Equation master

The physical sense of individual contributions into (2.3.37) using the joint density p2,i is illustrated in Fig. 2.20. 

The limitation in equation (2.3.36) that vectors rm+i and cannot coincide with f m, r m becomes unimportant in the continuous approximation containing integrals instead of cell sums. The above mentioned changes in the DFs due to recombination correspond exactly to those done by Dettmann [82]. In the case of the A-I-B - B reaction one must omit the last 

When dividing equation (2.3.42) by and ttiking into account equa- 

Equation (2.3.43) corresponds also to that in [82]. (iii) Particle motion. Diffusion is described by 

In the case of dynamical interaction the pair potentials U r), Umir) and UAB r) should be incorporated into equation (2.3.45). It could be done using the Smoluchowski equation [27, 83, 84] for a particle drift in the external potential W (r) and expressed in terms of single particle DF (or concentration of such non-interacting particles) 

Consider now the Hinshelwood model to describe a pressure-dependent unimolecular reaction  

J is the rotational quantum number for a given energy level and E is the energy. 

Starting with this model, the Master Equation has to be solved to express the rate constant. The master equation describes the differential equation of the distribution of the concentration as a function of all rotational quantum number. The time rate of change of the reactants is given by the following expression  

If the single-step transition probability only depends on the time interval f = f2 - fi, we can write more simply 

The Chapman-Kolmogorov equation can then be written in terms of transition probabilities as follows  

We define the transition probability per unit time W(.V3 x2) as follows  

(1 - Oot ) is the probability that no transition occurs during the interval t. The transition propensity ao is defined as 

Using Eq. 13.59 we write the differential form of the Chapman-Kolmogorov equation, or the master equation of the stochastic process X(f), as follows  

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Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

An important example for the application of general first-order kinetics in gas-phase reactions is the master equation treatment of the fall-off range of themial unimolecular reactions to describe non-equilibrium effects in the weak collision limit when activation and deactivation cross sections (equation (A3.4.125)) are to be retained in detail [ ]. 

Quack M 1979 Master equations for photochemistry with intense infrared iight Ber. Bunsenges. Phys. Chem. 83 757-75... 

A3.13.3.1 THE MASTER EQUATION FOR COLLISIONAL RELAXATION REACTION PROCESSES... 

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 ] ... 

A3.13.3.2 THE MASTER EQUATION FOR COLLISIONAL AND RADIATIVE ENERGY REDISTRIBUTION UNDER CONDITIONS OF GENERALIZED FIRST-ORDER KINETICS... 

Figure A3.13.2. Illustration of the analysis of the master equation in temis of its eigenvalues and example of IR-multiphoton excitation. The dashed lines give the long time straight line luniting behaviour. The fiill line to the right-hand side is for v = F (t) with a straight line of slope The intercept of the...

The master equation treatment of energy transfer in even fairly complex reaction systems is now well established and fairly standard [ ]. However, the rate coefficients kjj or the individual energy transfer processes must be established and we shall discuss some aspects of this matter in tire following section. 

Figure A3.13.14. Illustration of the quantum evolution (pomts) and Pauli master equation evolution (lines) in quantum level structures with two levels (and 59 states each, left-hand side) and tln-ee levels (and 39 states each, right-hand side) corresponding to a model of the energy shell IVR (liorizontal transition in figure...

A 3.13.6 STATISTICAL MECHANICAL MASTER EQUATION TREATMENT OF INTRAMOLECULAR ENERGY REDISTRIBUTION IN REACTIVE MOLECULES... 

Figure A3.13.15. Master equation model for IVR in highly excited The left-hand side shows the quantum levels of the reactive CC oscillator. The right-hand side shows the levels with a high density of states from the remaining 17 vibrational (and torsional) degrees of freedom (from [38]).

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.).

Oppenheim I, Shuler K E and Weiss G H 1977 Stochastic Processes in Chemicai Physics, The Master Equation (Cambridge, MA MIT Press)... 

Troe J 1977 Theory of thermal unimolecular reactions at low pressures. I. Solutions of the master equation J. Chem. Phys. 66 4745-57... 

Venkatesh P K, Dean A M, Cohen M H and Carr R W 1999 Master equation analysis of intermolecular energy transfer in multiple-well, multiple-channel unimolecular reactions. II. Numerical methods and application to the mechanism of the C. + O2 reaction J. Chem. Phys. Ill 8313... 

Tabor M, Levine R D, Ben-Shaul A and Steinfeld J I 1979 Microscopic and macroscopic analysis of non-linear master equations vibrational relaxation of diatomic molecules Mol. Phys. 37 141-58... 

Levitt M H and Di Bari L 1994 The homogeneous master equation and the manipulation of relaxation networks Bull. Magn. Reson. 16 94-114... 

C3.3.5.1 MASTER EQUATION ANALYSIS OF UNIMOLECULAR REACTION DYNAMICS... 

Knowledge of tire pairwise energy trairsfer rates fonrrs a basis for finding tire average rate of energy trairsfer in air ensemble of molecules. To tlris end, a system of master equations is commonly employed [15,16 aird 17]. Then, tire probability, to find excitation on molecule cair be calculated as ... 

Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

Master equation methods are not tire only option for calculating tire kinetics of energy transfer and analytic approaches in general have certain drawbacks in not reflecting, for example, certain statistical aspects of coupled systems. Alternative approaches to tire calculation of energy migration dynamics in molecular ensembles are Monte Carlo calculations [18,19 and 20] and probability matrix iteration [21, 22], amongst otliers. 

Paillotin G, Geaointov N E and Breton J 1983 A master equation theory of fluoresoenoe induotion, photoohemioal yield,... 

Jarzynski, 1997a] Jarzynski, C. Equilibrium free-energy differences from nonequilibrium measurements A master equation approach. Phys. Rev. E. 56 (1997a) 5018-5035... 

The relaxation of a system to equilibrium can be modeled using a master equation... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

Such globed master equations have recently gained attention in the study of protein folding. See for example C. De Dominicis, H. Orland and F. Lainee, J. Phys. Lett., 46 L463, 1985 E.I. Shakhnovich and A. M. Gutin, Eurphys. Lett., 9 569, 1989 J. G. Saven, J. Wang and P. G. Wolynes, J. Chem. Phys., 101 11037, 1994. 

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

Kinetic studies such as these use the master equation to follow the flow of probability between the states of the model. This equation is a basic loss-gain equation that describes the time evolution of the probability pi(t) for finding the system in state i [24]. The basic form of this equation is... 

Solving the master equation for the minimally frustrated random energy model showed that the kinetics depend on the connectivity [23]. Eor the globally connected model it was found that the resulting kinetics vary as a function of the energy gap between the folded and unfolded states and the roughness of the energy landscape. The model... 

Czerminski and Elber [64], who generated an almost complete map of the minima and barriers of an alanine tetrapeptide in vacuum. Using the master equation approach they were able to smdy aspects of this system s kinetics, which involve the crossing of barriers of different heights. 

Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

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