BATH INSTITUTE FOR COMPLEX SYSTEMS
"Multi-scale problems: modelling, analysis and applications"
University of Bath, 12th - 14th September 2005
This page will be updated regularly as the conference approaches.
Dr Grègoire Allaire
Title: Homogenization of the Schrödinger equation and effective mass theorems
Abstract: We study the homogenization of a Schrödinger equation with a large periodic potential: denoting by $\epsilon$ the period, the potential is scaled as $\epsilon^{-2}$. We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrödinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition. This is a joint work with Andrey Piatnitski.
Professor John Ball
Title: Phase nucleation in solids
Abstract:The talk will discuss some issues concerning the nucleation of
lower energy phases in solids undergoing phase transformations, and in
particular the roles played by compatibility and interfacial energy.
Dr. Alexei Beliaev
Title: Homogenization of pollution transport in random porous media accounting for hysteresis of sorption.
Abstract: Hysteresis of sorption and desorption processes in porous media is an important phenomenon in view of engineering problems of groundwater remediation, pollution trapping and environment protection. In this respect some recent advances in homogenization of partial differential equations with hysteretic terms will be presented in the talk. For a heterogeneous porous medium with random microstructure a homogenized transport equation with diffusion and sorption terms will be obtained. The analysis and proof of convergence is based on the methods of the non-linear semi-group theory.
Professor Gerard Ben Arous
Title: The many scales between bulk and extreme values of random media
Abstract: I will start by a general and elementary phenomenon: the most standard paradigm of probability theory , i.e the i.i.d sample, contains infinitely many scales, between its bulk and its extreme values. This simple fact explains an interesting multiscale structure of random media which induces an rich transition for different mechanisms in random media, like reaction-diffusion, trapping,
intermittency and aging.
This is based on joint works with L.Bogatchev, J.Cerny, S.Molchanov, A.Ramirez.
Prof. Leonid Berlyand
Title: Rise of correlations of transformation strains in laminated random polycrystals.
Abstract: We present a model of a laminated polycrystal with $n$ grains. The orientation of each grain is given by an uncorrelated random sequence of the orientation angles $\theta_i$, $i=1,\cdots, n$. Under imposed boundary conditions each grain undergoes stress free transformation, that depends on its orientation angle and result in transformation strains $\epsilon^T_i,\; i=1, \cdot, n$. The sequence of random variables $\epsilon^T_i,\; i=1, \cdot, n$ is obtain as the solution of a nonlinear variational (PDE) problem.
While the random variables $\theta_i$, $i=1,\cdots, n$ are uncorrelated, the the random variables $\epsilon^T_i,\; i=1, \cdot, n$ may or may not be correlated due to nonlinearity -- this is the central issue of our analysis. We investiogate this rise of correlations in three different scaling limits. Our proofs use the de Finetti's Theorem as well as the Riesz rearrangement inequality. This is a joint work with O. Bruno and A. Novikov.
Dr. Denis .I. Borisov
Title: Asymptotics for the spectrum of the Schroedinger operator perturbed by a fast oscillating potential
Abstract: We study the spectrum of one-dimensional Schrodinger operator perturbed by a fast oscillating potential. The oscillation period is a small parameter. The essential spectrum is found in an explicit form. The existence and multiplicity of the discrete spectrum are studied. The complete asymptotics expansions for the eigenvalues and the associated eigenfunctions are constructed. The work is partially supported by RFBR (05-01-97912-r_agidel) and by the programs 'Leading scientific schools'' (NSh-1446.2003.1) and ''Universities of Russia'' (UR.04.01.484).
Professor Sandra Chapman
Title: Scaling and intermittency in solar wind plasma turbulence- towards data
derived stochastic differential equations.
Abstract: The solar wind provides a natural laboratory for observations of MHD plasma turbulence, exhibiting scaling over extended temporal scales. We identify approximate self-similarity in the Probability Density Functions (PDF) of fluctuations in certain solar wind in- situ bulk plasma parameters. Whereas the fluctuations of speed for example are multi-fractal, we find that under
certain conditions the fluctuations in the ion density, field and particle energy densities and Poynting flux are approximately self-similar on timescales up to tens of hours. The intermittency of the system is then expressed in these parameters through the non-Gaussian nature of the single curve that describes the fluctuations PDF up to this timescale.
We first discuss generic methods for estimating the scaling exponents from these finite length observational timeseries. From the measured exponents we then construct a Fokker-Planck equation along with the associated Langevin equation. The steady state solutions to this predict the functional form of the PDF which we can then check against the observations.
