Coagulation and universal scaling limits for critical Galton–Watson processes

Iyer, Gautam; Leger, Nicholas; Pego, Robert L.

doi:10.1017/apr.2018.23

Cited by 7 publications

(2 citation statements)

References 47 publications

(96 reference statements)

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“…This result is essentially a restatement of Theorem 3.1 in [21], where different terminology is used. A simpler, direct proof will appear in [12], however.…”

Section: Bernstein Functions and Transformsmentioning

confidence: 99%

Coagulation–Fragmentation Model for Animal Group-Size Statistics

2016

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We study coagulation-fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H -theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent − 2 3 for small size to − 3 2 for large size, with an exponential cutoff.

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“…This result is essentially a restatement of Theorem 3.1 in [21], where different terminology is used. A simpler, direct proof will appear in [12], however.…”

Section: Bernstein Functions and Transformsmentioning

confidence: 99%

Coagulation–Fragmentation Model for Animal Group-Size Statistics

2016

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“…with g 0 (s) = s, where Ψ is the given branching mechanism, see Iyer et al [39]. For example, for the Smoluchowski coagulation case with constant kernel K = 1 we have considered hitherto, the function Ψ = Ψ (g) has the form Ψ (g) = − 1 2 g 2 ; see for example Menon and Pego [56].…”

Section: Example 5 (Multiple Mergers)mentioning

confidence: 99%

Applications of Grassmannian and graph flows to coagulation systems

Doikou¹,

Malham²,

Stylianidis³

et al. 2022

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We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as Grassmannian or nonlinear graph flows and are therefore linearisable, and/or integrable in this sense. We first prove that a general Smoluchowski-type equation with a constant frequency kernel, that encompasses a large class of such models, is realisable as a multiplicative Grassmannian flow, and then go on to establish that several other related constant kernel models can be realised as such. These include: the Gallay-Mielke coarsening model; the Derrida-Retaux depinning transition model and a general mutliple merger coagulation model. We then introduce and explore the notion of nonlinear graph flows, which are related to the notion of characteristics for partial differential equations. These generalise flows on a Grassmann manifold from sets of graphs of linear maps to sets of graphs of nonlinear maps. We demonstrate that Smoluchowski's coagulation equation in the additive and multiplicative frequency kernel cases, are realisable as nonlinear graph flows, and are thus integrable provided we can uniquely retrace the initial data map along characteristics. The additive and multiplicative frequency kernel cases correspond to inviscid Burgers flow. We explore further applications of such nonlinear graph flows, for example, to the stochastic viscous Burgers equation. Lastly we consider an example stochastic partial differential equation with a nonlocal nonlinearity that generalises the convolution form associated with nonlinear coagulation interaction, and demonstrate it can be realised as an infinite dimensional Grassmannian flow. In our companion paper, Doikou et al. [23], we consider the application of such infinite dimensional Grassmannian flows to classical noncommutative integrable systems such as the Korteweg-de Vries and nonlinear Schrödinger equations.

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