Abstract
We study diffusion-limited coalescence, A+A⇋A, in one dimension, in the presence of a diffusing trap. The system may be regarded as a generalization of von Smoluchowski's model for reaction rates, in that (a) it includes reactions between the particles surrounding the trap, and (b) the trap is mobile—two considerations which render the model more physically relevant. As seen from the trap's frame of reference, the motion of the particles is highly correlated, because of the motion of the trap. An exact description of the long-time asymptotic limit is found using the IPDF method and exploiting a “shielding” property of reversible coalescence that was discovered recently. In the case where the trap also acts as a sources—giving birth to particles—the shielding property breaks down, but we find an “equivalence principle”: Trapping and diffusion of the trap may be compensated by an appropriate rate of birth, such that the steady state of the system is identical with the equilibrium state in the absence of a trap.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
H. Haken, Synergetics (Springer-Verlag, Berlin, 1978).
G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Systems (Wiley, New York, 1980).
T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).
K. Kang and S. Redner, Fluctuation-dominated kinetics in diffusion-controlled reactions, Phys. Rev. A 32:435 (1985); V. Kuzovkov and E. Kotomin, Kinetics of bimolecular reactions in condensed media: critical phenomena and microscopic self-organization, Rep. Prog. Phys. 51:1479 (1988).
J. Stat. Phys. Vol. 65, Nos. 5/6 (1991); this issue contains the proceedings of a conference on Models of Non-Classical Reaction Rates, which was held at NIH (March 25-27, 1991) in honor of the 60th birthday of G. H. Weiss.
K. J. Laidler, Chemical Kinetics (McGraw-Hill, New York, 1965).
S. W. Benson, The Foundations of Chemical Kinetics (McGraw-Hill, New York, 1960).
M. von Smoluchowski, Z. Phys. Chem. 92:129 (1917).
R. Schoonover, D. ben-Avraham, S. Havlin, R. Kopelman, and G. H. Weiss, Nearest-neighbor distances in diffusion-controlled reactions modelled by a single mobile trap, Physica A 171:232 (1991).
D. ben-Avraham and G. H. Weiss, Statistical properties of nearest-neighbor distances in a diffusion-reaction model, Phys. Rev. A 39:6436 (1989).
A. D. Sánchez, M. A. Rodriguez, and H. S. Wio, Results in trapping reactions for mobile particles and a single trap, Phys. Rev. E 57:6390 (1998).
D. ben-Avraham, M. A. Burschka, and C. R. Doering, Statics and dynamics of a diffusion-limited reaction: Anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition, J. Stat. Phys. 60:695 (1990).
C. R. Doering, Microscopic spatial correlations induced by external noise in a reaction-diffusion system, Physica A 188:386 (1992).
H. Hinrichsen, K. Krebs, and M. P. Pfannmuller, Finite-size scaling of one-dimensional reaction-diffusion systems. Part I: Analytic results, J. Stat. Phys. 78:1429 (1995); B. Wehefritz, K. Krebs, and M. P. Pfannmüller, Finite-size scaling of one-dimensional reaction-diffusion systems. Part II: Numerical methods, J. Stat. Phys. 78:1471 (1995); H. Hinrichsen, K. Krebs, and I. Peschel, Solution of a one-dimensional reaction-diffusion model with spatial asymmetry, Z. Phys. B 100:105 (1996).
M. Henkel, E. Orlandini, and G. M. Schütz, Equivalence between stochastic systems, J. Phys. A 28:6335 (1995); M. Henkel, E. Orlandini, and J. Santos, Reaction-diffusion processes from equivalent integrable quantum chains, Ann. Phys. (N.Y.) 259:163 (1997).
H. Simon, Concentration for one and two species one-dimensional reaction-diffusion systems, J. Phys. A 28:6585 (1995).
D. Balboni, P.-A. Rey, and M. Droz, Universality of a class of annihilation-coagulation models, Phys. Rev. E 52:6220 (1995); P.-A. Rey and M. Droz, A renormalization group study of reaction-diffusion models with particles input, J. Phys. A 30:1101 (1997).
M. Bramson and D. Griffeath, Clustering and dispersion for interacting particles, Ann. Prob. 8:183 (1980); Z. Wahrsch. Geb. 53:183 (1980).
D. C. Torney and H. M. McConnell, Diffusion-limited reactions in one dimension, Proc. R. Soc. Lond. A 387:147 (1983).
J. L. Spouge, Exact solutions for a diffusion-reaction process in one dimension, Phys. Rev. Lett. 60:871 (1988).
H. Takayasu, I. Nishikawa and H. Tasaki, Phys. Rev. A 37:3110 (1988).
V. Privman, Exact results for diffusion-limited reactions with synchronous dynamics, Phys. Rev. E 50:50 (1994); Exact results for 1D conserved order parameter model, Mod. Phys. Lett. B 8:143 (1994); V. Privman, A. M. R. Cadilhe, and M. L. Glasser, Exact solutions of anisotropic diffusion-limited reactions with coagulation and annihilation, J. Stat. Phys. 81:881 (1995); Anisotropic diffusion-limited reactions with coagulation and annihilation, Phys. Rev. E 53:739 (1996).
D. ben-Avraham, Diffusion-limited coalescence, A + A ⇌ A, with a trap, Phys. Rev. E 58:4351 (1998); D. ben-Avraham, Inhomogeneous steady-states of diffusion-limited coalescence, A + A → A, Phys. Lett. A 249:415 (1998).
D. ben-Avraham, The method of interparticle distribution functions for diffusion-reaction systems in one dimension, Mod. Phys. Lett. B 9:895 (1995); D. ben-Avraham, The coalescence process, A + A → A, and the method of interparticle distribution functions, in Nonequilibrium Statistical Mechanics in One Dimension, V. Privman, ed., pp. 29-50 (Cambridge University Press, 1997).
D. ben-Avraham, Fisher waves in the diffusion-limited coalescence process A + A ⇌ A, Phys. Lett. A 247:53 (1998).
D. R. Nelson and N. Shnerb, Non-hermitian localization and population biology, Phys. Rev. E 58:1383 (1998).
K. A. Dahmen, D. R. Nelson, and N. M. Shnerb, Life and death near a windy oasis, http://xxx.lanl.gov/list/cond-mat, paper number 9807394. 112 Donev et al.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Donev, A., Rockwell, J. & ben-Avraham, D. Generalized von Smoluchowski Model of Reaction Rates, with Reacting Particles and a Mobile Trap. Journal of Statistical Physics 95, 97–112 (1999). https://doi.org/10.1023/A:1004573310526
Issue Date:
DOI: https://doi.org/10.1023/A:1004573310526