Master equations distributions

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.).

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]... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

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. 

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... 

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

According to the second law, (W) = Q) > 0, which implies that the average switching field is positive, (// ) > 0 (as expected due to the time lag between the reversal of the field and the reversal of the dipole). The work distribution is just given by the switching field distribution p H ). This is a quantity easy to compute. The probability that the dipole is in the down state at field H satisfies a master equation that only includes the death process,... 

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

The expansion method described above enables one to compute the spectral density of fluctuations in successive orders of O 1, provided the Master Equation is known.14 In the linear case, however, it was sufficient to know the macroscopic equation and the equilibrium distribution, as... 

On the other hand, (1.4) may be interpreted as to mean that any solution with initial condition P(y, t0) = 5(y — y0) is identical with the transition probability Pi i y> 11y0, to)- In fact, the master equation derived in the next chapter for Markov processes is of this type. Yet it cannot guarantee Markovian character, inasmuch as it still does not say anything about the higher distribution functions. ... 

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. 

V(t) is non-decreasing since U + V = const. Hence the corollary if initially (0) 0 for all n, then 4> (t) 0 for all t > 0. If the master equation would not have this property of conserving positivity, it could not be the correct equation for the evolution of a probability distribution. 

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... 

Exercise. Find the stationary distribution for the radioactive decay process described by the master equation (V.1.7). 

Exercise. In the preceding model let V2 and N go to infinity with finite density N/V2 = p. Write the master equation and show that the stationary solution is a Poisson distribution. (Could this have been known a priori ) What does detailed balance tell us in this case ... 

Exercise. A long-lived radioactive substance A decays into B through two shortlived intermediaries. A -> X - Y - B. When the amount of A is supposed constant, the joint distribution of the numbers n, m of nuclei X, Y obeys the bivariate master equation... 

Show that the master equation can, in principle, be solved by means of an r-variable generating function. In particular, prove that all solutions tend to a stationary joint Poisson distribution.510... 

Exercise. An n-type semiconductor, not heavily doped, can be described by (9.1), with M > N. Take the appropriate limit M - oo, find the corresponding master equation and equilibrium distribution. 

These equations determine both moments, provided their initial values are known. Our aim was to solve the master equation with initial delta distribution (2.8) and we took x0 as initial value of the macroscopic part, see (3.2). It follows that the initial fluctuations vanish,... 

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). 

Subdivide the total volume Q into cells A and call nk the number of particles in cell X. The cells must be so small that inside each of them the above mentioned condition of homogeneity prevails. Let P( nk, t) be the joint probability distribution of all nk. At t + dt it will have changed because of two kinds of possible processes. Firstly, the nk inside each separate cell X may change by an event that creates or annihilates a particle. In the master equation for P( nk, t) this gives a corresponding term for each separate cell. 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

Exercise. The neutrons in a nuclear reactor behave as free particles until they are absorbed, scatter, or cause fission and thereby produce more neutrons. The master equation for the joint probability distribution of the occupation numbers nk of the phase space cells X is... 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

In equation (4.7) the initial time is no longer mentioned, and it therefore applies to all distributions, regardless of the time the shutter opened, provided that occurred more than tc earlier than t. This proviso means that (4.7) is not a master equation (see the Warning in V.l). Yet it may be approximately treated as such if p does not change much during tc. 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

In terms of the master equation for the Markov process the formal kinetics is nothing but the mean-field theory where the fluctuation terms like that on the r.h.s. of equation (2.2.43) are neglected. Strictly speaking, the macroscopic description, equation (2.1.2), were correct if the fluctuation terms vanished as V —> oo. In a general case the function P(N, t) does not satisfy the Poisson distribution [16, 27] in particular, °N (N> ... 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

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