Annihilation of immobile reactants in the Bethe lattice

Majumdar, Satya N.; Privman, Vladimir

doi:10.1088/0305-4470/26/16/006

Cited by 29 publications

(59 citation statements)

References 27 publications

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“…Let us absorb the reaction rate R into the time scale by introducing the dimensionless time variable τ ≡ R t. Let us denote by P k (τ ) the probability that k randomly chosen nearest neighbor sites in a given lattice are all simultaneously occupied (k-site cluster). The evolution equations for the ensemble probabilities P k read [32] …”

Section: The Cr and The Ar Model In Bethe Lattices: Exact Solutiomentioning

confidence: 99%

On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond

Abad

2004

Phys. Rev. E

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We study the behavior of the chemical reactions A + A → A + S and A + A → S + S (where the reactive species A and the inert species S are both assumed to be immobile) embedded on Bethe lattices of arbitrary coordination number z and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact solutions for the coverage in the A species in terms of the initial condition are obtained. In particular, our results hold for the important case of an infinite one-dimensional (1D) lattice (z = 2). The method is based on an expansion in terms of conditional probabilities which exploits a Markovian property of these systems. Along the same lines, an approximate solution for the case of a 2D square lattice is developed. The effect of dilution in a random initial condition is discussed in detail, both for the lattice coverage and for the spatial distribution of reactants.

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Section: The Cr and The Ar Model In Bethe Lattices: Exact Solutiomentioning

confidence: 99%

On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond

Abad

2004

Phys. Rev. E

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“…This no-loops property may allow exact solvability for some models, for general coordination number ξ. Reaction diffusion models on the Cayley tree have been studied in, for example [12][13][14][15][16][17]. In [12,13,16] diffusion-limited aggregations, and in [14] two-particle annihilation reactions for immobile reactants have been studied.…”

Section: Introductionmentioning

confidence: 99%

Exactly solvable reaction diffusion models on a Cayley tree

2007

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The most general reaction-diffusion model on a Cayley tree with nearestneighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.

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“…is a solution to [13] and [14], and as the stationary solution is unique, this is the stationary solution.…”

Section: The Stationary Solutionmentioning

confidence: 98%

“…Note that the result is independent of coordination number n. So the stationary behavior of the system, reaction diffusion model on a Cayley tree with arbitrary coordination number n, is similar to that of a reaction diffusion model on a onedimensional lattice (n = 2), provided of course that there is no process which creates particles from two neighboring vacant sites. Using [17] for n = 0, and n = 1, together with the boundary condition [14], the constants C 1 and C 2 can be expressed in term of E s 1 r 6 ¼ 0 or r 0 6 ¼ 0 It is a difficult task to obtain a closed form for E s n . Things become, however, simpler for large n's.…”

Section: The Stationary Solutionmentioning

confidence: 99%

“…This no-loops property may allow exact solvability for some models, for general coordination number n Reaction diffusion models on the Cayley tree are studied in, for example [12][13][14][15][16][17]. In [12,13,16] diffusionlimited aggregations, and in [14] two-particle annihilation reactions for immobile reactants have been studied. There are also some exact results for deposition processes on the Bethe lattice [17].…”

Section: Introductionmentioning

confidence: 99%

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