Stochastic theory approximation

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

In the stochastic theory of branching processes the reactivity of the functional groups is assumed to be independent of the size of the copolymer. In addition, cyclization is postulated not to occur in the sol fraction, so that all reactions in the sol fraction are intermolecular. Bonds once formed are assumed to remain stable, so that no randomization reactions such as trans-esterification are incorporated. In our opinion this model is only approximate because of the necessary simplifying assumptions. The numbers obtained will be of limited value in an absolute sense, but very useful to show patterns, sensitivities and trends. 

Mann has proposed a stochastic theory based on a two-dimensional network of interconnecting pores with varying radii to explain hysteresis and entrapment in porosimetry. By assuming that filling of some of the larger radii pores is delayed until surrounding smaller pores are filled, a mechanism similar to the filling of ink-bottle pores, Mann s calculated porosimetry curves often approximate those from actual samples. 

The time required for a solute to flow past a given particle of stationary phase whose diameter is 5 pm is (5 pm)/(2.4 mm/s) = 2.1 ms. The stochastic theory predicts that the fraction of time that a molecule in the mobile phase will travel less than distance d is 1 — e <,/T" = 1 — e- 2-1 msW(3-5ms) = 0.55. That is, approximately half of the time, a solute molecule does not travel as far as the next particle of stationary phase before becoming adsorbed again to the same particle from which it just desorbed. If we lined up spherical particles of stationary phase, it would take 30 000 particles to cover the 15-cm length of the column. Each solute molecule binds —17 000 times as it transits the column, and half of those binding steps are to the same particle from which it just desorbed. 

The origin of each element composing the nuclear-ensemble approach can be traced back to decades ago, first with the works of Heller, Wilson and others in the 1980s, where absorption bands were computed based on molecular dynamics [9]. It is also influenced by the works of Skinner [10], which provided a useful link between Kubo s stochastic theory of the line shape [11] and molecular dynamics, and by the reflection principle [12], which approaches bound to continuum transitions from the nuclear-ensemble perspective. The intuitive character of the nuclear-ensemble approach has created a situation where although the method is frequently employed, there is no clear derivation of its formalism. This information gap makes difficult to understand the reasons for its limitations and to propose ways to improve the method. In this contribution, we derive equations for absorption cross sections and radiative decay rates based on the nuclear-ensemble method. The main approximations are made explicit, and improvements on the method are proposed, in particular ways to get rid of arbitrary parameters. 

We can also relate these two approximations through the stochastic theory of the line shape developed by Kubo [11] and applied to molecular line shapes by Saven and Skinner [10]. As shown by Kubo, the overlap function given by Eq. (13) is a general result for a Gaussian-dis-tributed random variable in a Markovian process [11]. In the limit of a very slow decay of the time-correlation function of this random variable, the overlap function reduces to Eq. (9) and the line has a Lorentzian shape. In the limit of a very fast decay of the time-correlation function, the overlap function reduces to Eq. (15) and the line has a Gaussian shape. Employing molecidar dynamics simulations of chromophores within non-polar fluids. 

A similar situation exists in the molecular-distribution function theory of liquids and one usually resorts to a superposition approximation. This amounts to assuming that, e.g., = 2 or something similar. It will be seen shortly that, contrary to unimolecular reactions, for bi-molecular reactions the stochastic mean is not the same as the classical kinetic expression for the concentration. 

In order to obtain numerical results it is necessary, on the one hand to know WVfl and Tv, on the other hand to solve (5.4). It is not our task to describe the various theories and approximations that concern the first problem, but it is fair to say that at best they lead to qualitative results. As a consequence there is a lot of leeway in choosing for WVfl and Tv expressions that facilitate the handling of the second problem. We mention two approaches, but it has to be admitted that they are more interesting as exercises in stochastic processes than useful for actually calculating unimolec-ular reaction rates. 

Stochastic approximations such as random walk or molecular chaos, which treat the motion as a succession of simple one- or two-body events, neglecting the correlations between these events implied by the overall deterministic dynamics. The analytical theory of gases, for example, is based on the molecular chaos assumption, i.e. the neglect of correlations betweeen consecutive collision partners of the same molecule. Another example is the random walk theory of diffusion in solids, which neglects the dynamical correlations between consecutive jumps of a diffusing lattice vacancy or interstitial. 

G. Adomian developed the decomposition method to solve the deterministic or stochastic differential equations.3 The solutions obtained are approximate and fast to converge, as shown by Cherrault.8 In general, satisfactory results can be obtained by using the first few terms of the approximate, series solution. According to Adomian s theory, his polynomials can approximate the... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

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