Finally, if the scaling exponents reflect a universal behaviour then they can be used to compare the phenomenology of turbulence in the solar wind with that found in other systems. From this we use the measured exponents to explore the role of passive scalars in solar wind turbulence.
Dr Kirill Cherednichenko
Title: Variational and asymptotic approaches to higher-order effects in periodic composites via homogenisation.
Abstract: Please click here for a PDF version of this abstract
Dr. Richard Craster
Title:Trapped modes in curved elastic waveguides
Abstract: We investigate the existence of trapped modes in elastic plates of constant thickness that possess bends of arbitrary curvature and flatten out at infinity; such trapped modes consist of finite energy localised in regions of maximal curvature. We present both an asymptotic model and numerical evidence to demonstrate the trapping. In the asymptotic analysis we utilise a dimensionless curvature as a small parameter, whereas the numerical model is based on spectral methods and is free of the small-curvature limitation. The two models agree with each other well in the region where both are applicable. Simple existence conditions depending on Poisson's ratio are offered and, finally, the effect of energy build-up in a bend when the structure is excited at a resonant frequency is demonstrated.
Joint work with D. Gridin, A. Adamou and R. Craster
Professor Weinan E
Title: Atomistic and continuum models of solids
Professor Alexander Figotin
Title: Slow light in photonic crystals
Abstract: Please click here for a PDF version of this abstract
Professor Gero Friesecke
Title: Variational problems in atomistic solid mechanics
Abstract: I discuss basic variational problems associated with (exact quantum mechanical or approximate) atomistic potential energy surfaces for crystals.Recent progress includes identification of the 3D fcc (face-centered cubic) lattice as a local minimizer of the celebrated Lennard-Jones pair interaction model (joint work with Florian Theil).
Dr Sebastian Guenneau
Title: Homogenisation of photonic quasi-crystals
Abstract: Please click here for a PDF version of this abstract
Professor Richard James
Title: Lessons on structure from the structure of viruses
Abstract: As the most primitive organisms, occupying the gray area between the living and nonliving, viruses are the least complex biological system. One can begin to think about them in a mathematical way, while still being at some level faithful to biochemical processes. We make some observations about their structure, formalizing in mathematical terms some rules-of-construction discovered by Caspar and Klug. We call the resulting structures objective structures. From the mathematical viewpoint it is then seen that objective structures include some of the most important structures studied in science today: carbon nanotubes, the capsids and tails of many viruses, the cilia of some bacteria, DNA octahedra, buckyballs, and severely bent and twisted beams. The rules defining them have rigorous quantum mechanical origins. One can also see that many of the simplifications people make about atomic calculations on crystal lattices – periodicity, the Cauchy-Born rule – have analogs for these more general structures, if only one accounts for the fact that different groups are involved. This common mathematical structure seems to pave the way toward many interesting calculations for such structures: the possibility of interesting electromagnetic properties, phase transformations between them, defects, nonlinear elastic properties.
Professor Peter Kuchment
Title: On some spectral problems arising in photonics and waveguide theory
Abstract: Please click here for a PDF version of this abstract
Dr Stefan Maier
Title: Effective mode volume in plasmonic nanoresonators – towards a common description of dielectric and metallic cavities
Abstract: Recently, there has been a great amount of research effort into the electromagnetics of metallic nanostructures, due to the fact that such structures support localized electromagnetic fields below the diffraction limit of light in the form of surface plasmon-polaritons (SPPs). This localization could allow the creation of waveguides for electromagnetic energy in the visible and near-infrared regime showing lateral mode profiles far below the diffraction limit of light , thus enabling the creation of a subwavelength photonic infrastructure for on-chip computing and sensing . The generation of light volumes far below the diffraction limit between closely spaced metallic interfaces has additionally been employed for applications such as single-molecule Raman sensing on rough metallic surfaces, which has thus far not been achieved using dielectric resonators.
However, the generation of light volumes below the diffraction limit in the dielectric space surrounding a metallic nanostructure supporting a well-defined surface-plasmon mode does not imply that the effective electromagnetic mode volume V eff itself is smaller than the diffraction limit, due to the fact that upon resonance a significant amount of the electromagnetic energy resides inside the metal itself. Thus, both from a theoretical (cavity quantumelectrodynamics) as well as from a practical point of view, it is not a priori clear under which circumstances plasmonic cavities are preferable to the use of dielectric cavities. While for the latter a description of their ability to both spectrally and spatially localize electromagnetic energy is well established in the form of quality factor Q and V eff , metallic cavities have to the best of our knowledge up to this point not been described in these terms.
