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2004-2005 Materials SeminarYitzhak Rabin (Bar-Ilan University, Israel rabin@mail.biu.ac.il) Modeling of bio-filaments: elasticity and fluctuations combined
Abstract: New studies of DNA molecules and protein filaments indicate the need to go beyond standard polymer models and construct a theory of fluctuating elastic filaments that would account for both bending and twist rigidity of these objects, as well as for their intrinsic shape. We discuss the intrinsic geometry, theory of elasticity and statistical mechanics of such objects, present a new type of Monte Carlo simulation based on this theory, and consider the implications of the theory for single molecule experiments on stretching, twisting and cyclization of double stranded DNA molecules. You can find papers on this research at: http://www.ph.biu.ac.il/fac.php?id=33 This talk is part of the Friday Focus Group: Soft Matter.
September 8, 2004, 11:15
am, Lind Hall 409 Stochastic conformal mapping and transport-limited aggregation Abstract: Conformal mapping provides elegant formulations of interfacial dynamics in two dimensions. Continuous conformal-map dynamics is well known for Laplacian growth, where the interfacial velocity is the gradient of a harmonic function, such as the pressure field in viscous fingering. Recently, stochastic Laplacian growth, which describes diffusion-limited aggregation and dielectric breakdown, has been formulated in terms of random, iterated conformal maps, by Hastings and Levitov (1998). Here, we extend these powerful analytical methods to a class of non-Laplacian growth phenomena, e.g. driven by advection-diffusion or electro-diffusion on flat or curved surfaces. As an example, we focus on the fractal aggregation of diffusing particles in a fluid flow. This talk is part of the Wednesday Focus Group: Singularities. September 17, 2004, 9.05 am,
Lind Hall 409 LC phases formed by banana-shaped molecules: possible consequences of tetrahedratic order Abstract: Some of the most interesting developments in liquid crystals over the last few years are related to the physical properties of the liquid crystalline phases exhibited by banana-shaped (bent-core) molecules. Experimental evidence to date suggests that this is indeed a new sub-field of thermotropic liquid crystals [1,2]. The biggest challenge is the condensed state known as B7 that grows into the isotropic liquid with many different growth patterns including spirals with both hands [3]. Neither the symmetry nor the physical properties of B7 are understood. Indeed, evidence so far is that B7 does not have a simple ground state [4] and shows many different optically anisotropic patterns when cooled from the isotropic phase. These patterns grow vigorously when the system is far from equilibrium e.g. after a temperature decrease or in an electric field, then shrink once equilibrium is reached leaving behind in many instances an isotropic liquid [5]. A tetrahedratic phase, T, is described by four unit vectors oriented along the four tetrahedral corners of a cube [6]. The tetrahedratic phase, T, is odd under parity. As tetrahedra are not space-filling objects, they can be viewed equally well as precursors of condensed lamellar and/or columnar mesophases. Tetrahedratics are thus a a model for studying frustrated lamellar and columnar liquid crystalline phases with many different condensed states that are energetically very close together [4]. Two years ago [7], in a first step, we analyzed the coupling of flow to electric fields and temperature gradients assuming that the symmetry of the tetrahedratic phase is unchanged by the external forces. Here we discuss what happens when the tetrahedral angle is allowed to change under the influence of external forces and fields [8]. We also analyze how an external electric field applied to an isotropic tetrahedratic phase can induce orientational order Q as well as smectic layering [9]. The relation of our analysis to recent experimental observations by Weissflog et al. [10] will be discussed. [1] H.R. Brand, H. Pleiner and P.E. Cladis, Eur. Phys. J. B 7, 347 (1998); 31, 147 (2003). [2] G. Pelzl, S. Diele and W. Weissflog, Adv. Mat. 11, 707 (1999). [3] G. Pelzl, S. Diele, A. Jakli, Ch. Lischka, I. Wirth and W. Weissflog, Liq. Cryst. 26, 135 (1999). [4] P.E. Cladis, H. Pleiner and H.R. Brand, Ferroelectrics 243, 221 (2000). [5] Y. Yusuf, Y. Hidaka, S. Kai, H.R. Brand, P.E. Cladis, W. Weissflog and G. Pelzl, Ferroelectrics 276, 171 (2002). [6] L.G. Fel, Phys. Rev. E52, 702 (1995). [7] H.R. Brand, H. Pleiner and P.E. Cladis, Eur. Phys. J. E 7, 163 (2002). [8] P.E. Cladis, H. Pleiner and H.R. Brand, Eur. Phys. J. E 11, 283 (2003). [9] H.R. Brand, P.E. Cladis and H. Pleiner, submitted. [10] W. Weissflog, M.W. Schroeder, S. Diele and G. Pelzl, Adv. Mat. 15, 630 (2003). This talk is part of the Friday Focus Group: Soft Matter. September 17, 2004, 11:15
