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Discrete models for materials:
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List of participants |
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Contacts: For further information contact Gill Walton progsec@maths.warwick.ac.uk |
We consider the overdamped inertial dynamics of a chain of particles with convex nearest-neighbour interactions in a tilted periodic potential. This is a model for nonlinear conductivity of charge density wave materials. For each spacial periodic class we prove existence of a globally attracting travelling wave if there are no equilibrium solutions. The key is a novel preserved partial order.
(Joint work with C. Le Bris (Ecole Nationale des Ponts et Chaussées) and P.-L. Lions (Université Paris 9 - Dauphine)) We present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models. The approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one. We carry out the corresponding calculations in the case of two-body energies, including higher order terms, and also in the case of Thomas-Fermi type models.
The study of dislocations in cristalline solids involves phenomena taking place at different interacting scales: from the nanoscale (a few atomic spacings, the width of dislocation cores) to the macroscale, where the collective behavior of dislocation densities controls properties such as the strength of the material. We address here the main issue at the atomic scale: the understanding of the structure and mobility of isolated dislocations. We present a discrete model for the dynamics of dislocations in cubic metals. Numerical solutions of the model in 3D suggest that dislocations can be identified with discrete nonlinear waves. In simplified 1D and 2D geometries, we are able to obtain information about the depinning thresholds (dynamic and static Peierls stress) and the speed of the defects thanks to the analysis of traveling wave solutions of the discrete models.
The time-dependent, dissipative, one-dimensional Frenkel-Kontorova model of dislocations with forcing is analysed in the continnuum limit. It is shown that while for the continuum model any force leads to dislocation motion, in the discrete case there is a critical force (the Peierls stress) which needs to be applied to get the dislocation to move, and that it moves by jumping from one atom position to the next. This critical force is exponentially small as the continuum limit is approached.
We can systematically convert the master equation into a Fokker-Planck equation by introducing a large parameter and expanding around it. This approach was first developed the van Kampen. The drift and diffusion parameters in the Fokker-Planck equation can be identified with the first and second moments of the master equation respectively . It has been shown by Kurtz that the two equations are identical when we allow large parameter to tend to infinity. With epitaxial systems the Large parameter is associate with ensemble time of the system. As such, we can apply the Langevin interpretation to the Fokker-Planck, equation to obtain stochastic differential equations that describe the evolution of epitaxial systems These equations can then be integrated, thus reducing algorithmic rules to a set of stochastic differential equations.
We apply this procedure to three basic models: the Family model, the Wolf-Villain model, and the surface diffusion model. In all cases, the results obtained from the numerical integration agree with those from KMC simulations. Although, stochastic integration is more costly computationally than KMC simulations, it puts a class of algorithms used in such simulations on a firm mathematical basis and provides a means for further analysis.
An overview of what is known about Gibbs states in the situation of lattice models with non-compact "spins". In particular, gradient Gibbs measures at nonzero temperatures will be discussed, with a view of an application (or a hope for an application) to elastic crystals.
One-dimensional Hamiltonian monatomic lattices with Hamiltonian consisting of the kinetic energy and nearest neighbour interaction energy of the particles, are known to carry localized travelling wave solutions, for generic nonlinear potentials. The asymptotic profile of these waves is derived in the high-energy limit $H\to\infty$ for Lennard-Jones type interactions. The limit profile is proved to be a universal, highly discrete, piecewise linear wave concentrated on a single atomic spacing.
Many natural and man-made materials are stable as two different solid phases under the same conditions of temperature and stress. Such materials are referred to as martensites and their mechanical behaviour is largely governed by the mobility of the interfaces between the different phases. Continuum theories for characterizing these materials have shown that the mobility of the interfaces is not completely determined by the balance laws of mass, momentum and energy. Extra constitutive information is needed to uniquely determine how the phase boundaries move. We adopt the point of view that this constitutive information is a consequence of the mechanics of rearrangement of the atoms at the phase boundary. We explore this possibility through a discrete mass and spring model of the solid and demonstrate that this constitutive information can indeed be extracted from such discrete simulations.
This will be a survey of various attempts to derive, from the statistical mechanics of interacting "particles", the qualitative (crystalline) structure of real materials in thermal equilibrium at low temperature and/or high pressure.
We study a prototypical model providing a unified description of dynamics for such crystalline defects as martensitic phase boundaries, dislocations and cracks. The defect is represented by a driven kink in an asymmetric on-site potential. To obtain an exact solution of the discrete problem we use a bi-quadratic approximation allowing one to fully expose the effects of different curvatures of the energy wells. Although the microscopic model is Hamiltonian it generates a nontrivial macroscopic kinetic relation which we present in the form of an explicit functional relation between the velocity of the defect and the conjugate configurational force.
Joint work with Peter Berg (Simon Fraser). The Bando model of highway traffic is a discrete model where each vehicle's displacement satisfies a second order ODE involving the relative displacement of the vehicle immediately ahead: this yields a 1D lattice differential equation without spatial reflexional symmetry. In the open lattice case, a wide variety of travelling and dispersive wave solutions may be driven by the $n\rightarrow\pm\infty$ boundary conditions. We describe how a related continuum model yields a classification of wave solutions by simple geometric techniques. An interesting question concerns the existence of nontrivial solutions, which are not driven by boundary conditions, and which co-exist with linearly stable uniform states. We investigate these solutions (i) via the continuum model wave classification, and (ii) by applying the numerical continuation package AUTO in the case where the lattice is a closed ring.
We will present various continuum limits in the regime of partial differential equations (PDEs) that arise from the modeling of epitaxial thin film growths. The goal is to connect microscopic and macroscopic descriptions of the growth processes. The hierarchy of models considered here can take into account the diffusion of terrace adatoms, attachment and detachment of edge adatoms, vapor diffusion and the effect of multiple species. The mathematical starting point is the Burton-Cabrera-Frank step flow model. The continuum PDEs take the form of a coupled diffusion equation for the adatom density and a Hamilton-Jacobi equation for the thin film height profile. Nucleation phenomena is considered by exploring possible boundary conditions for the PDE.
We prove rigorously that a large class of reaction-diffusion systems is described by stochastic rate equations with imaginary maultiplicative noise. These equations have been originally derived by Ben Lee and John Cardy in 1993 using field theoretic methods. We also show that non-rigorous analysis of these equations based on perturbative renormalization group method produces a decription of large fluctuation effects in reaction diffusion systems, which is in perfect agreement with numerical simulations.
For participation contact:
Mathematics Research Centre,
University of Warwick,
Coventry CV4 7AL
E-mail: peta@maths.warwick.ac.uk
Phone: +44 (0)24 7652 4403
Fax: +44 (0)24 7652 3548
