Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel
Abstract
Smoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version[Figure not available: see fulltext.] where c t = c 1 (t) is the concentration of l-particle clusters at time t. We prove that for initial data satisfying c 1(0)>0 and the condition 0 ⩽ c l (0) < A (1+ Δ)- l ( A Δ>0), the solutions behave asymptotically like c 1 (t)∼t -2≈c(lt-1) as t→∞ with lt -1 kept fixed. The scaling function ≈ c(ξ) is (1/gr)ξ, whereρ = sum _l lc_l (0), a conserved quantity, is the initial number of particles per unit volume. An analous result is obtained for the continuous version of Smoluchowski's coagulation equationpartial /{partial t}c(v,{{ }}t) = int_0^v {du{{ }}c(v - u,{{ }}t){{ }}c(u,{{ }}t) - 2c(v,{{ }}t)} int_0^infty {du{{ }}c(u,{{ }}t)} where c(v, t) is the oncentration of clusters of size v.
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- May 1994
- DOI:
- 10.1007/BF02186868
- Bibcode:
- 1994JSP....75..389K
- Keywords:
-
- Smoluchowski's coagulation equations;
- dynamical scaling;
- cluster growth;
- kinetics of first-order phase transitions