am, Lind Hall 409 Efficient sampling of molecular conformations Abstract: Molecular dynamics trajectories are of potential interest as sampling tools for applications in materials simulation and biomolecular modelling. I will explain a practical and purely dynamical approach to this problem based on (i) formal modifications to enable sampling from the canonical ensemble (e.g. Nos\'e's device), (ii) robust ergodicity enhancing extensions such as Recursive Multiple Thermostats, and (iii) the construction of efficient and stable symplectic numerical methods. This talk is part of the Friday Focus Group: Soft Matter. October 4, 2004, 11:15 am, Lind Hall 409Ellad B. Tadmor (Technion - Israel Institute of Technology tadmor@tx.technion.ac.il http://tx.technion.ac.il/~tadmor) Quasicontinuum challenges: finite temperature and dynamics This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. October 6, 2004, 11:15 am, Lind Hall 409 Analysis of superconductivity with inhomogeneities. Singularities focus group This talk is part of the Wednesday Focus Group: Singularities. October 8, 2004, 11:15 am, Lind Hall 409 Stereo-selective swelling of gels with imprinted phase chirality This talk is part of the Friday Focus Group: Soft Matter. October 11, 2004, 11:15 am,
Lind Hall 409 Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics Abstract: In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed, that aim at coupling anatomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical analysis for such a coupling in a one-dimensional setting. We study both the general case of a convex energy and a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs toaccount in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. October 13, 2004,
11:15 am, Lind Hall 409 Hall effect and pattern formation in superconductivity Abstract: I intend to shortly review the Hall effect and the time-dependent Ginzburg-Landau equations, and then present numeric results obtained for a 2D setup with applied current and magnetic field perpendicular to one another. We find situations in which the order parameter differs significantly from zero in a set of islands that appear to form a periodic structure. When the pattern of islands becomes irregular, it moves in or against the direction of the current and a Hall voltage is found. Tiny differences in the initial state may reverse the sign of the Hall voltage. When the average Hall voltage vanishes, the local Hall voltage does not necessarily vanish. This talk is part of the Wednesday Focus Group: Singularities. October 15, 2004, 11:15
am, Lind Hall 409 Paper crumpling and other topics This talk is part of the Friday Focus Group: Soft Matter. October 18, 2004, 11:15 am,
Lind Hall 409 A framework for dynamic scaling Abstract: Classical (1930s!) methods from probability theory work amazingly well to understand "dynamic scaling" in some toy models of coalescence. They give optimal results, interesting counterexamples, and fit nicely with dynamical systems ideas. This talk will be a leisurely introduction to these methods. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. October 20, 2004, 11:15
am, Lind Hall 217 Note room change! Weak Jacobians as a tool in analyzing Ginzburg-Landau vortices Abstract: I will describe the motion of a weak Jacobian, focusing on the theory developed by Jerrard-Soner, and describe an application to establishing existence of permanent currents. I will also survey some non-existence results for permanent currents. This talk is part of the Wednesday Focus Group: Singularities. October 22, 2004, 11:15 am,
Lind Hall 217 Note room change! What's wrong with the Langevin equations? This talk is part of the Friday Focus Group: Soft Matter. November 1, 2004, 11:15
am, Lind Hall 409 Mathematical strategies for stochastic multiscale problems: coarse-graining, loss of information and adaptivity Abstract: Hybrid deterministic/stochastic systems, arising as couplings of microscopic models and deterministic macroscopic equations are commonplace in a wide array of applications, ranging from catalysis and deposition processes to stochastic models for tropical and open ocean convection. A major challenge in all these problems arises in the direct numerical simulation of realistic size systems due to scale and model disparities, while due to nonlinear interactions across a wide range of scales, the stochasticity inherited from the microscopic model can play a subtle but important role in the dynamic behavior of the overall system. In this talk we attempt to address directly or indirectly these issues; one of the primary tools we have developed for this purpose is a new mathematical framework for the hierarchical stochastic coarse-graining of microscopic dynamics. Computational comparisons of coarse-grained and microscopic simulations along with accompanying rigorous estimates on the loss of information between the time-dependent coarse-grained and microscopic probability distribution functions highlight the validity regimes of the method. Furthermore we discuss spatial adaptivity for microscopic simulations constructed using the coarse-grained stochastic processes tools we have already developed. The adaptivity criterion is based, in analogy to PDE finite element methods, on a posteriori estimates on the loss of information between the coarse-grained and the microscopic pdf. The presented results are joint work with A. J. Majda (Courant), P. Plechac (Warwick), A. Sopasakis (UMass), J. Trashorras (Paris IX) and D.G. Vlachos (Chem. Eng. Delaware). This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. November 3, 2004, 11:15
