Asymptotic Behavior of Densities in Diffusion-Dominated Annihilation Reactions

Bramson, Maury; Lebowitz, Joel L.

doi:10.1103/physrevlett.61.2397

Cited by 242 publications

(230 citation statements)

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“…We allow the particle and traps to have different diffusion constants, D ′ and D respectively. Our results take the forms originally derived by Bramson and Lebowitz [14], as expressed in Eq. The results are obtained by deriving upper and lower bounds for λ d , and showing these coincide for d ≤ 2.…”

Section: Discussion and Summarymentioning

confidence: 73%

“…In fact, since the exposition of the process as a model of monopole-antimonopole annihilation in the early universe nearly twenty years ago [12], only a few results are known exactly. Most notably, Bramson and Lebowitz [14] proved rigorously that, at large times, the density of the minority species (which we will take to be the A particles) behaves as…”

Section: Introductionmentioning

confidence: 99%

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Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

2003

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The problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d. We consider the case where the traps are initially randomly distributed in space, with uniform density ρ, and derive upper and lower bounds for the probability Q(t) (averaged over all particle and trap trajectories) that the particle survives up to time t. We show that, for 1 ≤ d ≤ 2, the bounds converge asymptotically to giveρ and D is the diffusion constant of the traps, and that Q(t) ∼ exp(−4πρDt/ ln t) for d = 2. For d > 2 bounds can still be derived, but they no longer converge for large t. For 1 ≤ d ≤ 2, these asymptotic form are independent of the diffusion constant of the particle. The results are compared with simulation results obtained using a new algorithm [V. Mehra and P. Grassberger, Phys. Rev. E 65, 050101 (2002)] which is described in detail. Deviations from the predicted asymptotic forms are found to be large even for very small values of Q(t), indicating slowly decaying corrections whose form is consistent with the bounds. We also present results in d = 1 for the case where the trap densities on either side of the particle are different. For this case we can still obtain exact bounds but they no longer converge.

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Section: Discussion and Summarymentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

2003

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“…It has been first shown [2] and subsequently proven [3,4] that as t → ∞ the mean density n(t) follows …”

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confidence: 99%

“…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] ln…”

mentioning

confidence: 99%

“…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 …”

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confidence: 99%

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

et al. 2002

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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 )...

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Subdiffusion‐Limited Reactions

Yuste¹,

Lindenberg²,

Ruiz-Lorenzo³

2008

Anomalous Transport

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IntroductionAnomalous diffusion processes are ubiquitous in nature [1][2][3][4][5][6]. Their occurrence is usually associated with complex systems that induce spatial and/or temporal correlations in the diffusion process. The signature of normal diffusion is the linear asymptotic dependence of the mean-square displacement of the diffusing entity (hereafter called "the particle") on time, r 2 ∼ t, t → ∞. The signature of anomalous diffusion is a nonlinear dependence on time. In particular, if the growth with time is sublinear, so that(13.1) the particle is said to be subdiffusive. (The process is superdiffusive when the limit goes to infinity.) In this chapter, we focus on the important class of subdiffusive processes for whichand where the (anomalous) diffusion exponent γ satisfies 0 < γ < 1. An interesting class of diffusive processes are so-called diffusion-limited reactions. These are processes in which diffusion is the dominant mixing mechanism and, furthermore, where the time for reactants to find one another is much longer than the time it takes for a reaction to occur following such an encounter. Therefore, in these systems diffusion is the key factor that determines the spatial distribution of reactants and the resultant reaction rate. Since diffusion is not a particularly effective mixing mechanism, diffusionlimited reactions often present extremely interesting spatial as well as temporal characteristics. In this context, it is especially appropriate to point to the pioneering work of Turing on pattern formation in reaction-diffusion systems [7]. Diffusion-limited reactions show up in a vast number of applications including not only chemical (see, e.g., [8]) but also biological (e.g., [9]), ecological (e.g., [10]) and economic processes (e.g., [11]) that have been studied over many decades.

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Asymptotic Behavior of Densities in Diffusion-Dominated Annihilation Reactions

Cited by 242 publications

References 23 publications

Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

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

Subdiffusion‐Limited Reactions

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