Trapping reactions with randomly moving traps: Exact asymptotic results for compact exploration

Abstract

In a recent Letter Bray and Blythe have shown that the survival probability PA(t) of an A particle diffusing with a diffusion coefficient DA in a 1D system with diffusive traps B is independent of DA in the asymptotic limit t → ∞ and coincides with the survival probability of an immobile target in the presence of diffusive traps. Here we show that this remarkable behavior has a more general range of validity and holds for systems of an arbitrary dimension d, integer or fractal, provided that the traps are "compactly exploring" the space, i.e. the "fractal" dimension dw of traps' trajectories is greater than d. For the marginal case when dw = d, as exemplified here by conventional diffusion in 2D systems, the decay form is determined up to a numerical factor in the characteristic decay time. 02.50Ey, Trapping A+B → B and recombination A+B → 0 reactions (TR and RR) involving randomly moving A and B particles which react "when they meet" at a certain distance b are ubiquitous in nature. A few stray examples include quenching of delocalized excitations, coagulation, recombination of radicals, charge carriers or defects, or biological processes related to population survival [1].In recent years there has been much interest in the long-time behavior of these processes, following a remarkable discovery [2,3,4,5,6,7,8,9,10] of many-particle effects, which induce essential departures from the conventionally expected behavior [1].A pronounced deviation from the text-book predictions was found for the diffusion-controlled RR in case when initially the particles of the A and B species are all distributed at random with strictly equal mean densities n 0 . It has been first shown [2] and subsequently proven [3,4] that as t → ∞ the mean density n(t) follows K S (τ ) being the d-dimensional Smoluchowski-type "constant", defined as the flux of diffusive particles through the surface of an immobile sphere of radius b.For the TR two situations were most thoroughly studied: the case when As diffuse while Bs are static, and the situation in which the As are immobile while Bs diffuse -the so-called target annihilation problem (TAP). In the case of static, randomly placed (with mean density ρ) traps the A particle survival probability P A (t) shows a non-trivial, fluctuation-induced behavior [3,4,5,6,7,8,9,10] lnwhich is intimately related to many fundamenal problems of statistical physics [3,4,5,6,7,8,9,10,11]. Survival probability P target (t) of an immobile target A of radius b in presence of point-like diffusive traps B (TAP) can be calculated exactly for any d (see Refs. [14] and [3,12,13]):where φ On contrary, the physically most important case of TR when both As and Bs diffuse was not solved exactly. It has been proven [4] that here P A (t) obeyswhich equation defines its time-dependence exactly. On the other hand, the factor λ d (D A , D B ) remained as yet an unknown function of the particles' diffusivities and d. Since the time-dependence of the function on the rhs of Eq.(4) follows precisely the behavior of t dτ K S (τ )...

Intermittent random walks for an optimal search strategy: one-dimensional case

“…An interesting extension of our model may be a situation in which the target itself moves randomly. It is known, however, that if this motion is diffusive (or sub-diffusive) in low dimensions, the long-time asymptotic form of the probability P N is exactly the same as when the target is immobile (see [22] and also [23,24]). Consequently, in low dimensions in the large-N limit, the search for a diffusive (or sub-diffusive) target by searchers performing intermittent random walks will proceed in exactly the same way as determined in this paper for the case of an immobile target.…”

Lattice theory of trapping reactions with mobile species

“…Recently, some very interesting and unexpected results have been established for trapping A + B → B reactions involving randomly moving species [25,26], which have resolved, at least in part, this problem. It has been shown that in one or two dimensions [25], or more generally in systems, in which the fractal dimension of the B particle trajectories is greater than the dimension d of the embedding space [26], (i.e.…”

Survival probability of a ballistic tracer particle in the presence of diffusing traps

“…For example, if the tracer particle is static, this problem is known as the target annihilation problem [7]. Of particular interest is the case when the tracer particle itself has a diffusive motion, a problem that was first studied by Bramson and Lebowitz [8] and has recently seen a flurry of activity [9][10][11][12][13]. It is, however, somewhat frustrating that, despite various new developments, this diffusive target annihilation problem has defied a direct exact solution.…”

“…(βR), (12) where R = | R|, φ is the angle between R and the z axis, P l (x) is the Legendre polynomial of degree l and K ν (x) is the modified Bessel function of index ν. The unknown coefficients b l are determined from the boundary condition P st (R = a) = 1.…”