am, Lind Hall 409 Multiple time scale dynamics for a coupled nonlinear Schrodinger equations This talk is part of the Wednesday Focus Group: Singularities. November 3, 2004, 2:30
pm, Vincent Hall 570 (note room change!) What's wrong with the Langevin equations? Part II This talk is part of the Wednesday Focus Group: Singularities. November 8, 2004, 11:15 am,
Lind Hall 217 (Note: room change!) Mathematical questions in the transitions between scales (1) The emergent laws are often continuum laws which break down at defects or singularities. The continuum evolution laws must be supplemented with evolution laws for the motion and properties of the defects. Is there a way to classify the possible continuum evolution laws for these defects in a systematic way? Brief examples will include laws for crack growth laws, dislocation dynamics, and facet edge evolution under etching. (2) The emergent laws often involve many parameters or entire unknown functions. Fitting these to data often lead to ill--determined problems: the Hessian of the cost near the best fit often has eigenvalues spaced logarithmically roughly equally over ranges that can span ten or more orders of magnitude. (One such problem leads to the Hilbert matrix, a classic example of badly conditioned matrices.) Our group calls these "sloppy models", reflecting the slop in the parameter values that are consistent with the data. Is there an understanding of this qualitative behavior? Brief examples will include deriving error estimates for models of signal transduction networks and for interatomic potentials in materials. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. November 10, 2004, 11:15 am,
Room 570 Vincent Hall (note room change!) Viscoelastic fluids: a energetic variational approach This talk is part of the Wednesday Focus Group: Singularities. November 12, 2004, 11:15
am, Lind Hall 409 Title: TBA This talk is part of the Friday Focus Group: Soft Matter. November 15, 2004, 11:15 am,
Lind Hall 409 A multiscale model for the dynamics of solids at finite temperature Abstract: I will present a multiscale method for the modeling of dynamics of solids at finite temperature. In this method, the molecular dynamics is reformulated to the form of conservation laws, which are to be coupled with macroscale descriptions. I will specifically discuss the following issues: 1. extracting constitutive or kinetic relations from atomistic models, This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. November 16, 2004, 11:15 am,
Lind Hall 409 Superfluid passing an obstacle This talk is part of the Wednesday Focus Group: Singularities. November 22, 2004, 11:15 am,
Lind Hall 409 Effective Hamiltonians for transforming materials This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. November 29, 2004, 11:15
am, Lind Hall 409 Title: TBA This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. December 1, 2004, 11:15
am, Lind Hall 409 Diffused interface motion with surface tension and transport effect Abstract: We discuss a problem of phase boundary
motion with hydrodynamic effect via a diffused interface model. The energy law
leads us to study the problem where the velocity of the phase boundary is determined
by the sum of mean curvature and fluid velocity. Via the Allen-Cahn type equation
coupled with a viscous flow equation, we aim to establish the existence of a
global weak solution in the setting of geometric measure theory. An analogue
of Serrin's condition for the Navier-Stokes turns up for the existence. This talk is part of the Wednesday Focus Group: Singularities. December 3, 2004, 10:10
am, Lind Hall 409 Mathematical and computational models for dislocation dynamics This is a special presentation. December 6, 2004, 11:15
am, Lind Hall 409 Long time simulaton of stochastic differential equations Abstract: I will discuss some methods for the long time simulation of stochastic ordinary differential equations. I will show the short comings of the forward Eular method and give some remedies. In perticular, I will discuss some simple addaptive ideas. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. Note: The speaker is also scheduled to give a talk on Friday, 12/10/2004 at the UMN Probability seminar. The title of his talk is "Ergodicity of the 2D Navier-Stokes equations under very degenerate forcing." The abstract will appear in "What's Happening Now" accessible on the sidebar of the IMA webpage. January 11, 2005, 10:10
am, Lind Hall 409 Introduction: One mode viscoelasticity January 12, 2005, 11:15
am, Lind Hall 409 Schrodinger Vortices II Abstract: Under the limit of large Ginzburg-Landau parameter, Schrodinger vortices condense down to point particles and satisfy a simple Kirchhoff's Law - just like Euler point vortices. Given some technical variational results, I will describe the proof of vortex dynamics for the Schrodinger-Ginzburg-Landau equations.(This talk is a continuation of the December 8 lecture.) This talk is part of the Wednesday Focus Group: Singularities. January 14, 2005, 11:15
am, Lind Hall 409 Coupled transport: Two-fluid model January 18, 2005, 3:35 pm, Lind
Hall 409. Note time change! Modeling and simulation of materials under highly dynamic loads Abstract: The simulation of the mechanical response
of materials has been the task of computational mechanics since its inception.
