The master equation

The master equation will be given. Let (/) be a random variable denoting the quantity of X Dj, and are the (deterministic) initial conditions for X, Y and Z and p and X are the forward and backward rate constants. Then for 

The most fundamental description of the dynamics of the socio-configuration (3.5) including its fluctuations is given by a master equation for its probability distribution. 

In other words, (3.9) is the probabihty distribution Pn in t + r) with w = (no + k) starting from an initial distribution 

Obviously this initial distribution is concentrated at the point n = no. The Markov assumption is now made that the conditional probability does not depend on the motion before time t which led to the initial configuration no. The expression (3.9) is zero by definition if no + or no contain negative numbers n ai + kai or n ah these are invalid configurations. 

Since one of the (valid) configurations n = Hq + k will always be attained at time t + r when starting from any arbitrary (valid) configuration no, 

Further, since any probability p (n + r) is generated by transitions from all configurations n + A occupied with probabilities p n + k t) at time t, it follows by definition of the conditional probability that 

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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 [ ]. 

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

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 A3.13.16. 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... 

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. 

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. 

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

For a macroscopic variable A Sj), the time evolution of the expectation value, Eq. (11), is obtained by the master equation explicitly as... 

Mean Field Approximation as a first order approximation, we will ignore all correlations between values at different sites and parameterize configurations purely in terms of the average density at time t p. The time evolution of p under an arbitrary rule [Pg.73]

However, the transition rates down and up are equal, as in Eq. (4.19), only in the high-temperature limit. In general the master equations are... 

RNApolymerase molecules are involved in the process. If the system is well stirred so that spatial degrees of freedom play no role, birth-death master equation approaches have been used to describe such reacting systems [33, 34]. The master equation can be simulated efficiently using Gillespie s algorithm [35]. However, if spatial degrees of freedom must be taken into account, then the construction of algorithms is still a matter of active research [36-38]. 

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... 

We may also introduce the transition probability per unit time at fixed values of the coordinates of slower subsystems, Wlf(q9 Q) and WfXq, < ), and consider the master equations for the corresponding probability densities RXq, Q) and Rf(q, Q), etc. 

The form of the operators of evolution involved in these equations depends on the way in which they are described. The solution of the master equations enables us, in principle, to find the average rate of transition for both small and large values of the transition probabilities Wlfi Wlf and Wfl, Wfl. 

The master equation approach considers the state of a spur at a given time to be composed of N. particles of species i. While N is a random variable with given upper and lower limits, transitions between states are mediated by binary reaction rates, which may be obtained from bimolecular diffusion theory (Clifford et al, 1987a,b Green et al., 1989a,b, 1991 Pimblott et al., 1991). For a 1-radi-cal spur initially with Ng radicals, the probability PN that it will contain N radicals at time t satisfies the master equation (Clifford et al., 1982a)... 

The master equation methodology can be readily generalized to multiradical spurs, but it is not easy to include the reactions of reactive products (Green et al, 1989 Pimblott and Green, 1995). This approach is therefore limited to spur reactions where the reaction scheme is relatively simple. 

We have already commented that the master equation method is not suitable, at present, to handle reactive products because, inter alia, the dimensionality of the problem increases with reactions of products. There is no difficulty, in principle, to including reactive products in Monte Carlo simulation, since the time of reaction and the positions of the products can be recorded. In practice, however, this requires a greatly expanded computational effort, which is discouraging. 

Clifford et ah (1987a,b) considered acid spurs (primary radicals H and OH) and computed the evolution of radical and molecular products by the master equation (ME) and IRT methods. Reasonable values were assumed for initial yields, diffusion constants, and rate constants, and a distribution of spur size was included. To be consistent with experimental yields at 100 ns, however, they found it necessary that the spur radius be small—for example, the radius of H distribution (standard deviation in a gaussian distribution) for a spur of one dissociation was only in the 0.4—0.75 nm range. Since in acid spurs H atoms inherit the distribution of eh, this is considered too low. This preliminary finding has later been revised in favor of spurs of much greater radius. 

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

In Eqs. (II. 1)—(II.4) we have assumed that there is only one system oscillator. In the case where there exists more than one oscillator mode, in addition to the processes of vibrational relaxation directly into the heat bath, there are the so-called cascade processes in which the highest-frequency system mode relaxes into the lower-frequency system modes with the excess energy relaxed into the heat bath. These cascade processes can often be very fast. The master equations of these complicated vibrational relaxation processes can be derived in a straightforward manner. 

The master equation affects the evolution of the distribution function of all the velocities and is written ... 

Stecki and Taylor also remark that the master equation (40) (where one neglects the destruction fragment) can be expanded in a series about t = 0 and can be then put in the form ... 

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