Gaseous diffusion as a correlated random walk

The mean-square displacement per collision of a molecule immersed in a gas at equilibrium is given by its mean-square displacement between two consecutive collisions (mean-square free path) corrected by a prefactor in the form of a series. The $n\mathrm{th}$ term of the series is proportional to the mean value of the scalar product $r_1·r_n$, where $r_i$ is the displacement of the molecule between the $(i-1)$-th and $i\mathrm{th}$ collisions. Simple arguments are used to obtain approximate expressions for each term. The key finding is that the ratio of consecutive terms in the series closely approximates the so-called mean persistence ratio. Exact expressions for the terms in the series are considered and their ratios for several consecutive terms are calculated for the case of hard spheres, showing an excellent agreement with the mean persistence ratio. These theoretical results are confirmed by solving the Boltzmann equation by means of the direct simulation Monte Carlo method. By summing the series, the mean-square displacement and the diffusion coefficient can be determined using only two quantities: the mean-square free path and the mean persistence ratio. A simple and an improved expression for the diffusion coefficient $D$ are considered and compared with the so-called first and second Sonine approximations to $D$ as well as with computer simulations of the Boltzmann equation. It is found that the improved diffusion coefficient shows very good agreement with simulation results over all intruder and molecule mass ranges. When the intruder mass is smaller than that of the gas molecules, the improved formula even outperforms the first Sonine approximation.