65 research outputs found
Justification of the Cauchy-Born approximation of elastodynamics
We present sharp convergence results for the Cauchy-Born approximation of general classical atomistic interactions, for static problems with small data and for dynamic problems on a macroscopic time interval
Surface energy and boundary layers for a chain of atoms at low temperature
We analyze the surface energy and boundary layers for a chain of atoms at low
temperature for an interaction potential of Lennard-Jones type. The pressure
(stress) is assumed small but positive and bounded away from zero, while the
temperature goes to zero. Our main results are: (1) As at fixed positive pressure , the Gibbs measures and
for infinite chains and semi-infinite chains satisfy path large
deviations principles. The rate functions are bulk and surface energy
functionals and
. The minimizer of the surface functional
corresponds to zero temperature boundary layers. (2) The surface correction to
the Gibbs free energy converges to the zero temperature surface energy,
characterized with the help of the minimum of
. (3) The bulk Gibbs measure and Gibbs
free energy can be approximated by their Gaussian counterparts. (4) Bounds on
the decay of correlations are provided, some of them uniform in
Convergence of the -Means Minimization Problem using -Convergence
The -means method is an iterative clustering algorithm which associates
each observation with one of clusters. It traditionally employs cluster
centers in the same space as the observed data. By relaxing this requirement,
it is possible to apply the -means method to infinite dimensional problems,
for example multiple target tracking and smoothing problems in the presence of
unknown data association. Via a -convergence argument, the associated
optimization problem is shown to converge in the sense that both the -means
minimum and minimizers converge in the large data limit to quantities which
depend upon the observed data only through its distribution. The theory is
supplemented with two examples to demonstrate the range of problems now
accessible by the -means method. The first example combines a non-parametric
smoothing problem with unknown data association. The second addresses tracking
using sparse data from a network of passive sensors
Lattice dynamics on large time scales and dispersive effective equations
We investigate the long time behavior of waves in crystals. Starting from a linear wave equation on a discrete lattice with periodicity , we derive the continuum limit equation for time scales of order . The effective equation is a weakly dispersive wave equation of fourth order. Initial values with bounded support result in ring-like solutions, and we characterize the dispersive long time behavior of the radial profiles with a linearized KdV equation of third order
Distribution of cracks in a chain of atoms at low temperature
We consider a one-dimensional classical many-body system with interaction potential of Lennard--Jones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(-β e surf /2) with e surf > 0 a surface energy
Recommended from our members
Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature
We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature β- 1 goes to zero. Our main results are: (1) As β→ ∞ at fixed positive pressure p> 0 , the Gibbs measures μβ and νβ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals E¯ bulk and E¯ surf. The minimizer of the surface functional corresponds to zero temperature boundary layers; (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of E¯ surf; (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts; (4) Bounds on the decay of correlations are provided, some of them uniform in β. © 2020, The Author(s)
- …