We consider low-energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e. the so-called kissing number.

The aim of this paper is to study relaxation rates for the Cahn-Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish differential and algebraic relationships between the energy, the dissipation, and the squared H -1 distance to a kink.

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form E[V](X):=?1?i<j?NV(|X(i)-X(j)|),where X(j) ? R represents the position of the particle j and V(r) ? R is the pairwise interaction energy potential of two particles placed at distance r. We show that under suitable assumptions on the single-well potential V, the ground state energy per particle converges to an explicit constant E¯ [V] , which is the same as the energy per particle in the square lattice infinite configuration.

The COVID-19 pandemic triggered a global research effort to define and assess timely and effective containment policies. Understanding the role that specific venues play in the dynamics of epidemic spread is critical to guide the implementation of fine-grained non-pharmaceutical interventions (NPIs). In this paper, we present a new model of context-dependent interactions that integrates information about the surrounding territory and the social fabric.

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on I"-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity.

This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ? > 0 representing the lattice spacing of the crystal, we remove a disc of radius ? around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ?

In Alicandro et al. (J Mech Phys Solids 92:87-104, 2016) a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in Cermelli and Gurtin (Arch Ration Mech Anal 148, 1999) and predicts motion along the glide directions of the crystal.

We consider 2D discrete systems, described by scalar functions and governed by periodic interaction potentials. We focus on anisotropic nearest neighbors interactions in the hexagonal lattice and on isotropic long range interactions in the square lattice. In both these cases, we perform a complete Gamma-convergence analysis of the energy induced by a configuration of discrete topological singularities. This analysis allows to prove the existence of many metastable configurations of singularities in the hexagonal lattice.

The diffusive behaviour of simple random-walk proposals of many Markov Chain Monte Carlo (MCMC) algorithms results in slow exploration of the state space making inefficient the convergence to a target distribution. Hamiltonian/Hybrid Monte Carlo (HMC), by introducing fictious momentum variables, adopts Hamiltonian dynamics, rather than a probability distribution, to propose future states in the Markov chain. Splitting schemes are numerical integrators for Hamiltonian problems that may advantageously replace the St¨ormer-Verlet method within HMC methodology.