Calculus of Variations and Geometric Measure Theory

L. Lussardi - A. Melchor Hernandez - M. Morandotti

$\Gamma$-convergence of discrete energies modeling self-aggregation of stochastic particles

created by morandott on 08 May 2023
modified by lussardi on 23 Jan 2024


Submitted Paper

Inserted: 8 may 2023
Last Updated: 23 jan 2024

Pages: 17
Year: 2023

ArXiv: 2305.05761 PDF


In this work, we demonstrate that a functional modeling the self-aggregation of stochastically distributed lipid molecules can be obtained as the $\Gamma$-limit of a family of discrete energies driven by a sequence of independent and identically distributed random variables. These random variables are intended to describe the asymptotic behavior of lipid molecules that satisfy an incompressibility condition. The discrete energy keeps into account the interactions between particles. We resort to transportation maps to compare functionals defined on discrete and continuous domains, and we prove that, under suitable conditions on the scaling of these maps as the number of random variables increases, the limit functional features an interfacial term with a Wasserstein-type penalization.

Keywords: Gamma-convergence, Wasserstein distance, lipid bilayer, self-aggregation, transportation maps