Statistical Models of Chemical Kinetics in Liquids

Abstract

Statistical methods are used increasingly in theoretical chemistry. Applications range from the use of stochastic relaxation techniques in determining minimum energy molecular configurations to dynamical Monte Carlo simulations of molecular motion in liquids. This paper focuses on diffusion-controlled reactions in radiation chemistry. Here the interest is in describing the evolution of isolated clusters, containing a few chemically active particles, resulting from the passage of ionising radiation through a liquid. The subsequent chemistry is determined by the rate at which these particles can encounter each other, pairwise, in the course of their random motion. Classically the trajectories of the particles are described as sample paths of a continuous stochastic process, which in many cases can be assumed to be a diffusion. We have devised an approximate theory based on the approximation that pair distances evolve independently. The effect of this geometric distortion has been studied for small systems by simulation. The extension of this work to a wider class of realistic chemical processes poses many problems. Analytical progress is difficult and methods of simulating large spatial systems are not well developed. Problems which remain to be solved relate to the random geometry of the cluster as its constituent particles diffuse, the development of good approximations to first passage time distributions for diffusions with inhomogeneous drift and methods for the analysis of random motion in a liquid on the basis of detailed non-Markovian models of molecular movement.

Probability Approximations via the Poisson Clumping Heuristic

“…Berman (1983b) Section 6 treats a special case of Example 18; Baxendale (1984) gives the associated integral test. Clifford et al (1987) discuss some "near miss" models in the spirit of Example 19 and give references to the chemistry literature.…”

Minima of Independent Bessel Processes and of Distances Between Brownian Particles

“…Hence, if a reaction between two particles causes them to become inert, so they will not react with other particles they meet, this factor is irrelevant to the number of reactions up to time tJ2\ the same applies if two particles react by annihiliation. So such a reaction (A + A -> 0) is 'second order' (see [4,Section 1]) in the sense that in this limiting regime, the mean number of reactions up to a fixed time t x /2 is proportional to the square of the initial concentration of molecules of type A.…”

“…We shall also discuss the case of m(m-1)/2 non-independent Bes(d) processes obtained by taking the pairwise distances between m independent particles performing Brownian motion in U d . This is motivated by a connection with chemical kinetics (see Clifford, Green and Pilling [4]). If the presence of a chemical reaction between two particles depends on their being close, then the minimum interparticle distance tells us whether a reaction is taking place.…”

“…On the other hand, we could assume that the particles are initially scattered over U d in 'geminate pairs' (see [4,Section 2]), that is, m/2 pairs of particles are independently scattered over U d , in such a way that interparticle distances are much greater between pairs than within pairs. If the particles then proceed to perform independent Brownian motions, then over a short time scale, the pairs will evolve independently.…”

Computer algebra in probability and statistics

“…) We shall concentrate on itovsn3 for the remainder of this section, to provide a further example of the use of computer algebra in a structural manner. It first saw the light of day in response to results from stochastic chemistry (CLIFFORD and GREEN, 198.5; CLIFFORD, GREEN and PILLING, 1987;KENDALL, 1989, and was developed to deal with problems in the statistical theory of shape, KENDALL (1988), KENDALL (1990a). A moderately extensive development is available, and more details are to be found in KENDALL (1990b), KENDALL (1991a), KENDALL (1991b).…”