Self-similar behavior of the exchange-driven growth model with product kernel

Eichenberg, Constantin; Schlichting, André

doi:10.1080/03605302.2020.1845205

Cited by 4 publications

(7 citation statements)

References 31 publications

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“…However, the proof of Theorem 4 relies on the fact that we have in this case an explicit ODE for f which implies exponentially fast convergence to the equilibrium value 1 κ . This allows us to adapt a strategy of [ES21] where self-similar long-time behaviour is proved for a model of exchange-driven growth.…”

Section: Suppose That ∞mentioning

confidence: 99%

“…We remark that in the case ρ > 1 κ and f (0) > 1 κ , we can conclude without the additional assumption ∞ n=1 n 2−λ c n (0) < ∞ by bounding ∞ n=1 c n (t) from below by the solution of the pure diffusion problem (10). One can then use a result from [ES21] that characterizes the time decay of ∞ n=1 cn (t). We omit the details here.…”

Section: Clusters On the Membrane: Proof Of Prop 2 Partmentioning

confidence: 99%

“…The key tool in the rigorous analysis will be to compare the rescaled tails to solutions of the spatially continuous homogeneous equation ( 14). Our approach is inspired by and makes use of the results in [ES21], who show a self-similarity result for solutions of the homogeneous equation (13). Our situation is made more complicated by the presence of the inhomogeneity G k (t), since estimates for the fundamental solution of (13) as derived in [ES21] are degenerate at t = 0.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Our approach is inspired by and makes use of the results in [ES21], who show a self-similarity result for solutions of the homogeneous equation (13). Our situation is made more complicated by the presence of the inhomogeneity G k (t), since estimates for the fundamental solution of (13) as derived in [ES21] are degenerate at t = 0. However, expressing C k (t) in terms of this fundamental solution involves a convolution in time and space.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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A Becker–Döring Type Model for Cell Polarization

Pohl,

Niethammer

2023

J Stat Phys

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We propose a model for cell polarization based on the Becker–Döring equations with the first coagulation coefficient equal to zero. We show convergence to equilibrium for power-law coagulation and fragmentation rates and obtain a loss of mass in the limit $$t \rightarrow \infty $$ t → ∞ depending on the initial mass and the relative strengths of the coagulation and fragmentation processes. In the case of linear rates, we further show that large clusters evolve in a self-similar manner at large times by comparing limits of appropriately rescaled solutions in different spaces.

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Section: Suppose That ∞mentioning

confidence: 99%

Section: Clusters On the Membrane: Proof Of Prop 2 Partmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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A Becker–Döring Type Model for Cell Polarization

Pohl,

Niethammer

2023

J Stat Phys

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“…We will here focus on the continuous case, keeping in mind that often one can use the continuous selfsimilar solutions also to approximate large-time evolution of similarly scaled solutions to the discrete equation [7,9]. In this work, we assume that the coagulation kernel is homogeneous, i.e., there exists a γ ∈ R such that K (ax, ay) = a γ K (x, y) , x, y > 0, a > 0.…”

Section: Discrete and Continuous Coagulation Equation With Sourcementioning

confidence: 99%

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

Ferreira

Lukkarinen

Nota

et al. 2021

Commun. Math. Phys.

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We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

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Coagulation equations with source leading to anomalous self-similarity

Ferreira,

Franco,

Lukkarinen

et al. 2023

J. Phys. A: Math. Theor.

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We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity γ ⩽ − 1 , and behaves like x γ + λ y − λ when y ≪ x with γ + 2 λ > 1 . Our analysis shows that the long-time behaviour of the solutions depends on the parameters γ and λ. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of γ + λ and on whether γ = − 1 or γ < − 1 . In all these cases, the scaling differs from the usual one that has been previously obtained when γ + 2 λ < 1 or γ + 2 λ ⩾ 1 , γ > − 1 . In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case γ + 2 λ = 1 , γ ⩽ − 1 .

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Self-similar behavior of the exchange-driven growth model with product kernel

Cited by 4 publications

References 31 publications

A Becker–Döring Type Model for Cell Polarization

A Becker–Döring Type Model for Cell Polarization

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

Coagulation equations with source leading to anomalous self-similarity

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