Gelation in cluster coagulation processes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908480423591936 |
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| author | Andreis, Luisa Iyer, Tejas Magnanini, Elena |
| author_facet | Andreis, Luisa Iyer, Tejas Magnanini, Elena |
| contents | We consider the problem of gelation in the cluster coagulation model introduced by Norris [\textit{Comm. Math. Phys.}, 209(2):407-435 (2000)], where pairs of clusters of types $(x,y)$ taking values in a measure space $E$, merge to form a new particle of type $z\in E$ according to a transition kernel $K(x,y, \mathrm{d} z)$. This model possesses enough generality to accommodate inhomogeneities in the evolution of clusters, including variations in their shape or spatial distribution. We derive general, sufficient criteria for stochastic gelation in this model. As particular cases, we extend results related to the classical Marcus--Lushnikov coagulation process, showing that reasonable `homogenous' coagulation processes with exponent $γ>1$ yield gelation; and also, coagulation processes with kernel $\bar{K}(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +ε}$ for $ε>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10232 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gelation in cluster coagulation processes Andreis, Luisa Iyer, Tejas Magnanini, Elena Probability Mathematical Physics Analysis of PDEs 60K35, 82C22, 05C90 We consider the problem of gelation in the cluster coagulation model introduced by Norris [\textit{Comm. Math. Phys.}, 209(2):407-435 (2000)], where pairs of clusters of types $(x,y)$ taking values in a measure space $E$, merge to form a new particle of type $z\in E$ according to a transition kernel $K(x,y, \mathrm{d} z)$. This model possesses enough generality to accommodate inhomogeneities in the evolution of clusters, including variations in their shape or spatial distribution. We derive general, sufficient criteria for stochastic gelation in this model. As particular cases, we extend results related to the classical Marcus--Lushnikov coagulation process, showing that reasonable `homogenous' coagulation processes with exponent $γ>1$ yield gelation; and also, coagulation processes with kernel $\bar{K}(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +ε}$ for $ε>0$. |
| title | Gelation in cluster coagulation processes |
| topic | Probability Mathematical Physics Analysis of PDEs 60K35, 82C22, 05C90 |
| url | https://arxiv.org/abs/2308.10232 |