Only now, however, can we seriously envision reliable and accurate simulations
for materials undergoing highly dynamic loads, such as those generated by impacts,
blast and detonation of energetic materials. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing but since January 17 coincides with Martin Luther King Day it is rescheduled for Tuesday instead. January 19, 2005, 11:15
am, Lind Hall 409 On the motion of boundary vortices Abstract: In a certain thin-film limit of the micromagnetic energy, the magnetization develops singularities on the boundary, so-called vortices, that share many features with the interior Ginzburg-Landau vortices as described by Bethuel-Brezis-Helein. For example, the positions of the minimizing vortices are governed by the minimization of a finite-dimensional renormalized energy. I will present ongoing work to show that also the gradient flow motion is essentially given by the gradient flow of the renormalized energy, using the method of Gamma-convergence of gradient flows of Sandier-Serfaty. This talk is part of the Wednesday Focus Group: Singularities. January 21, 2005,
11:15 am, Lind Hall 409 Modeling under constraints: liquid crystals January 24, 2005, 11:15 am,
Lind Hall 409 Computations of multivalued solutions of nonlinear PDEs Abstract: Many physical problems arising from high frequency waves, dispersive waves or Hamiltonian systems require the computations of multivalued solutions which cannot be described by the viscosity methods. In this talk I will review several recent numerical methods for such problems, including the moment methods, kinetic equations and a level set method. Applications to the semiclassical Schroedinger equation and Euler-Piosson equations with applications to modulated electron beams in Klystrons will be discussed. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. January 26, 2005,
11:15 am, Lind Hall 409 Elliptic and parabolic spectral problem with the p-Laplacian This talk is part of the Wednesday Focus Group: Singularities. January 28, 2005,
10:00 am, Lind Hall 409 Non-homogeneous systems: surface effects January 31, 2005, 11:15 am, Lind Hall 409 Pinning and de-pinning phenomena in materials interfacial problems This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. February 2, 2005, 11:15 am,
Lind Hall 409 Singular solutions to a regular problem Abstract: The n-dimensional (quasi)conformal mappings are defined by a first-order partial differential relation (pdr): u(x) K, with a set K of n n matrices that is very regular in the sense of Morrey's quasiconvexity. However, such a pdr can have very singular solutions if considered outside the natural Sobolev space. In this talk, I will discuss how Müller- Sverák's idea of the Gromov convex integration method can be applied to construct the singular solutions with a dense set of singularities. This talk is part of the Wednesday Focus Group: Singularities. February 4, 2005, 11:15
am, Lind Hall 409 A multiscale model for the dynamics of solids Abstract: At the atomic scale, solids can be modeled by molecular mechanics or molecular dynamics, which have become very useful tools in studying crystal structure, defect dynamics and material properties. However due to the computational complexity, the application of these models are usually limited to very small spatial and temporal scales. On the other hand continuum models, such as elasticity, elastodynamics and their finite element (or finite volume) formulations, have been widely used to study processes at much larger scales. But the constitutive relation involved in these continuum models may be ad hoc, and fails to account for the presence of microstructure in the material. In this talk I will present a multiscale model, which couples the atomistic and continuum models concurrently. The macroscale model evolves the system at continuum scale, and the atomistic model, which only involves a small number of atoms, estimate the constitutive data and defect structure. I will show the estimate of the modeling error as well as various applications of this new model. This talk is part of the Friday Focus Group: Soft Matter. Informal
Discussion: Wednesday, February 9, 2005, 3:35 pm, Lind Hall 409 Ab initio molecular dynamics from solutions to surface chemistry Abstract: Molecular dynamics, the technique in which Newton's equations of motion for a given system are solved numerically subject to initial and boundary conditions, has become one of the most important methods for theoretical investigations of complex condensed phase processes. Newton's equations specify how the individual atoms in a system move under the action of the forces between them, and hence it is necessary to specify these forces in order to perform a molecular dynamics calculation. This can be accomplished either by postulating an empirical force law, in which simple functional forms are used to describe chemical bond, electrostatic and Van der Waals interactions, or by computing the forces directly from the electronic Schrödinger equation at each step in the calculation. The former method allows very large systems, such as proteins and other biological macromolecules in solution to be studied over very long time scales but are inherently limited. The latter method, known as ab initio molecular dynamics (AIMD), allows chemical reactions, where bonds are broken and formed, to be studied with a high degree of accuracy but requires large amounts of computer time while permitting access only to very short time scales. Despite these limitations, AIMD has had a very signicant impact in a number of application areas of chemical, biological and technological importance. In this talk, I will discuss the AIMD technique and describe several such applications. I will demonstrate how AIMD has elucidated the underlying microscopic mechanisms of long-range proton transport in hydrogen-bonded liquids and solids, a problem of importance in the design of proton-exchange membranes for fuel cells, and I will show how AIMD has yielded new insights into how conjugated dienes interact with semi-conductor surfaces, a problem of current interest in molecular electronics. This talk is part of the Molecular Dynamics and Sampling Focus Group despite the day difference. February 11, 2005, 2:30
