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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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