The Annals of Applied Probability

Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions

Maury Bramson and Joel L. Lebowitz

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Abstract

Consider the system of particles on Zd where particles are of two types, A and B, and execute simple random walks in continuous time. Particles do not interact with their own type, but when a type A particle meets a type B particle, both disappear. Initially, particles are assumed to be distributed according to homogeneous Poisson random fields, with equal intensities for the two types. This system serves as a model for the chemical reaction ABinert. In Bramson and Lebowitz [7], the densities of the two types of particles were shown to decay asymptotically like 1/td/4 for d<4 and 1/t for d > 4, as t → ∞. This change in behavior from low to high dimensions corresponds to a change in spatial structure. In d<4, particle types segregate, with only one type present locally. After suitable rescaling, the process converges to a limit, with density given by a Gaussian process. In d>4, both particle types are, at large times, present locally in concentrations not depending on the type, location or realization. In d=4, both particle types are present locally, but with varying concentrations. Here, we analyze this behavior in d<4; the behavior for d=4 will be handled in a future work by the authors.

Article information

Source
Ann. Appl. Probab., Volume 11, Number 1 (2001), 121-181.

Dates
First available in Project Euclid: 27 August 2001

Permanent link to this document
https://projecteuclid.org/euclid.aoap/998926989

Digital Object Identifier
doi:10.1214/aoap/998926989

Mathematical Reviews number (MathSciNet)
MR1825462

Zentralblatt MATH identifier
1016.60088

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
diffusion limited reaction annihilating random walks asymptotic densities spatial structure

Citation

Bramson, Maury; Lebowitz, Joel L. Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions. Ann. Appl. Probab. 11 (2001), no. 1, 121--181. doi:10.1214/aoap/998926989. https://projecteuclid.org/euclid.aoap/998926989


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References

  • [1] Arratia, R. (1979). Coalescing Brownian motions on the line. Ph.D. thesis, Univ. Wisconsin, Madison.
  • [2] Arratia, R. (1981). Limiting point processes for rescalings of coalescing and annihilating random walks on d. Ann. Probab. 9 909-936.
  • [3] Bhattacharya, R. N. and Rao, R. R. (1986). Normal Approximation and Asymptotic Expansions. Krieger, Malabar.
  • [4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [5] Billingsley, P. (1971). Weak Convergence of Measures: Applications in Probability. SIAM, Philadelphia.
  • [6] Bramson, M. and Griffeath, D. (1980). Asymptotics for interacting particle systems on d. Z. Wahrsch. Verw. Gebiete 53 183-196.
  • [7] Bramson, M. and Lebowitz, J. L. (1991). Asymptotic behavior of densities for two-particle annihilating random walks. J. Statist. Phys. 62 297-372.
  • [8] Bramson, M. and Lebowitz, J. L. (1991). Spatial structure in diffusion limited two-particle reactions. J. Statist. Phys. 65 941-952.
  • [9] Bramson, M. and Lebowitz, J. L. (2001). Spatial structure in high dimensions for diffusion limited two-particle reactions. In preparation.
  • [10] Chung, K. L. (1974). A Course in Probability Theory. Academic Press, New York.
  • [11] Kang, K. and Redner, S. (1985). Fluctuation-dominated kinetics in diffusion-controlled reactions. Phys. Rev. A 32 435-447.
  • [12] Kesten, H. and van den Berg, J. (2000). Asymptotic density in a coalescing random walk model. Ann. Probab. 28 303-352.
  • [13] Kuelbs, J. (1973). The invariance principle for Banach space valued random variables. J. Multivariate Anal. 3 161-172.
  • [14] Lee, B. P. and Cardy, J. (1995). Renormalization group study of the A + B diffusion-limited reaction. J. Statist. Phys. 80 971-1007.
  • [15] Lee, B. P. and Cardy, J. (1997). Erratum: Renormalization group study of the A + B diffusion-limited reaction. J. Statist Phys. 87 951-954.
  • [16] Orey, S. and Pruitt, W. (1973). Sample functions of the N-parameter Wiener process. Ann. Probab. 1 138-163.
  • [17] Ovchinnikov, A. A. and Zeldovich, Ya. B. (1978). Role of density fluctuations in bimolecular reaction kinetics. Chem. Phys. 28 215-218.
  • [18] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • [19] Toussaint, D. and Wilczek, F. (1983). Particle-antiparticle annihilation in diffusive motion. J. Chem. Phys. 78 2642-2647.