pm, EE/CS 3-180 Note room change! Enhanced conformational sampling via novel variable transformations and very large time-step molecular dynamics Abstract: One of the computational grand challenge problems is to develop methodology capable of sampling conformational equilibria in systems with rough energy landscapes. If met, many important problems, most notably protein folding, could be significantly impacted. In this talk, I will present two new approaches for addressing this problem. First, I will show how molecular dynamics can be combined with a novel variable transformation designed to warp configuration space in such a way that barriers are reduced and attractive basins stretched. This method rigorously preserves equilibrium properties while leading to very large enhancements in sampling efficiency. Next, a new very large time-step molecular dynamics method will be introduced that overcomes the resonances which plague many molecular dynamics algorithms. The performance of the methods is demonstrated on a variety of systems including liquid water, long polymer chains simple protein models, and oligopeptides. This talk is part of the Molecular Dynamics and Sampling Focus Group despite the day difference. February 14, 2005, 11:15
am, Lind Hall 409 Multiscales and waves in high contrast photonic crystals Abstract: Photonic crystals are special materials designed for guiding waves along optical fibers, being periodic media displaying the "band gap" effect. If the "component" materials have highly contrasting properties, then the band gaps can be characterized explicitly using "non-standard" homogenization theory of "double porosity" type. This allows to obtain rigorous results on the location of the gaps and on the existence and the pattern of localisation of of the eigenstates in the gaps. We review recent results, as well as other associated "multiscale" effects for current and future study: effects of various microgeometries, effects of "finite periodicity" on leaks, "small" inclusions, etc. This project is a part of activities of recently established Bath Institute for Complex Systems (BICS), within a large initiative of Mathematics Program of British Engineering and Physical Sciences Research Council (EPSRC) on "Critical Mass in Interdisciplinary Mathematical Research", with BICS focussion on multiscales and networks, see http://www.bath.ac.uk/math-sci/BICS/themes.htm This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. February 18, 2005, 11:15
am, Lind Hall 409 Investigation of a Cholesteric Liquid Crystal Film with Negative Dielectric
Anisotropy: 1-D Analysis Cholesteric liquid crystals have an intrinsic tendency to form twisting configurations of the average orientation of the elongated liquid-crystal molecules. The resulting spatially periodic patterns can be used to advantage in certain technological applications. However, this intrinsic twisting tendency makes it difficult to control the orientational properties of these materials. We are in the process of analyzing such a system. It consists of a thin film of a cholesteric liquid crystal with a negative dielectric anisotropy. In this system, three main equilibrium configurations are observed experimentally: a uniform director field (constant throughout the film), a translationally independent cholesteric texture (which is a function only of the position across the narrow film thickness), and a cholesteric finger texture (which is a function of two space variables, 1-D periodic in the plane of the film). We will talk about the bifurcation and phase behavior of the first two configurations (uniform vs translation independent cholesteric), which can be analyzed by fairly simple means but already hold some surprises. The more difficult exploration of the cholesteric finger texture is in progress. This talk is part of the Friday Focus Group: Soft Matter. February 22, 2005, 1:25 am
pm, Lind Hall 409 What makes molecular dynamics work? Abstract: The numerical treatment of molecular dynamics (MD) is problematic due to the exponential growth of error with time. The technique of shadowing can show that the computed solution is very nearly the exact trajectory for a slightly different initial value. This type of result would be adequate, and it has been shown to work for fairly complicated problems. However, it is highly unlikely that a (useful) shadowing result is possible for highly elliptic dynamical systems such as MD. Rather, it is suggested that a general treatment for MD be based on randomness in the initial values and applying the concept of weak convergence from stochastic differential equations. Weak convergence requires that expectations be accurately computed for smooth distributions of initial values. In this setting it is plausible that accurate solutions can be obtained for very long intervals of time. There remain questions concerning the accuracy of different numerical integrators in this weak sense, and these questions are explored. In the case of ergodic Hamiltonian systems, evidence is presented suggesting m that consistent integrators give convergent results on very long intervals if the integrator nearly conserves energy on very long intervals and conserves volume in phase space. However, for certain practical reasons it seems that the stronger property of being symplectic is needed, and this is explained. This talk is part of the Molecular Dynamics and Sampling Focus Group. February 23, 2005 11:15 am,
