BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Enrico Bibbona (Politecnico di Torino) DTSTART:20241024T153000Z DTEND:20241024T160000Z DTSTAMP:20260421T124902Z UID:MoRN/107 DESCRIPTION:Title: N ew deterministic scaling limits of models of nanoparticle growth\nby E nrico Bibbona (Politecnico di Torino) as part of Seminar on the Mathematic s of Reaction Networks\n\n\nAbstract\nWe consider the following nanopartic le growth model\, where the species $M$\nrepresents monomers\, and $P_i$ a re nanoparticles of size $i$:\n\\[ \n \\left\\{ \n \\begin{array}{l}\ n mM \\xrightarrow{\\nu N^{1-m}} P_m\, \\\\\n M + P_i \\xr ightarrow{\\nu N^\\theta} P_{i+1}\, \\quad i \\in \\{1\,\\ldots\, N\\}\n \\end{array}\n \\right.\n\\]\n$M(0) = N$\, $P_i(0) = 0$ for all $i$. W e demonstrate that the proportion of particles\, at time $N^αt$\, with a size within the interval $N^{1−β} [a\, b]$ (for any positive $a$\,\n$b$ )\, approaches a deterministic limit for large $N$. This is subject to the scaling\nconditions $\\theta + \\alpha + \\beta = 0$ and $1 + \\alpha = \ \beta$. We fully characterize such a scaled size distribution and establis h its satisfaction\, in terms of Schwartz distributions\, of the Lifshitz- Slyozov transport partial differential equation (PDE). We remark\nthat the special case $\\beta = 1$ (which implies $\\alpha = −1$ and $\\theta = 0$) is the so-called classical scaling. In this case the convergence of th e stochastic model to an infinite system of ODEs\, named after Becker and Döring\, is a classical result\, see e.g.\n[3]. Moreover\, after a furthe r coarsening step\, the ODE model was shown to be\nwell approximated by th e solution of the above mentioned PDE [2\,5]. We show\nin a single step ho w this PDE solution limit arise directly from the stochastic\nmodel\, both under the classical scaling and in a wider range of scalings. To\nprove t he 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 Hernande z\, Gabor Lente\, Elena Sabbioni\, Paola Siri\, and Rebeka Szabo.\n\n[1] D . Cappelletti and G. A. Rempala\, Individual molecules dynamics in reactio n\nnetwork models\, SIAM Journal on Applied Dynamical Systems\, 22 (2023)\ , pp.\n1344– 1382.\n\n[2] E. Hingant and R. Yvinec\, Deterministic and s tochastic Becker-Döring equations: Past and recent mathematical developme nts\, in Stochastic Processes\,\nMultiscale Modeling\, and Numerical Metho ds for Computational Cellular Biology\, Springer International Publishing\ , 2017\, pp. 175–204.\n\n[3] I. Jeon\, Existence of gelling solutions fo r coagulation-fragmentation equations\, Communications in Mathematical Phy sics\, 194 (1998)\, pp. 541–567.\n\n[4] E. Sabbioni\, R. Szabó\, P. Sir i\, D. Cappelletti\, G. Lente\, and E. Bibbona\,\nFinal nanoparticle size distribution under unusual parameter regimes. ChemRxiv. 2024.\n\n[5] A. Va sseur\, F. Poupaud\, J.-F. Collet\, and T. Goudon\, The Becker-D¨oring\ns ystem and its Lifshitz–Slyozov limit\, SIAM Journal on Applied Mathemati cs\,\n62 (2002)\, pp. 1488– 1500.\n LOCATION:https://researchseminars.org/talk/MoRN/107/ END:VEVENT END:VCALENDAR