As a canonical example allowing analytical tractability, we analyze the electromagnetic mode confinement in terms of Q and V eff for a generic one-dimensional metal-insulator-metal heterostructure. For small enough separation of the metallic half-spaces, such a structure can indeed support modes with V eff far below the diffraction limit, taking proper account of both dispersion and the energy inside the metallic regions. Using finite-difference time-domain simulations, we show that three-dimensional metallic nanoresonators can show Q/ V eff surpassing those of their dielectric counterparts for infrared frequencies. Additionally, we present a description of surface enhanced Raman scattering in terms of Q/ V eff , which can guide the design of plamonic nanostructures for sensing purposes. We also carefully place plasmonic confinement into context with other approaches to create electromagnetic resonators with large Q/ V eff .
Stefan A. Maier, Photonics and Photonic Materials Group, Department of Physics, University of Bath , Bath BA2 7AY , UK
Oskar Painter, Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
Professor Vladimir Mazya
Title: Hadamard-type variational formulae for Green's kernels in singularly perturbed domains.
Abstract: Please click here for a PDF version of this abstract.
Dr Christof Melcher
Title: Remarks on Ginzburg-Landau approximation and applications to regularity
Professor Alexander Mielke
Title: Evolution of microstructure in shape-memory alloys: analysis and numerics
Abstract: We consider a rate-independent model for the evolution of gradient Young-measures in single crystals of SMAs like CuAlNi under a slowly varying external loading. The energetic formulation is based on an energy-storage functional which encodes the different phases and a dissipation functional which encodes energetic losses during the transformation between the different phases.
The model is formulated in terms of the macroscopic deformation and the gradient-Young measure for the microscopic deformation gradient. We provide existence result for the model and show that a suitably relaxed space-time discretization produces discrete approximations which contain
subsequences converging to solutions. The discretization if the gradient Young-measures is based on piecewise constant nested laminations as introduced by Ortiz et al., but the mean values are only taken to be close to the macroscopic deformation gradient.
M. Kruzik, A. Mielke, T. Roubicek: Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi, Meccanica 2005, to appear.
Professor Graeme Milton
Title: Anomalous localized resonance, a proof of superlensing in the quasistatic regime, and limitations of superlenses
Abstract: One of the interesting features of composite materials is that they can exhibit properties unlike any known material, such as a negative Poisson's ratio. This decade there has been a lot of work in designing and fabricating materials with an effective negative refractive index. Veselago (1968) realized that a slab of material with a refractive index of -1 would act as a lens. Pendry (2000) went further and suggested that the Veselago lens would act as a superlens having the remarkable property of providing a perfect image of an object in contrast to conventional lenses which are diffraction limited and only able to focus a point source to an image having a diameter of the order of the wavelength of the incident radiation. This is now a rapidly expanding field, attracting controversy and interest from theorists, experimentalists and numerical modelers.
Enlarging upon work of Nicorovici, McPhedran and Milton (1994) we have a rigorous proof that in the quasistatic regime a cylindrical superlens can successfully image a dipolar line source in the limit as the loss in the lens tends to zero. In this limit it is proved that the field blows up to
infinity in two sometimes overlapping annular anomalously locally resonant regions, one of which extends inside the lens and the other of which extends outside the lens. If the source is in the resonant region then surprisingly the total energy absorbed by the lens goes to infinity as the
loss in the lens goes to zero. If the object being imaged responds to an applied field it is argued that it must lie outside the resonant regions to be successfully imaged. If the image is being probed it is argued that the resonant regions created by the probe should not interfere with either the probe itself or the object being imaged, if that object responds to an applied field. Perfect imaging in a cylindical superlens is shown to extend to the static equations of magnetoelectricity or thermoelectricity provided they have a special structure which makes these equations equivalent to the quasistatic equations, as follows from previous work of Cherkaev and Gibiansky (1994) and Milton (2002).
Dr Roger Moser
Title: A Ginzburg-Landau type theory arising in thin-film micromagnetics
Abstract: We consider the asymptotic behavior of variational problems from micromagnetics, in particular evolution problems given by the Landau-Lifshitz equation, in a special thin-film regime. The most remarkable aspect of the limiting theory is the appearance of boundary vortex structures. They can be regarded as a variant of Ginzburg-Landau vortices, and the known tools from that theory prove also useful for the micromagnetic thin-film limit.
