Smoluchowski flux and Lamb-Lion Problems for Random Walks and Lévy Flights with a Constant Drift

Abstract

We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index $1<\mu \le 2$) in the presence of a constant drift $c$. Initially these random walkers are uniformly distributed over the positive real line $z\ge 0$ with a density ${\rho }_0$. At the origin $z=0$ there is an immobile absorbing trap (or a lamb), such that when a particle crosses the origin, it gets absorbed there. Our main focus is on (i) the flux of particles ${\Phi }_c(n)$ out of the system (the "Smoluchowski problem") and (ii) the survival probability $S_c(n)$ of the trap or lamb (the "lamb-lion problem") until step $n$. We show that both observables can be expressed in terms of the average maximum $\mathbb{E}[M_c(n)]$ of a single random walk or Lévy flight after $n$ steps. This allows us to obtain the precise asymptotic behavior of both ${\Phi }_c(n)$ and $S_c(n)$ analytically for large $n$ in the two problems, for any value of $1<\mu \le 2$ and $c\in \mathbb{R}$. In particular, for $c>0$, we show the rather counterintuitive result that for $1<\mu <2$, $S_{c>0}(n\to \infty )$ vanishes as $S_{c>0}(n\to \infty )\approx \exp \left(-\lambda n^{2-\mu }\right)$, where $\lambda$ is a $\mu$-dependent positive constant, while for standard random walks (i.e., with $\mu =2$), $S_{c>0}(n\to \infty )\to K_{RW}>0$, as expected. Our analytical results are confirmed by numerical simulations.

28 pages, 4 figures

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