Exact Occupation Kinetics for One-Dimensional Arrays of Dumbbells

Abstract

By utilizing a branch‐dependent counting technique, exact relationships are developed which describe the ensemble average of the kinetics of occupation by dumbbells of finite one‐dimensional arrays of compartments. It is shown that, as the number of compartments per array tends to infinity, $\mathrm{〈\theta (t)〉}$ , the ensemble average of the fraction of an array which is occupied, is given by $$\text{〈θ(t)〉=1−}\exp \text{{−2[1−}\exp \mathrm{(-vt)]\}}$$ , where v is the striking frequency and t is time. By contrasting these results with the statistics of one‐dimensional arrays of dumbbells, it is demonstrated that it is inappropriate to employ classical statistical‐mechanical methods (e.g., the Bethe approximation) to treat the kinetic aspects of occupation where configurational correlation exists. (Here we define configurational correlation to be the situation in which the occupation of a compartment precludes the occupation of one of its nearest neighbors.)