New deterministic scaling limits of models of nanoparticle growth

Abstract: We consider the following nanoparticle growth model, where the species $M$ represents monomers, and $P_i$ are nanoparticles of size $i$: \[ \left\{ \begin{array}{l} mM \xrightarrow{\nu N^{1-m}} P_m, \\ M + P_i \xrightarrow{\nu N^\theta} P_{i+1}, \quad i \in \{1,\ldots, N\} \end{array} \right. \] $M(0) = N$, $P_i(0) = 0$ for all $i$. We demonstrate that the proportion of particles, at time $N^αt$, with a size within the interval $N^{1−β} [a, b]$ (for any positive $a$, $b$), approaches a deterministic limit for large $N$. This is subject to the scaling conditions $\theta + \alpha + \beta = 0$ and $1 + \alpha = \beta$. We fully characterize such a scaled size distribution and establish its satisfaction, in terms of Schwartz distributions, of the Lifshitz-Slyozov transport partial differential equation (PDE). We remark that the special case $\beta = 1$ (which implies $\alpha = −1$ and $\theta = 0$) is the so-called classical scaling. In this case the convergence of the stochastic model to an infinite system of ODEs, named after Becker and Döring, is a classical result, see e.g. [3]. Moreover, after a further coarsening step, the ODE model was shown to be well approximated by the solution of the above mentioned PDE [2,5]. We show in a single step how this PDE solution limit arise directly from the stochastic model, both under the classical scaling and in a wider range of scalings. To prove the result we use a the framework originally developed for epidemic models in [1]. Preliminary results based on simulations alone are available at [4]. This is joint work with Daniele Cappelletti, Anderson Melchor Hernandez, Gabor Lente, Elena Sabbioni, Paola Siri, and Rebeka Szabo.

[1] D. Cappelletti and G. A. Rempala, Individual molecules dynamics in reaction network models, SIAM Journal on Applied Dynamical Systems, 22 (2023), pp. 1344– 1382.

[2] E. Hingant and R. Yvinec, Deterministic and stochastic Becker-Döring equations: Past and recent mathematical developments, in Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology, Springer International Publishing, 2017, pp. 175–204.

[3] I. Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Communications in Mathematical Physics, 194 (1998), pp. 541–567.

[4] E. Sabbioni, R. Szabó, P. Siri, D. Cappelletti, G. Lente, and E. Bibbona, Final nanoparticle size distribution under unusual parameter regimes. ChemRxiv. 2024.

[5] A. Vasseur, F. Poupaud, J.-F. Collet, and T. Goudon, The Becker-D¨oring system and its Lifshitz–Slyozov limit, SIAM Journal on Applied Mathematics, 62 (2002), pp. 1488– 1500.

algebraic geometrydynamical systemsprobability

Audience: researchers in the topic

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Seminar on the Mathematics of Reaction Networks

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This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs twice a month, typically on the 2nd and 4th Thursday of the month, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

We look forward hearing about new work and meeting many of you over zoom! Many of the talks are recorded; to see the recording, from Past Talks, open details of the listed talk for a video link.

The organizers.

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