Spatial Statistics and Digital Image Analysis (1991) / Chapter Skim
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9. Stochastic Modeling in Physical Chemistry
9. Stochastic Modeling in Physical Chemistry
Pages 159-180
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From page 159... ...
From a statistician's perspective, a familiar thread within the densely woven fabric of physical theory, mathematical development, and experimental technique is the constant concern with finding simple and expedient models, frequently of a stochastic nature (van Kampen, 1981; Wax, 1954~. Thus, although physical theory may in principle provide a complete microscopic description of the problem at hand, in practice the intractability of the mathematical development prevents useful predictions from being made.
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From page 160... ...
Chemical reactions occur when reactive species approach each other as a result of molecular motion. As in the case of Brownian motion it is natural to Took for a stochastic approximation to the reaction process, but here we must track the motion of a large number of atomically small reactive species.
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From page 161... ...
. For the latter case, where the Smoluchowski theory might be expected to break down because the intermolecular distances are highly correlated, the survival probability is related to the volume of the Wiener sausage, swept out by the diffusing molecule in the course of its trajectory.
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From page 162... ...
9.2.1 Racliation Spurs Diffusion controlled reactions are the fastest reactions that occur in solution. Experimental observations of the rate of reaction contain information about the initial spatial distribution of the reactive species.
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From page 163... ...
The two spatially dependent concentrations (which are identical) obey macroscopic continuum equations of the form d Itch = d FACE = DA V CA—kA A CA—kA B CA CB DB V2 CB—kBB CB—kAB CA CB where the first terms on the right-hand sides represent diffusive spreading of the concentration profiles, and the remaining terms represent local depletion by reaction; the rate coefficients k are given by Smoluchowski's theory (cf.
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From page 164... ...
The continuum approximation always predicts a ratio of 1:2:1 since it corresponds to the case of infinitely many particles. The independent pairs approximation can be used to provide a stochastic theory of spur kinetics.
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From page 165... ...
Initial configuration spherical Gaussian. Monte CarIo simulation (o o o)
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From page 166... ...
Monte CarIo simulation (o 0 o) and simulation using the {P approximation (—)
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From page 167... ...
What we would like to do is to find stochastic rules that indicate where the molecules will be, and how fast they will be traveling, as time goes on. For example, in radiation chemistry, where the effect of radiation is to create reactive species distributed throughout the liquid, we are interested in the time taken for such species to encounter and react with each other.
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From page 168... ...
where p and r are the vectors of momenta and positions of the molecules. The kinetic energy is given by N (p)
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From page 169... ...
They illustrated their method by simulations using both hard sphere and square well potentials. Their paper is the first example of molecular dynamics simulation.
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From page 170... ...
The first is to build a dynamic description of molecular motion that will move through the state space according to acceptable physical principles. This is the approach of molecular dynamics.
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From page 171... ...
they notice that the shape of the curve closely follows that of the velocity autocorrelation function. Stratifying the molecular trajectories by initial kinetic energy, I,ynden-Bell et al.
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From page 172... ...
To throw further light on the causes of the nearly uniform distribution of R at the time lag for which the velocity autocorrelation function is zero,
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From page 173... ...
The continuous lines refer to the 7.2~o of particles with the highest kinetic energy, and the dashed lines to the 12~o with the lowest energy. Reprinted, by permission, from I,ynden-Bell et al.
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From page 174... ...
9.4.1 Physical Properties of Matter in Equilibrium The properties of bulk matter are reasonably well understood on a quaTitative level, and if the substance is made up of simple molecules or atoms, such as the liquid inert gases, numerical simulations of small systems, based on the known intermolecular forces and involving of the order of a thousan(1 molecules, are quite successful in reproducing the observed properties. Molecular dynamics simulations become more complicated when the molecules are neither spherical nor rigid.
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From page 175... ...
Attention has turned in recent years to the study of interfaces between different phases of matter. Monte Cario and molecular dynamics simulations have concentrated on bubbles and droplets, and a number of experimental techniques have been devised for studying the gas-solid, gas-liquid, and solidliquid interfaces.
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From page 176... ...
9.4.4 Statistical Mechanics The fundamental molecular properties and their interactions as measured by spectroscopists are the data required by statistical mechanics for the description of bulk matter. Statistical mechanics is the central unifying theory of physical chemistry as it relates the properties of isolated molecules with the bulk.
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From page 177... ...
The initiation and establishment of the critical conditions for detonation have been subject to little detailed realistic investigation; although of course there are obvious analogies with the stochastic theories of spatial epidemics. The necessary cooperative interaction between small numbers of initiating sites suggests
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From page 178... ...
A significant proportion of the chemical reaction following radiation occurs on a short time scale, when the localized distribution has not yet relaxed by diffusion. The chemical process can be treated successively using stochastic methods.
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From page 179... ...
Clifford, Stochastic models of diffusion controlled ionic reactions in radiation induced spurs: (ii) Low permittivity solvents, ~7.
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From page 180... ...
[18] RueDe, D., Statistical Mechanics: Rigorous Results, W
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