Lattice kinetics of diffusion-limited coalescence and annihilation with sources

Abad, E.; Masser, Thomas; ben‐Avraham, Daniel

doi:10.1088/0305-4470/35/7/301

Cited by 5 publications

(14 citation statements)

References 37 publications

(91 reference statements)

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“…The resulting equation is similar to ones occurring in coalescence models [15][16][17] . From this we expect that a large time and small ǫ limit discussed subsequently is equivalent to the model in Ref.…”

Section: B Simulationssupporting

confidence: 56%

“…From this we expect that a large time and small ǫ limit discussed subsequently is equivalent to the model in Ref. 15 . Our approach, which exploits the generating function method, becomes equivalent, in the scaling limit, though in a conjugate space, to continuum approximations used in the coalescence studies of Refs.…”

Section: B Simulationsmentioning

confidence: 59%

“…II and III, respectively. Our models share some aspects with diffusion limited coalescence models [15][16][17] and with fragmentationaggregation models 13,14 . They are similarly described in terms of cluster or interval probabilities, and like the fragmentation models they are amenable to analytic investigation based on an independent cluster approximation (the independent interval approximation to the joint cluster length probability occurring in the master equation).…”

Section: Introductionmentioning

confidence: 71%

“…Our approach, which exploits the generating function method, becomes equivalent, in the scaling limit, though in a conjugate space, to continuum approximations used in the coalescence studies of Refs. 15 and 17 . Now the generating function (Eq.…”

Section: B Simulationsmentioning

confidence: 99%

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Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment, and deposition

Reis

Stinchcombe

2004

Phys. Rev. E

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Monolayer cluster growth in far-from-equilibrium systems is investigated by applying simulation and analytic techniques to minimal hard core particle (exclusion) models. The first model (I), for post-deposition coarsening dynamics, contains mechanisms of diffusion, attachment, and slow activated detachment (at rate ǫ ≪ 1) of particles on a line. Simulation shows three successive regimes of cluster growth: fast attachment of isolated particles; detachment allowing further (ǫt)1/3 coarsening of average cluster size; and t −1/2 approach to a saturation size going like ǫ −1/2 . Model II generalizes the first one in having an additional mechanism of particle deposition into cluster gaps, suppressed for the smallest gaps. This model exhibits early rapid filling, leading to slowing deposition due to the increasing scarcity of deposition sites, and then continued power law ((ǫt) 1/2 ) cluster size coarsening through the redistribution allowed by slow detachment. The basic (ǫt) 1/3 domain growth laws and ǫ −1/2 saturation in model I are explained by a simple scaling picture involving the time for a particle to detach and diffuse to the next cluster. A second, fuller approach is presented which employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. This allows, through the steady state solution of the corresponding equation for a cluster probability generating function, quantitative results for the saturation of model I in excellent agreement with the simulation results. For model II, it provides a one-variable scaling function solution for the coarsening probability distribution, and in particular quantitative agreement with the cluster length scaling and its amplitude.

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Section: B Simulationssupporting

confidence: 56%

Section: B Simulationsmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 71%

Section: B Simulationsmentioning

confidence: 99%

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Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment, and deposition

Reis

Stinchcombe

2004

Phys. Rev. E

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“…We note again that time-dependent quantities in our model have no analog in the mapping to the 1d models we discussed above. The emergence of density-dependent long time scales in reversible coalescence [16,48,49] therefore has no relation to the long time scales in our model, which are due to a very slow reorganization of fascicle basins (Sec. V D) -a consequence of axon turnover.…”

Section: B In Presence Of Detachmentmentioning

confidence: 91%

Model of fasciculation and sorting in mixed populations of axons

2011

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We extend a recently proposed model [Chaudhuri et al., Europhys. Lett. 87, 20003 (2009)] aiming to describe the formation of fascicles of axons during neural development. The growing axons are represented as paths of interacting directed random walkers in two spatial dimensions. To mimic turnover of axons, whole paths are removed and new walkers are injected with specified rates. In the simplest version of the model, we use strongly adhesive short-range inter-axon interactions that are identical for all pairs of axons. We generalize the model to adhesive interactions of finite strengths and to multiple types of axons with type-specific interactions. The dynamic steady state is characterized by the position-dependent distribution of fascicle size and fascicle composition. With distance in the direction of axon growth, the mean fascicle size and emergent time scales grow monotonically, while the degree of sorting of fascicles by axon type has a maximum at a finite distance. To understand the emergence of slow time scales, we develop an analytical framework to analyze the interaction between neighboring fascicles.

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Crossover between diffusion-limited and reaction-limited regimes in the coagulation–diffusion process

Shapoval¹,

Dudka²,

Durang³

et al. 2018

J. Phys. A: Math. Theor.

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The change from the diffusion-limited to the reaction-limited cooperative behaviour in reaction–diffusion systems is analysed by comparing the universal long-time behaviour of the coagulation–diffusion process on a chain and on the Bethe lattice. On a chain, this model is exactly solvable through the empty-interval method. This method can be extended to the Bethe lattice, in the ben-Avraham–Glasser approximation. On the Bethe lattice, the analysis of the Laplace-transformed time-dependent particle-density is analogous to the study of the stationary state, if a stochastic reset to a configuration of uncorrelated particles is added. In this stationary state logarithmic corrections to scaling are found, as expected for systems at the upper critical dimension. Analogous results hold true for the time-integrated particle-density. The crossover scaling functions and the associated effective exponents between the chain and the Bethe lattice are derived.

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Lattice kinetics of diffusion-limited coalescence and annihilation with sources

Cited by 5 publications

References 37 publications

Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment, and deposition

Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment, and deposition

Model of fasciculation and sorting in mixed populations of axons

Crossover between diffusion-limited and reaction-limited regimes in the coagulation–diffusion process

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