Professor Stefan Mueller
Title: A hierarchy of theories for thin elastic bodies
Abstract: The search for simplified theories for thin elastic bodies is as old as elasticity theory itself. In fact by now there exist a large variety of such theories, usually derived on the basis of certain apriori assumptions on the behaviour of the solutions of the full three-dimensional theory, and their precise status and range of validity is often unclear. It was only about ten years ago that
a lower dimensional theory was derived rigorously from the full, nonlinear three-dimensional theory.
Recently Friesecke, James and M\"uller have obtained a full hierachy of limiting theories (ordered by the scaling of the elastic energy as a function of thickness) by variational methods. These
include Kirchhoff's nonlinear plate theory as well as the von K\'arm\'an plate theory. In the talk I will review some of the central ideas behind these results, in particular a geometric rigidity estimate. I will also give an outlook on largely unexplored scaling regimes (which include conjectured
relations to the crumpling of elastic sheets) and on a first result beyond the variational setting.
Professor Michael Ortiz
Title: Discrete dislocations in crystals and continuum plasticity
Dr Felix Otto
Title: The onset of switching in thin film ferromagnetic elements: A bifurcation
analysis by Gamma--convergence
Abstract: Motivation for this joint work with Ruben Cantero-Alvarez is the following experimental observation for thin film ferromagnetic elements. Elements with elongated rectangular cross--section are saturated along the longer axis by a strong external field. Then the external field is slowly
reduced. At a certain field strength, the uniform magnetization buckles into a quasiperiodic domain pattern which resembles a concertina. Our hypothesis is that the period of this pattern is the frozen--in period of the unstable mode at critical field.
Starting point for the analysis is the micromagnetic model which has three length scales. We rigorously identify four scaling regimes for the critical field. One of the regimes, which has been overseen by the physics literature. is displays an oscillatory unstable mode, which we identify
asymptotically. In this parameter regime, we identify a ``normal form'' for the bifurcation. It turns out to be coercive, whence the bifurcation is supercritical.
The analysis amounts to the combination of an asymptotic limit with a bifurcation argument. This is carried out by a suitable ``blow--up'' of the energy landscape in form of Gamma--convergence. Numerical Simulation of the normal form visualizes a homotopy from the unstable mode with its
harmonic oscillation to the strongly nonlinear concertina pattern.
Professor Andrey Piatnitski
Title: Homogenization of singular random structures and random measures
Abstract: The talk will focus on scaling up problems for a family of random stationary measures. We will discuss the main properties of functional spaces introduced in terms of these measures, and then consider homogenization problem for differential operators and variational functionals defined on these spaces.
Professor Grigory Panasenko
Title: The partial homogenization:continuous and discrete models
Abstract: The partial homogenization method will be discussed for some PDEs and for some discrete models. This method is related to the method of asymptotic partial domain decomposition (see G.Panasenko "Multi-scale Modelling for Structures and Composites", Springer, 2005). The equation is homogenized in some subdomain and the special asymptotically exact interface conditions are set between the homogenized and non-homogenized parts. The error estimates are proved for the difference of the exact solution and the solution of the partially homogenized model.
Professor Etienne Pardoux
Title: Homogenization of periodic degenerate PDEs
Abstract: Please click here for a PDF version of this abstract
Dr Grigoris Pavliotis
Title: A multiscale approach to Brownian Motors
Abstract: The problem of Brownian motion in a periodic potential, under the influence of external forcing, which is either random or periodic in time, is studied. Multiscale techniques are used to derive general formulae for the steady state particle current and the effective diffusion tensor. These
formulae are then applied to calculate the effective diffusion coefficient for a Brownian particle in
a periodic potential driven simultaneously by additive Gaussian white and coloured noise. Our theoretical findings are supported by numerical simulations.
Professor Pedro Ponte Castaneda
Title: Macroscopic behavior, microstructure evolution and loss of ellipticity for elastomeric composites
Abstract: This work presents the application of a recently proposed “second-order” homogenization method [1,2] to generate estimates for effective behavior, microstructure evolution and loss of ellipticity in hyperelastic composites with random microstructures that are subjected to finite deformations. Two extreme cases are considered that are illustrative of the physics of the problem: porous elastomers and fiber-reinforced elastomers. The main concept behind the method is the introduction of an optimally selected “linear comparison composite,” which can then be used to convert available linear homogenization estimates into new estimates for the nonlinear hyperelastic composite. In this paper, explicit results are provided for matrix materials with isotropic and strongly elliptic constitutive behavior. In spite of the strong ellipticity of the matrix phase, the homogenized “second-order” estimates for the overall behavior are found to lose ellipticity at sufficiently large deformations corresponding to the possible development of shear band-type instabilities [3]. The reasons for this result have been linked to the evolution of the microstructure, which, under appropriate loading conditions, can induce “geometric softening” leading to overall loss of ellipticity. Furthermore, for the rigidly reinforced case, the homogenization procedure recovers the exact overall incompressibility constraint in the limit of incompressible behavior for the matrix. Correspondingly, for the porous case, the homogenization procedure recovers the exact evolution of the porosity under an arbitrary finite-deformation history.
