Extended detailed balance

Being applied for the relaxation of populations (k = 0), this equality expresses the demands of the detailed balance principle. This is simply a generalization of Eq. (4.25), which establishes the well-known relation between rates of excitation and deactivation for the rotational spectrum. It is much more important that equality (5.21) holds not only for k = 0 but also for k = 1 when it deals with relaxation of angular momentum J and the elements should not be attributed any obvious physical sense. The non-triviality of this generalization is emphasized by the fact that it is impossible to extend it to the elements of the four-index... 

The quantities K0 and Kt thus define the solution. As indicated in Appendix A, the result, Eqs. (5)-(9), is identical with the familiar statistical mechanical solution for the case of nearest-neighbor interactions, summarized for example, by Schwarz.2 We note the ease with which the results have been obtained here. The procedure could be extended to other cases, for example, a copolymer (i.e., a linear lattice with two types of sites) distributed in a prescribed manner and undergoing a transition to two other types of sites. For the finite chain, however, the use of nearest-neighbor conditional probabilities and detailed balancing will not yield the complete solution.3... 

Remark. The detailed balance relation (4.2) or (6.1) asserts that the matrix W is virtually symmetric and will be seen in the next section to guarantee that W can be diagonalized. The relation (6.14) is also a property of W but does not by itself guarantee diagonalizability, and wil be referred to as extended detailed balance . The relations (6.12) and (6.13) are not properties of W but relate the transition probabilities in one system to those in another system. They will therefore not be honored with the name detailed balance. The extended detailed balance property will be important in XI.4. 

This difficulty does not arise in the case of closed, isolated physical systems, because there the stationary solution is known to be the thermal equilibrium distribution Pe(x), as given by ordinary statistical mechanics. This knowledge implies some information about At and Bij9 but more information is available if also detailed balance (V.6.1) or extended detailed balance (V.6.14) holds. In the following we shall therefore examine the situation specified by the following stipulations. 

Warning. The idea of using a nonlinear Fokker-Planck equation as a general framework for describing fluctuating systems has attracted many authors. Detailed balance, in its extended form, was a useful aid, but the link with the deterministic equation caused difficulties. It may therefore be helpful to emphasize three caveats. 

The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. 

This section is mainly concerned with the formal theory of relaxation processes, so it will be supposed that the molecules A undergo transitions among a set of states due to collisions with X, that the binary collision conditions are satisfied, and that over this set of states the transition probability is complete and satisfies detailed balance. An integral written without indicated limits will be understood to extend over all the states in question. 

The form of Pst (w ) in (4.49) together with (4.50) clearly comprises (4.44) but extends it, as a first approximation, to the case without detailed balance. 

Although the balance feature, o-rings, and the cartridge concept were discussed in detail in the previous chapter, here is a brief review why this seal design will give your pumps their best chance for extended leak-free service with reduced maintenance costs. 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

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