Abstract
In the present paper we continue the investigation of the so-called coalescing ideal gas in one dimension, initiated by Ermakov. The model consists of point-like particles moving with velocities ±1 which coalesce and choose a fresh velocity with the same distribution when colliding. In the previous paper various identities in law were derived for the infinitely extended system. In the present paper we consider the scaling limit of the model in its various guises. The main result is the derivation of the scaling limit (invariance principle) for the joint law of an arbitrary finite number of individual particle trajectories in this system. We also obtain the scaling limit of the density profile of the system, which strongly resembles earlier results of Belitsky and Ferrari.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
R. Arratia, Limiting point processes for resealing of coalescing and annihilating random walks on ℤd, Ann. Prob. 9:909–936 (1981).
V. Belitsky and P. A. Ferrari, Ballistic annihilation and deterministic surface growth, J. Stat. Phys. 80:517–543 (1995).
P. Billingsley, Convergence of Probability Measure (Wiley, New York, 1968).
M. Bramson and D. Griffeath, Asymptotics for interacting particle systems on ℤd, Z. Wahrsch. verw. Geb. 53:183–196 (1980).
M. Bramson and D. Griffeath, Clustering and dispersion rates for some interacting particle systems on ℤ1, Ann. Probab. 8:183–213 (1980).
R. Durrett, Probability: Theory and Examples (Wadsworth, 1991).
A. Ermakov, Exact probabilities and asymptotics for the one-dimensional coalescing ideal gas, Stoch. Processes and Their Applications 71:275–284 (1997).
R. Fisch, Clustering in the one-dimensional three-color cyclic cellular automaton, Ann. Prob. 20:1528–1548 (1992).
D. Griffeath, Additive and cancellative interactive particle systems, Lect. Notes in Math., Vol. 724, Springer, New-York, 1979.
T. E. Harris, On a class of set-valued Markov processes, Ann. Probab. 4:175–194 (1976).
K. Itô and H. P. McKean Jr., Diffusion Processes and Their Sample Paths, 2nd ed. (Springer, 1974).
J.-P. Kahane, Some Random Series of Functions (Cambridge Univ. Press, 1968).
T. M. Liggett, Interacting Particle Systems (Springer, 1985).
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion (Springer, 1991).
W. Whitt, Weak convergence of probability measures on the functional space C[0, ∞), Ann. Math. Statist. 41:939–944 (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ermakov, A., Tóth, B. & Werner, W. On Some Annihilating and Coalescing Systems. Journal of Statistical Physics 91, 845–870 (1998). https://doi.org/10.1023/A:1023071714672
Issue Date:
DOI: https://doi.org/10.1023/A:1023071714672