Solution of the master equation

Solutions of the Master Equation.—In the low-pressure limit of a thermal unimolecular 

El Eo) non-dissociative states (dissociative states are rapidly depopulated by the fast intramolecular dissociation process). As is well known, the time evolution of the populations [A(i)] is given by a series of exponentially decaying terms which ctHTiespond to an initial rovibrational relaxation, a subsequent incubation period with overlap of vibrational rriaxation of upper levels and dissociation, and the final dissociation period with steady-state of all populations [A(i)]. Explicit solutions of the master equation for the dissociation of diatomic molecules have been extensively reviewed by H. O. Pritchard in Volume 1 of this series. Such 

Flgare 4 Initial period of the dissociation and recombination reaction 2H in He. Model calculation for 2000 K, 10 atm, and [H2]/[He] = 10. (Upper curve and left scale recombination. Lower curve and right scale dissociation) 

Figare 6 Dissociation, incubation, and vibrational relaxation rate constants of the reaction N2O N2 + 0 at low pressures in At. Relaxation and inadtation data from ref. 55, dissociation data from refs. SS—S8 

The second method permits analytical solutions to be obtained for a limited dass of transition probabilities, such as the exponential models (8) and (9). The nonequilibrium factors HE) approach 1 at energies Eo, marked deletion HE) 1 extends from Eo down to Eo minus a few RT. From tte solution for HE), the low-pressure rate constant (ito) follows directly from equatitm (15), and 

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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.).

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

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 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 transition probabilities W% C C) cannot be arbitrary but must guarantee that the equilibrium state P C) is a stationary solution of the master equation (5). The simplest way to impose such a condition is to model the microscopic dynamics as ergodic and reversible for a fixed value of X ... 

In equilibrium x (t) = x is constant in time. In this case, the stationary solution of the master equation is the equihbrium solution... 

One basic difficulty with the nonlinear equation arises from the following. Consider a physical situation where a source of particles is composed of many emitters, each emitting a particle at a time. If considered alone, each particle would be described by a localized wave /,- solution of the master equation. Now, what happens if, instead of emitting the particles one by one, the source emits many particles at the same time If the master equation were a linear equation, like the usual Schrodinger equation, the answer would be trivial. The general solution would be simply the sum of all particular solutions. 

Of course, there may be more than one. Each 0 is a time-independent solution of the master equation. When normalized it represents a stationary probability distribution of the system, provided its components are nonnegative. In the next section we shall show that this provision is satisfied. But first we shall distinguish some special forms of W. 

We first establish two lemmas. Let ( ) be any solution of the master equation, not necessarily positive or normalized. At a given time t the positive, negative and zero components are distinguished by using an index u, v, w, respectively ... 

Throughout we use the superscript e for the thermodynamic equilibrium, and s for any stationary, i.e., time-independent solution of the master equation. 

The present proof is more limited than the one in 3 because we have to assume beforehand that there is a stationary solution that is everywhere positive. For closed, isolated, physical systems one knows that that is so, and we therefore use here the symbol pi for that stationary solution of the master equation. Yet the proof also applies to other cases provided they have no transient states, but the proof does not require detailed balance of any other symmetry relation of the type (4.2). 

Suppose one has found all eigenvalues and eigenvectors obeying (7.1). The question then is whether (7.2) is complete, i.e., whether it is sufficient to represent all solutions of the master equations. In other words, is it possible to find for every initial distribution p(0) suitable constants cx such that... 

This equation has the same form as (3.7) for an isolated system the stationary solution of the master equation ps is identical with the thermodynamic equilibrium pe. 

When r(n) and g(n) are both linear in n it is usually impossible ) to give an explicit solution of the master equation other than the stationary solution. An approximate treatment is given in chapter VIII and a systematic approximation method will be developed in chapter X. We here merely list a few typical examples. 

To facilitate the discussion it is helpful to specify three of the numerous meanings of the word state . We shall call a site any value of the stochastic variable X or n. We shall call a macrostate any value of the macroscopic variable . A time-dependent macrostate is a solution of the macroscopic equation (X.3.1), a stationary macrostate is a solution of (X.3.3). We shall call a mesostate any probability distribution P. A time-dependent meso-state is a solution of the master equation, the stationary mesostate is the time-independent solution PS(X). 

On the other hand, it is clear from (3.4) that n = 0 is an absorbing site, so that the stationary solution can be only psn = 5n)0- All other solutions of the master equation tend towards it, i.e., with probability one the population will ultimately die out A moment s reflection resolves this paradox. The... 

