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Proof of dynamical scaling in Smoluchowski’s coagulation equation with constant kernel. (English) Zbl 0828.60093
Summary: Smoluchowski’s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version $\dot c_l = \sum^{l - 1}_{k = 1} c_{l - k} c_k - 2c_l \sum^\infty_{k = 1} c_k,$ where $$c_l = c_l(t)$$ is the concentration of $$l$$- particle clusters at time $$t$$. We prove that for initial data satisfying $$c_1 (0) > 0$$ and the condition $$0 \leq c_l (0) < A(1 + \Delta)^{- l}$$ $$(A, \Delta > 0)$$, the solutions behave asymptotically like $$c_l (t) \sim t^{-2} \widetilde c(lt^{-1})$$ as $$t \to \infty$$ with $$lt^{-1}$$ kept fixed. The scaling function $$\widetilde c (\xi)$$ is $$(1/ \rho) \exp [(-1/ \rho) \xi]$$, where $$\rho = \sum_l lc_l (0)$$, a conserved quantity, is the initial number of particles per unit volume. An analogous result is obtained for the continuous version of Smoluchowski’s coagulation equation ${\partial \over \partial t} c(v,t) = \int^v_0 du c (v - u,t) c(u,t) - 2c (v,t) \int^\infty_0 du c (u,t),$ where $$c(v,t)$$ is the concentration of clusters of size $$v$$.

##### MSC:
 60K40 Other physical applications of random processes 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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