Stochastic master equation

In closing this section, we note that the stochastic master equation, Eq. (16), can be used to study the effect of boundary conditions on transport equations. If a(x, y, t) is sufficiently peaked as a function of x — y, that is if transitions occur from y to states in the near neighborhood of y, only, then the master equation can be approximated by a Fokker-Planck equation. The effects of the boundary on the master equation all appear in the properties of a(x, y, t). However, in the transition to the Fokker-Planck differential equation, these boundary effects appear as boundary conditions on the differential equation.7 These effects are prototypes for the study of how molecular boundary conditions imposed on the Liouville equation are reflected in the macroscopic boundary conditions imposed on the hydrodynamic equations. 

Stochastic analysis presents an alternative avenue for dealing with the inherently probabilistic and discontinuous microscopic events that underlie macroscopic phenomena. Many processes of chemical and physical interest can be described as random Markov processes.1,2 Unfortunately, solution of a stochastic master equation can present an extremely difficult mathematical challenge for systems of even modest complexity. In response to this difficulty, Gillespie3-5 developed an approach employing numerical Monte Carlo... 

Suppose the geometry of a compartmentalized system is described by a lattice of integral or fractal dimension of given size and shape, and characterized by N discrete lattice points (sites) embedded in a Euclidean space of dimension d = de and local connectivity or valency v. At time t = 0, assume that the diffusing coreactant A is positioned at a certain site j with unit probability. For f > 0 the probability distribution function p(f) governing the fate of the diffusing particle is determined by the stochastic master equation... 

In Section III the temporal behavior of diffusion-reaction processes occurring in or on compartmentalized systems of various geometries, as determined via solution of the stochastic master equation (4.3), is studied. Also, in Sections III-V, results are presented for the mean walklength (n). From the relation (4.7), and the structure of the solutions (4.6) to Eq. (4.3), the reciprocal of (n) may be understood as an effective first-order rate constant k for the process (4.2) or (n) itself as a measure of the characteristic relaxation time of the system it is, in effect, a signature of the long-time behavior of the system. 

Instead of mobilizing large scale Monte Carlo simulations of the visitation probability P as a function of walklength n, Pk can be evaluated (as a function of time) using the stochastic master equation [60]. Suppose at time f = 0 a random walker is positioned with unit probability at a site m in the interior of the lattice (away from the boundary of the system). For t > 0 this probability evolves among the lattice sites as determined by Eq. (4.3). An entropy-like quantity... 

The above approach, based on the solution of Eq. (4.3), is numerically superior and more accurate than one based on conventional Monte Carlo simulations. For comparison, Pitsianis et al. [59b] performed Monte Carlo simulations using 100,000 random walks on a Sierpinski gasket of 29,526 sites and obtained a value of 1.354 for dj (the exact value is 1.365). In the approach elaborated above, the value 1.367 was obtained by solving the stochastic master equation on a gasket of only 366 sites. 

In solving the stochastic master equation, two classes of initial conditions are specified. Consider first a coreactant diffusing in an ambient environment (e.g., a solution) and colliding with a colloidal particle (or cellular assembly) at some site k on its surface. From that moment t = 0) on, the coreactant is assumed to diffuse randomly on the surface (only) from site to site until it reacts irreversibly with a target molecule anchored at one site. 

Regarding the dependence of the reaction efficiency on the dimensionality of the compartmentalized system, the studies reported in Sections III.B.3 and III.B.4 on processes taking place on sets of fractal dimension showed that, consistent with the results found for spaces of integer dimension, the higher the dimensionality of the lattice, the more efficient the trapping process, ceteris paribus. Processes within layered diffusion spaces, which can be characterized using an approach based on the stochastic master equation (4.3), show a gradual transition in reaction efficiency from the behavior expected in c( = 2 to that in = 3 as the number k of layers increases from fe = 1 to k = 11. 

The steady-state concentration of KJHsCFg molecules corresponding to the ith energy interval ([ K HgCFsli) can be calculated via die steady-state stochastic master equation (Equations 16 and 17). Here... 

Stochastic methods simulate the dynamic changes that occur in the structure of the adlayer of catalytic surface and thus model the elementary surface kinetics [82-ioo] j jjg temporal changes of a system can be followed by solving the stochastic master equation which simulates the dynamic changes in the system as it moves from one state i) to another state (j). The master equation, which can written as... 

Equistability of a homogeneous stable stationary state on the upper branch of the hysteresis loop, labelled I in Fig. 11.1, with a homogeneous stable stationary state on the lower branch, labelled II, occurs at one value of the influx coefficient k within the loop. Say that point occms at the location of line A. The predictions of the stationary solution of the master stochastic master equation are (a) the minimmn of the bimodal stationary probability distribution is located on the separatrix, and (b) at equistability the probability of fluctuations P(c) obeys the condition... 

Further insight into the quantity D x) can be obtained by introducing the age T of a fluctuation state, that is the time interval between the last transition X 1 —> X and the moment of observation. The age t is determined by a succession of random events and hence is a random variable, and obeys a stochastic master equation ([4], p. 7273). Prom the stationary form of that equation we derive the relation... 

T)) is given by a zero mean Frisson white noise characterized by TX- W-) the stochastic master equation is meaningful for X-w. 

That is, we have recovered Eqs. (6.29H6.32). Recall that we previously obtained these equations from the stochastic (master equation) and deterministic (chemical kinetics) approaches, confirming once more the unity of the different existing formulations to studying the dynamics of chemical-reaction systems. 

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