Lind Hall 409 Title: TBA This talk is part of the Wednesday Focus Group: Singularities. February 23, 2005 1:25 pm, Lind Hall 409 Using classical density functional theory for long-time simulation in materials science Abstract: Molecular Dynamics is a method for simulating materials at microscopic length and time scales. Often we are happy to resolve atomic length scales, but would rather not resolve such fine time scales. Indeed, the time step-length used in MD is typically restricted by the smallest time scale in the system. This makes long-time simulation, of the type necessary to observe grain-boundary motion (for example), unfeasible. I will present a general framework for overcoming this difficulty while still maintaining detailed microscopic information about the system. The strategy is to coarse-grain the system in time: this leads to a time-averaged density field, for which we can derive approximate equations of motion. The motion of this field is described by a non-local partial differential equation which no longer contains the undesired time-scales of the original system. I will demonstrate this procedure for two model systems: the Lenard-Jones model of Argon and the Stillinger-Weber model of Silicon. This talk is part of the Molecular Dynamics and Sampling Focus Group. February 24, 2005 10:10am, Lind Hall 409 Joint molecular dynamics discussion Abstract: Can we calculate time correlation functions accurately well beyond timescales for which trajectories are accurate? The discussion is part of the Molecular Dynamics and Sampling Focus Group. February 25, 2005,
11:15 am, Lind Hall 409 Microscale flow and transport problems arising in surfactant rheology, surface patterning, and polymer electrophoresis Abstract: Fluid flow and transport processes occurring on length scales of microns or less often involve phenomena which are unimportant at larger length scales. Although such phenomena can complicate our ability to understand and design microscale flow and transport processes, they also offer opportunities to engineer novel and useful effects. Three examples will be presented in this talk in support of this idea. In the first example, we consider an instability that arises when a fluid flows past a soft elastic solid. Experiments and theoretical calculations suggest that this instability is responsible for certain rheological phenomena observed in surfactant solutions, and that it may also be useful for enhancing mixing in microscale flows. In the second example, we consider a thin liquid film dewetting near a polymer gel. Numerical simulations using a lubrication-theory-based model which couples the fluid and gel dynamics indicate that the dewetting process can be used to template topographical structures on the gel surface. In the third example, we consider polymer electrophoresis through a narrow slit. Brownian dynamics simulations show that the relationship between the chain transit velocity and chain length depends in a sensitive way on slit dimensions, and suggest the existence of an optimum slit width for electrophoretic separations. This talk is part of the Friday Focus Group: Soft Matter. February 28, 2005,
11:15 am, Lind Hall 409 Natural patterns and minimizers of the regularized Cross-Newell energy Abstract: The Cross-Newell equation is a phase equation that describes the large scale dynamics of a class of pattern forming systems. This equation is variational, and the long time behavior is given by the minimizers of the associated Cross-Newell (Aviles-Giga) energy. However, the "phase" is not a function in the usual sense, since it is multiple valued, and it's gradient is a "two-valued vector field." I will present some recent ideas on how to account for these factors in the minimization of the energy, and also some analytical/numerical results from this approach. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. March 2, 2005,
11:15 am, Lind Hall 409 Stable and finite Morse index solutions on bounded domains with small diffusion This talk is part of the Wednesday Focus Group: Singularities. March 4, 2005,
11:15 am, Lind Hall 409 Mathematical models for biaxial liquid crystals phases The search for thermotropic biaxial phases has recently found some firm evidence of their existence. It has rightly been remarked that this "announcement has created considerable excitement, for it opens up new areas of both fundamental and applied research. It seems that a Holy Grail of liquid-crystal science has at last been found" (see G.R. Luckhurst, Nature 430, 413 (2004)). In this lecture, I shall present a mean-field model that has the potential to describe such an evanescent phase of matter. More specifically, I show the outcomes of a bifurcation analysis of the equilibrium equations and I illuminate the complete phase diagram, which exhibits two tricritical points. The predictions of this analysis are also qualitatively confirmed by a Monte Carlo simulation study. One of the main conclusions is that two order parameters suffice to label all equilibrium phases, though they exhibit different bifurcation patterns. This talk is part of the Friday Focus Group: Soft Matter. March 8, 2005,
2:30 pm, Lind Hall 409 Direct calculation of crystal-melt interfacial free energies from molecular simulation Note date change! Abstract: The crystal-melt interfacial free energy, the work required to create a unit area of interface between a crystal and its own melt, is a controlling property in the kinetics and morphology of crystal growth and nucleation, especially in the case of dendritic growth. Despite the technological importance of this quantity, accurate experimental data is difficult to obtain. The paucity of experimental measurements has motivated the development of a variety of novel computational methods to determine the interfacial free energy via molecular simulation. After a short tutorial on thermodynamic integration techniques for free energy calculation, I will introduce our method of cleaving walls for the calculation of the crystal-melt interfacial free energy, and a competing method based on fluctuation spectra. Results for a variety of simple systems will be presented to give a broad picture of the interaction and crystal structure dependence of the interfacial free energy. The results will be discussed in relation to popular empirical theories of the interfacial free energy. This talk is part of the Molecular Dynamics and Sampling Focus Group. March 9, 2005, 11:15
