlo simulations of a process eliminate the quantitatively less significant combinatorial elements of the solution by artificial realization. Thus, the discussion here is of largely conceptual value. 

Figure A3.13.16. Illustration of the level populations (eorresponding to the C-C oseillator states) from various treatments in the model of figure A3.13.15 for C2Hg at a total energy E = (he) 41 000 em and a tlneshold energy = (he) 31 000 em The pomts are mieroeanonieal equilibrium distributions. The erosses result from the solution of the master equation for IVR at steady state and the lines are thennal populations at the temperatures indieated (from [38] quant, is ealeulated with quantum densities of states, elass. with elassieal meehanieal densities.).

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... 

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). 

Exercise. In the preceding model let V2 and N go to infinity with finite density N/V2 = p. Write the master equation and show that the stationary solution is a Poisson distribution. (Could this have been known a priori ) What does detailed balance tell us in this case ... 

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 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. 

There is another interesting connection [1]. We define P (Xi,fi Xo,fo) to be the probability density of observing Xj molecules in V at time tj given that there are Xq molecules at to- This function is the solution of the master equation (2.29) for the initial condition... 

The theory presented here is limited to the transport of incoherent excitations, describable with a Pauli master equation. In addition, chromophore diffusion resulting from Brownian forces on the chains is assumed to be negligible on the time scale of excitation transport. However, no assumptions are made about the density of the material, so the theory should be applicable to homogeneous melts and to polymers dispersed in amorphous solids, as well as to chains in solution. 

Derive the chemical master equation for the reversible reaction in Eq. 13.1. Solve for the probability density and compare to the deterministic solution. 

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