Mechanical behaviour of heterogeneous nanochains in the $\Gamma$-limit of stochastic particle systems

TL;DR

This paper develops a mathematical model for the mechanical behavior of heterogeneous nanochains with stochastic properties, using $ ext{Gamma}$-convergence and ergodic theorems to analyze their effective behavior as particle spacing diminishes.

Contribution

It introduces a stochastic homogenization framework for nanochains with heterogeneous interactions and addresses mathematical challenges posed by singular potentials.

Findings

Effective behavior characterized via $ ext{Gamma}$-convergence.

Inclusion of stochastic distribution of material parameters.

Handling of finite-range interactions with arbitrary neighbors.

Abstract

Nanochains of atoms, molecules and polymers have gained recent interest in the experimental sciences. This article contributes to an advanced mathematical modeling of the mechanical properties of nanochains that allow for heterogenities, which may be impurities or a deliberately chosen composition of different kind of atoms. We consider one-dimensional systems of particles which interact through a large class of convex-concave potentials, which includes the classical Lennard-Jones potentials. We allow for a stochastic distribution of the material parameters and investigate the effective behaviour of the system as the distance between the particles tends to zero. The mathematical methods are based on $\Gamma$-convergence, which is a suitable notion of convergence for variational problems, and on ergodic theorems as is usual in the framework of stochastic homogenization. The allowed singular structure of the interaction potentials causes mathematical difficulties that we overcome by an approximation. We consider the case of $K$ interacting neighbours with $K\in \mathbb{N}$ arbitrary, i.e., interactions of finite range.