  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 $n$-th term of the series is proportional to the mean value of the
scalar product $\rb_1 \cdot \rb_{n}$, where $\rb_i$ is the displacement of the
molecule between the $(i-1)$-th and $i$-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.
