The two-particle problem in comb-like structures

Abstract

Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comb-like structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comb-like architecture can be strongly different according to whether both reactants are moving (no matter their relative diffusivities) or only one, and according to the density of short-cuts among the branches.

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• Statistics of diffusive encounters with a small target: Three complementary approaches
• First encounters on Bethe Lattices and Cayley Trees
• Interplay between particle trapping and heterogeneity in anomalous diffusion
• First-encounter time of two diffusing particles in two- and three-dimensional confinement
• Random walks on uniform and non-uniform combs and brushes
• Mean first-encounter times of simultaneous random walkers with resetting on networks
• Displacement of transport processes on networked topologies
• Smart Walkers in Discrete Space