References
[1] P. Ponte Castañeda. “Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory.” J. Mech. Phys. Solids 50 (2002): 737-757.
[2] O. Lopez-Pamies and P. Ponte Castañeda. “Second-order estimates for the macroscopic response and loss of ellipticity in porous rubbers at large deformations.” J. Elasticity 76 (2004): 247-287.
[3] G. Geymonat, ,S. Müller and N. Triantafyllidis. “Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity.” Arch. Rational Mech. Anal. 122 (1993): 231—290.
Professor Philip Russell
Title: Light and Matter in Small Spaces
Abstract: Periodically structured materials, scaled suitably so that the number of resonances in each repeating unit is strongly restricted, permit waves (acoustic, electromagnetic) to be manipulated in dramatic ways. When designed so that both the low and high density units are anti-resonant over a range of frequencies, a forbidden band forms and the material becomes "dark", i.e., it cannot support travelling waves. A highly successful example of such a structurally "crystalline" material is photonic crystal fibre (PCF) - a hair-thin thread of glass with a lattice of hollow channels running along its length. PCF makes it possible, e.g., for light to be trapped inside an empty micro-tube that can be loaded with particles, liquids or gases. Gigantic enhancements result in the interaction of light with matter - and matter with light. The strong discontinuity in acoustic properties between air and glass gives rise to phononic band gaps and families of multi-GHz guided acoustic modes, which themselves interact strongly with light, creating unusual multi-peaked Brillouin back-scattering spectra. Through its unique and varied characteristics, PCF has escaped the confines of conventional fibre optics, in the process creating a renaissance of new possibilities in many diverse areas of fundamental and applied research. And of course, the efficient numerical analysis of the optical and acoustic properties of PCF - a vital component in the research - poses an interesting and non-trivial mathematical challenge.
Dr Florian Theil
Title: Crystallization in two dimension
Abstract: Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type
E(y)= sum_{0<i<j<=N} V(|y(i)-y(j)|)
where y(i) in R^2 is the position of particle i and V(r) is the pair-interaction energy of two particles which are placed at distance r. Due to the Mermin-Wagner theorem it can't be expected that at finite temperature this system exhibits long-range ordering.
Under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential, it can be shown that the ground states form a triangular lattice.
Dr. Stanislav Volkov
Title:Modelling a polling system with 3 queues and 1 server in an overload regime
Abstract: We consider a queuing system with three queues
and just one server, in the regime when the system overall is overloaded,
but no individual queue is. The service discipline is as follows. Once the
server is at node j, it stays there until it serves all customers in that queue.
After this, the server moves to the "more expensive" of the two
queues. We will show that a.s. there will be a periodicity in the order of
services, as suggested by the behavior of the corresponding dynamical systems;
we also study the cases (of measure 0) when the dynamical system is chaotic, and
prove that then the stochastic one cannot be periodic either.
(Based on a joint work with M.Menshikov, I.MacPhee and S. Popov)
Professor Steve Wiggins
Title: High Dimensional Dynamical Systems: Theory and Computational Realisation for Molecular Dynamics
Abstract: Please click here for a PDF version of this abstract.
Professor Vassili Zhikov
Title: Uniqueness and Non-uniqueness for Linear Elliptic Equations and Stochastic Homogenization
Abstract: We consider an elliptic equation $-\div(\nabla u + a u)=f$ with solenodial vector $a$, which is related to diffucion in incompressible flow. We give examples of non-uniqueness and then define approximation solutions. Then we consider a more general elliptic equation $-\div(\nabla u + A \nabla u)=f$ with a skew-symmetric matrix $A$ (related to diffusion in turbulent flows). We give a sufficient condition of uniqueness and define so-called variational solutions. The above uniqueness questions originally emerged from homogenisation theory, so we briefly consider homogenisation problems and related Nash-Aronson type estimates and the Central Limit Theorem. For comparison, we consider also symmetric problems, namely equation $-\div(\rho u)=f$ with Lavrentiev phenomenon, define for them $H$-solutions, $W$-solutions, variational solutions, accessible or approximation solutions, and give model examples.
