Smoluchowski flux and Lamb-Lion Problems for Random Walks and L\'evy Flights with a Constant Drift

Abstract

We consider non-interacting particles (or lions) performing one-dimensional random walks or L\'evy flights (with L\'evy index $1 < \mu \leq 2$) in the presence of a constant drift $c$. Initially these random walkers are uniformly distributed over the positive real line $z\geq 0$ 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 L\'evy 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 $1<\mu\leq 2$ 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.