Size Independence of Statistics for Boundary Collisions of Random Walks and Its Implications for Spin-Polarized Gases

A bounded random walk exhibits strong correlations between collisions with a boundary. For a one-dimensional walk, we obtain the full statistical distribution of the number of such collisions in a time $t$. In the large $t$ limit, the fluctuations in the number of collisions are found to be size independent (independent of the distance between boundaries). This occurs for any interboundary distance, from less to greater than the mean free path, and means that this boundary effect does not decay with increasing system size. As an application, we consider spin-polarized gases, such as 3-helium, in the three-dimensional diffusive regime. The above results mean that the depolarizing effect of rare magnetic impurities in the container walls is orders of magnitude larger than a Smoluchowski assumption (to neglect correlations) would imply. This could explain why depolarization is so sensitive to the container’s treatment with magnetic fields prior to its use.

Figure 1

Sketches of bounded random walks: (a) a diffusive walk and (b) a quasiballistic walk. In (a) the boundary collisions are clustered. In (b) the boundary collisions are anticlustered (predominantly equally spaced), because random changes of direction (highlighted with stars) are rare.

Figure 2

The evolution of the distribution of boundary collisions with time, for three different $\epsilon$, given by numerically inverting the Laplace transformed distribution in Eqs. (6, 7, 8). We plot the distribution of $N_{+}$, for a particle initially at the midpoint between the boundaries, $z_0=a=1/2$. To make the time evolution of the shape of the distribution clearly visible, we multiply the vertical scale by the width of the distribution, $V_t^{1/2}$, at each time $t$, where $V_t=\mathrm{var}[N_{+}(t)]$. The Gaussian is that given below Eq. (8) for $t/t_{\mathrm{trav}}=10$.

Figure 3

The random walk of a ${}^3\mathrm{He}$ atom due to collisions with other ${}^3\mathrm{He}$ atoms. On the left, collisions with the container walls are marked by ellipses. On the right, we show the effect of one such boundary scattering on the atom’s spin.