Vacancy Annihilation for One-Dimensional Dumbbell Kinetics

Expressions are developed which describe the ensemble average of the annihilation of groups of contiguous vacant compartments when spatially random attempts are made to place dumbbells on a linear array of N compartments. It is shown that in the limit, as the number of compartments tends to infinity, <θ_p(t)>, the ensemble average of the fraction of the compartments which is composed of p contiguous vacant compartments, is given by $${\mathrm{〈\theta }}_p{\mathrm{(t)〉=pe}}^{\text{−2}}\left(\text{1−}{\displaystyle \sum _{\text{n=2}}^{\infty }}{C}_n{e}^{\mathrm{-nvt}}\right)e^{\text{−(p−1)vt}}$$ , where v is the striking frequency of the dumbbells, t is time, and the C_n's are appropriately defined coefficients.