Smoluchowski flux and lamb-lion problems for random walks and Lévy flights with a constant drift. (21st August 2019) - Majumdar, Satya N; Mounaix, Philippe; Schehr, Grégory
Abstract: We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index ) in the presence of a constant drift c . Initially these random walkers are uniformly distributed over the positive real line with a density . 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 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 of a single random walk or Lévy flight after n steps. This allows us to obtain the precise asymptotic behavior of both and S c ( n ) analytically for large n in the two problems, for any value of and . In particular, for c > 0 and, we show that vanishes as, where is a -dependent positive constant, in contrast with the case of standard random walks (i.e. with ) for which . Our analytical results are confirmed by numerical simulations.