91Ó°ÊÓ

The space outside a sphere of radius \(R\) is filled with a medium in which there are \(\rho\) independently diffusing particles per unit volume. Find for the rate at which these particles are absorbed into the sphere*) $$ J(t)=4 \pi R D\left(1+\frac{R}{\sqrt{\pi D t}}\right) \rho $$

For the random walk of VI.2 find that a walker who starts at \(m\) first reaches the left or the right wall with probabilities $$ \mathrm{n}_{L, m}=\frac{R-m}{R-L} \text { and } \pi_{R, m}=\frac{m-L}{R-L} $$ For \(m=0, L<0\) this answers the question: If two gamblers with initial capitals \(R\) ) and \(-L\) toss a coin until one is ruined, what are their chances?

For the random walk, \(\pi_{k, m}\) has been computed in (2.3). Find \(3_{R . m}\) and subsequently the conditional mean first-passage time*) $$ \tau_{R, m}=\frac{1}{6}(R-m)(R+m-2 L) \quad(L \leqslant m \leqslant R) $$ The equations we have found for \(\pi_{R, m}, \tau_{R, m}, 3_{R, m}\) can be solved explicitly. First take equation (2.2) for \(\pi_{R, m}\). To solve it set \(\pi_{R_{t m+1}}-\pi_{R, m}=\Delta_{m}\) so that the equation reduces to a one- step recursion: $$ g_{m} \Delta_{m}=r_{m} \Delta_{m-1} \quad(L+1 \leqslant m \leqslant R-1) $$ This can be solved to give $$ \Delta_{m}=\frac{r_{m}}{g_{m}} \frac{r_{m-1}}{g_{m-1}} \frac{r_{m-2}}{g_{m-2}} \cdots \frac{r_{L+1}}{g_{L+1}} \Delta_{L^{-}} $$ Note that \(\Delta_{L}=\pi_{R, L+1}\), but it is still undetermined. Subsequently $$ \pi_{k, m}=\sum_{n=1}^{m-1} \Delta_{g}=\pi_{R, L+1}+\sum_{u=L+1}^{m-1} \frac{r_{u} r_{u-1} r_{\mu-2} \cdots r_{L+1}}{g_{\mu} g_{\mu-1} g_{p-2} \cdots g_{L+1}} \pi_{R, L+1} $$ The condition that this must be compatible with \(\pi_{\mathbb{q}, \boldsymbol{R}}=1\) determines \(\pi_{R, L+1}\). Inserting the result one finally gets $$ \pi_{R, m}=\frac{1+\sum_{\mu=L+1}^{m-1} \frac{r_{p} r_{\mu-1} \cdots r_{L+1}}{g_{\mu} g_{\mu-1} \cdots g_{L+1}}}{1+\sum_{\mu-L+1}^{n-1} \frac{r_{u} r_{j-1} \cdots r_{L+1}}{g_{\mu} g_{\mu-1} \cdots g_{L+1}}} $$