Asymptotics for the wiener sausage

Donsker, M. D.; Varadhan, S. R. S.

doi:10.1002/cpa.3160280406

Cited by 506 publications

(415 citation statements)

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“…Furthermore, if the diffusion constants of the A and B particles are the same, one can also view ρ A (t) as the fraction of particles that have not met any other particles. Thus the reaction A+ B → ∅ in the limit of a low density of A particles has been discussed under the guises of uninfected walkers [17] in which random walkers infect each other on contact, diffusion in the presence of traps [16,18] in which the B particles are considered as traps for the A particles, and predator-prey models [19] in which one asks for the survival of a prey (the A particle) being 'chased' by diffusing predators (the B particles). To avoid confusion, we shall adopt only the trapping terminology in our discussion.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

2003

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“…Convincing arguments for the validity of Lifshitz' conjecture were given by Friedberg and Luttinger [18], [32]. Its rigorous proof [15], [36], [37] relies on Donsker's and Varadhan's involved large-deviation results [15], [14,Section 4.3] about the long-time asymptotics of certain Wiener integrals.…”

Section: Introductionmentioning

confidence: 99%

The fate of lifshits tails in magnetic fields

et al. 1995

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We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.

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“…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%

“…which 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.…”

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confidence: 99%

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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Asymptotics for the wiener sausage

Cited by 506 publications

References 5 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

The fate of lifshits tails in magnetic fields

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

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