am, Lind Hall 409 Critical intensities for phase transitions in a 3D Smoluchowski equation Abstract: We study the structure of equilibrium solutions to a Smoluchowski equation on a sphere, which arises in the modelling of rigid rod-like molecules of polymers. A complete classification of intensities for phase transitions to equilibrium solutions is obtained. It is shown that the number of equilibrium solutions hinges on whether the potential intensity crosses two critical values _{1} 6.731393 and _{2}7.5. Furthermore, we present explicit formulas for all equilibrium solutions. These solutions consist of a set of axially symmetric functions and all those which are obtained from this set by rotation. In this joint work with Hui Zhang and Pingwen Zhang, we solve the Onsager's 1949 conjecture on phase transitions in rigid rodlike polymers. This talk is part of the Friday Focus Group: Soft Matter. March 9, 2005,
2:30 pm, Lind Hall 409 Multiscale approaches to molecular dynamics and sampling Note date change! In the first part of this talk, I will survey some approaches for producing multiscale models for molecular dynamics (MD) and sampling. I will consider two parts of the problem: finding coarsened variables, and then integrating or propagating the coarsened model. I will discuss the approach of Brandt and collaborators to semi-automatically determine the coarsened variables, and the more ad-hoc approach of Gear and collaborators, who assume a reaction-coordinate is known which produces a natural separation of scales. Both methods attempt to sample the fast scales, and then to do an accurate integration of the slow scales. Related approaches will be mentioned, such as Leimkuhler's and Reich's reversible integrators. This talk is part of the Molecular Dynamics and Sampling Focus Group. March
10, 2005, 2:30 pm, Lind Hall 409 Informal discussion of molecular dynamics Note date change! This talk is part of the Molecular Dynamics and Sampling Focus Group. March 14, 2005, 11:15 am,
Lind Hall 409 A variational model of dislocations in the line tension limit Abstract: We study the limit of a dislocation model proposed by Ortiz et al., in which slip occurs only on one plane. Mathematically the core is an extension of the Alberti-Bouchitte-Seppecher results for 1/eps nonconvex two-well energy + H^{1/2} norm squared to an periodic array of wells (hence no naive coercivity). From the analysis point of view H^{1/2 } is interesting since it leads to a logarithnmic rescaling. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. March 15, 2005,
1:25 pm, Lind Hall 409 Macroscopic simulation of microstructures in finite-strain elastoplasticity Abstract: The computer simulation of the evolution of microstructures in finite-strain elastoplasticity requires a time-space discretization. The resulting mathematical model of each time-step yields a minimization problem with a nonconvex energy density W. Therein, the energy minimizing (better called infimizing) sequences of deformations develop enforced finer and finer oscillations in the deformation gradients called microstructures. The infimal energy is not attained and in the limit of those infimizing sequences, the deformation gradients yield a measure to describe statistically the oscillations. This gradient Young measure (GYM) acts as a generalized solution and conveys several pieces of information about the energy infimizing process such as the macroscopic deformation (i.e. the expected value of the GYM) or the stress field (GYM applied to derivative DW of energy density). The presentation gives a simple example in finite elastoplasticity with a single-slip mechanism and then explains the effect of nonconvexity and the relaxation theory from modern calculus of variations in 1D, 2D, and the vector case in a series of Examples related to Bolza, Young, Tartar plus one benchmark and a phase-transition. The numerical analysis of the relaxed formulation with adaptive finite element schemes and their stabilization is briefly discussed. In general, however, the quasiconvex hull is not known by some closed form expression. Instead a new computational challenge, numerical quasiconvexification, is in order and some new attempts towards this are discussed. The relaxation theory allows for a macroscopic simulation and only allows limited insight in the underlying microstructure patterns (through the GYM). More insight in the context of finite elastoplasticity is promised by energies extended by some surface energy. The mathematical model of which is less obvious in finite elastoplasticity and the presentation briefly discusses severe difficulties even with much simpler examples which lead to curved needles and branching structures near interfaces. March 16, 2005,
11:15 am, Lind Hall 409 The 2-Wasserstein metric and its applications to
PDEs Abstract: We introduce the 2-Wasserstein metric on the set of probabilities and study several constrained variational problems in that metric. We analyze the induced geometry of the set of densities satisfying the constraint on the variance and means and we determine all the geodesics on it. These analysis were motivated by questions in kinetic theory. The evolution of many mechanical systems can be represented by paths on the set of probability measures. These paths may consist of measures which are not absolutely continuous. It is necessary have a notion of infinite dimensional Hamiltonian systems on the whole set of measures. We give examples of evolutive systems that have a Hamiltonian structure according to that new concept. This talk is part of the Wednesday Focus Group: Singularities. March 18, 2005,
