Generalized Master Equations

The fimdamental kinetic master equations for collisional energy redistribution follow the rules of the kinetic equations for all elementary reactions. Indeed an energy transfer process by inelastic collision, equation (A3.13.5). can be considered as a somewhat special reaction . The kinetic differential equations for these processes have been discussed in the general context of chapter A3.4 on gas kmetics. We discuss here some special aspects related to collisional energy transfer in reactive systems. The general master equation for relaxation and reaction is of the type [H, 12 and 13, 15, 25, 40, 4T ] ... 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

Remark. The properties of (2.4) for small 0 are very similar to those of a general master equation for large Q. Many methods and ideas apply to both. In fact, the distinction between both cases is often ignored. Yet, apart from their different physical meaning, there is an important mathematical difference as well. 

B. Long-Tailed Waiting Times Processes and the Generalized Master Equation... 

It has been shown that the FFPE (19) is equivalent to the generalized master equation [58]... 

Eq. (2.25) is often referred to as generalized master equation (GME). It should be noted that Eq. (2.20) or (2.25) describes the time evolution of an isolated system. 

From Eq. (3.34), one can obtain the generalized master equation (GME), which describes the dynamics of populations and coherences (or phases) of the systems. 

In this section, the general case of non-adiabatic electronic transition is considered. To describe the dynamical processes of the system, the generalized master equation (GME) shall be used [3,10-16] ... 

Since it is the dynamics of the system that is of interest, it would be convenient to preaverage over the environment variables and obtain an equation of motion for ps(t), the system component of the density matrix. Formal work of this kind [161,. .. 162] yields the so-called generalized master equation. Deriving the generalized f master equation, and extracting the various approximations utilized, goes well "astray of the central focus of this book. For this reason we just sketch the models id direct the reader to suitable review articles [161, 162] that provide an appropriate pview. 

TJf the correlation time of the environment is much shorter than the typical time scale for the variation of the system, then the generalized master equation is of the ten... 

As such, a variety of approximations to the generalized master equations have yielded equations that are used to model the effect of the environment on the system dynamics. 

THE CONTINUOUS-TIME RANDOM WALK VERSUS THE GENERALIZED MASTER EQUATION... 

III. The Generalized Master Equation and the Zwanzig Projection Method... 

III. THE GENERALIZED MASTER EQUATION, AND THE ZWANZIG PROJECTION METHOD... 

Equation (20) is the central result of the Zwanzig projection method, and it is one of the two theoretical tools under scrutiny in this chapter, the first being the Generalized Master Equation (GME), of which Eq. (20) is a remarkable example, and the second being the Continuous Time Random Walk (CTRW) [17]. It must be pointed out that to make Eq. (20) look like a master equation, it is necessary to make the third term on the right-hand side of it vanish. To do so, the easiest way is to make the following two assumptions ... 

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