Lattice theory of trapping reactions with mobile species

Abstract

We present a stochastic lattice theory describing the kinetic behavior of trapping reactions A + B → B, in which both the A and B particles perform an independent stochastic motion on a regular hypercubic lattice. Upon an encounter of an A particle with any of the B particles, A is annihilated with a finite probability; finite reaction rate is taken into account by introducing a set of two-state random variables -"gates", imposed on each B particle, such that an open (closed) gate corresponds to a reactive (passive) state. We evaluate here a formal expression describing the time evolution of the A particle survival probability, which generalizes our previous results. We prove that for quite a general class of random motion of the species involved in the reaction process, for infinite or finite number of traps, and for any time t, the A particle survival probability is always larger in case when A stays immobile, than in situations when it moves.

Exact eigenvalue spectrum of a class of fractal scale-free networks

“…Then, interesting questions arise: What happens if the target is not immobile? Would it be possible in this case to use the so-called Pascal Principle [13,52,53] to obtain a rigorous upper bound? Although the questions go beyond the scope of the present paper, they are worth studying in the future.…”

Symmetric rearrangements around infinity with applications to Lévy processes

“…This type of result, where the optimal trajectory is the constant trajectory, has been called the Pascal principle in the physics literature. For the lattice version of the trapping problem, the Pascal principle was established in [MOBC04], see also [DGRS10, Corollary 2.1]. In the continuum setting above where the spherical hard traps follow independent Brownian motions, it was first established in dimension 1 in [PSSS11], assuming that f is continuous.…”

“…[BB02,MOBC04] and the references therein). It has also been studied as a detection problem in a mobile communication network (see e.g.…”

Subdiffusivity of a Random Walk Among a Poisson System of Moving Traps on ℤ $\mathbb {Z}$

“…However, when the traps are mobile, surprisingly little is known. In a previous work [DGRS12] (see also [PSSS13]), the long-time asymptotics of the annealed and quenched survival probabilities were identified in all dimensions, extending earlier work in the physics literature [MOBC03,MOBC04]. The goal of the current work is to investigate the path behavior of the one-dimensional random walk conditioned on survival up to time t in the annealed setting, which is the first result of this type to our best knowledge.…”