Fluctuations in nonequilibrium systems have been studied mainly through two approaches the master equation approach16 and more recently the Ginsburg-Landau functional approach.17 In the master equation approach, the microscopic transition probabilities for chemical reactions and diffusion are taken to be given, and a master equation for the spatiotemporal variation of the probability distribution is obtained. Though the explicit solution of the master equation is difficult to obtain, some important general features could be deduced from it. One can show... 

If the solution of a deterministic reaction rate equation differs from the first moment corresponding to the solution of the master equation, it can generally be considered as a differently conditioned average of the same random variable.144... 

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... 

Direct solution of the master equation is impractical because of the huge number of equations needed to describe all possible states (combinations) even of relatively small-size systems. As one example, for a three-step linear pathway among 100 molecules, 104 such equations are needed. As another example, in biological simulation for the tumor suppressor p53, 211 states are estimated for the monomer and 244 for the tetramer (Rao et al., 2002). Instead of following all individual states, the MC method is used to follow the evolution of the system. For chemically reacting systems in a well-mixed environment, the foundations of stochastic simulation were laid down by Gillespie (1976, 1977). More... 

UNIMOL Calculation of Rate Coefficients for Unimolecular and Recombination Reactions. R. G. Gilbert, T. Jordan, and S. C. Smith, Department of Theoretical Chemistry, Sydney, NSW 2006, Australia, 1990. FORTRAN computer code for calculating the pressure and temperature dependence of unimolecular and recombination (association) rate coefficients. Theory based on RRKM and numerical solution of the master equation. See Theory of Unimolecular and Recombination Reactions, by R. G. Gilbert and S. C. Smith, Blackwell Scientific Publications, Oxford, 1990. 

The kinetics of chemical reactions on surfaces is described using a microscopic approach based on a master equation. This approach is essential to correctly include the effects of surface reconstruction and island formation on the overall rate of surface reactions. The solution of the master equation using Monte Carlo methods is discussed. The methods are applied to the oxidation of CO on a platinum single crystal surface. This system shows oscillatory behavior and spatio-temporal pattern formation in various forms. 

The usual approach to dynamic Monte Carlo simulations is not based on the master equation, but starts with the definition of some algorithm. This generally starts, not with the computation of a time, but with a selection of a site and a reaction that is to occur at that site. We will show here that this can be extended to a method that also leads to a solution of the master equation, which we call the random-selection method (RSM). [31]... 

Solution of the master equation has related dc to the average energy (AF) transferred per collision [1] by... 

The solution of the master equation can provide the time evolution of the occupation probabilities of all the minima, but it is not immediately obvious how to extract interfunnel rate constants from this information when there are many different pathways between the funnels. To proceed we consider a general scheme for equilibrium between two funnels A and B, i.e., A B with forward and reverse rate constants k+ and fe. The rate of change of the occupation probability of funnel A is accordingly... 

The analytic solution of the master equation decomposes the flow of probability into a series of exponentially decaying modes, each of which has a characteristic decay constant. It is instructive to look at how the contributions from these modes vary across the spectrum of time scales. From Eq. (1.41) mode j makes an important contribution to the probability evolution of minimum / if is large in magnitude. The mode... 

Cracial to the simulations presented here is the inclusion of surface reconstmction, together with correct time-dependence of the reactions. As such, the method provides an extension of earlier important computer simulations of CO oxidation on Pt surfaces " . A dynamic Monte Carlo method is used based on the solution of the master equation of the reaction system. The reaction system consists of a regular grid with periodic boundary conditions. The largest grid used in our simulations contained ca. eight million reaction sites. A short description of the model is presented in Fig. 3 and in Table I, that shows the parameters of the rate constants considered. 

Fig. 7 Concentration profile 0 over the site number i for different values of the particle-particle interaction parameter co as resulting from the solution of the master equations. From [72] with permission...

For nonequilibrium systems the probability of a fluctuation is given by the solution of the master equation. The absolute and relative fluctuations, at least for some nonequilibrium systems, are of similar magnitude to equilibrium systems. 

Figure 9.11. Time evolution of (a) nonequilibrium rate constant k 12(f)and (b) electronic population in the donor state Pi(t) for V= 120cm . The parameter Icijlt) was obtained from the molecular dynamics simulation data for the model back ET reaction in a rigid collinear triatomic molecule with equivalent donor and acceptor sites separated by a neutral spacer in a polar solvent. The parameter Pfi) is a result of solution of the master equation. (Reproduced from [62c] with permission. Copyright (1996) by the American Institute of Physics.)...

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