11:15 am, Lind Hall 409 Stretching of polymers on sub-Kolmogorov scales in a turbulent flow Abstract: First results on numerical studies of the stretching of Hookean dumbbells on scales below the viscous length of the advecting turbulent flow are presented. Direct numerical simulations of the Navier-Stokes turbulence are combined with Brownian dynamics simulations for simple polymer chains. The role of extreme stretching events on the overall statistics is discussed. Our findings are compared with recent analytical models for the polymer advection in Gaussian random flow without time-correlation. March 21, 2005,
11:15 am, Lind Hall 409 Computation of geometric PDEs and aplications This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. March 22, 2005, 11:15
am, Lind Hall 409 Hydrogels with excellent mechanical performance: An approach to understand the secret of cartilages Abstract: A hydrogel is a polymer network swollen with large amount of water. It is a solid on the macroscopic scale: having a definite shape and does not flow. At the same time, it behaves like a solution on the molecular scale: water-soluble molecules can diffuse in a hydrogel with various diffusion constants reflecting sizes and shapes of the molecules. Because of its specific structure, a gel exhibits a variety of unique behaviors such as phase-transition, specific adsorption equilibrium, presence of unfrozen water, chemomechanical behavior, etc. Due to the unique properties, a wide range of industrial, medical, pharmaceutical, and prosthetic applications have been proposed. Application of a hydrogel as a mechanical device is fairly limited due to its lack in mechanical strength. Many gel researchers have thought that the mechanical weakness is unavoidable because of its solution-like nature, i.e., low density of polymer chains and small friction between the chains. Furthermore, it is well known that in synthetic gels are inhomogeneous in structure, which is considered as a factor to decrease the mechanical strength. However, if we pay attention to biological systems, we find some hydrogels, such as a cartilage, with excellent mechanical performances. It is a challenging problem in modern gel science to fill the gap between the man-made gels and the biological gels. Another interesting problem of a gel is its surface property. Few is known of the surface properties of a gel although we observe fascinating surface behavior of bio-organs. For example, the extracellular mucins, which comprise a family of high molecular weight, extensively glycosylated glycoproteins, are crucial to the biological activity, which relates to lubrication and protection of cell surfaces from damage. Another example is the animal cartilage, which sustains a daily compression of 100kg/cm2 and has an extremely low friction coefficient. Two topics regarding to the mechanical properties of a gel, as a soft and wet matter, will be addressed in the seminar. The first is how to produce a hydrogel with an excellent mechanical roughness, and the second is what is the friction law that a gel obeys. This talk is part of the Friday Focus Group: Soft Matter. March
23, 2005, 11:15 am, Lind Hall 409 Compatibility and phase nucleation Abstract: The talk will discuss the problem of martensitic phase nucleation, and related connections between the quasiconvexity condition of the calculus of variations and compatibility of gradients. This talk is part of the Wednesday Focus Group: Singularities. March 25, 2005,
10:10 am, Lind Hall 409 Mathematical models of lipid membranes Abstract: Lipid membranes are aggregates of amphiphilic molecules, which consist of a hydrophilic head and one or more hydrophobic tails. Living in an aqueous environment, these molecules tend to form bilayers where the hydrophobic parts are hidden by the hydrophilic ones, and so their contact with water is reduced. A further reduction is obtained when the bilayer closes itself to form a vesicle, which is modelled as a compact, two-dimensional surface. We will first survey the classical results concerning the analysis of the elastic energy functional which determines the equilibrium vesicle shapes when both their area and enclosed volume are fixed. Proteins, thought of as rigid bodies, are usually modelled as small cones. When embedded in a lipid bilayer, they modify the membrane configuration by fixing the direction of the surface normal at the contact points. In the two-dimensional approximation, where the membrane shape is modelled by a closed curve, we determine the exact equilibrium shape of the membrane in the presence of one or more proteins. The excess of elastic energy induced by the proteins gives rise to a mediated interaction between them. The interaction may be either attractive or repulsive, depending on the protein shape and relative distance. In the three-dimensional case, however, the panorama changes: the shape perturbations induced by the proteins are strongly localized and decay within a characteristic lengthscale of the order of the protein diameter. Asymptotic methods allow to derive the analytical shape of the perturbation. This talk is part of the Friday Focus Group: Soft Matter. March 25, 2005,
11:10 am, Lind Hall 409 Homogenization of a Ginzburg-Landau model for a nematic liquid crystal
with inclusions Abstract: We consider a nonlinear homogenization problem for a Ginzburg-Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that inclusions are separated by distances of the same order as their size, we find an effective functional in the limit of small inclusions. We generalize the variational method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg-Landau functional. As a byproduct of our analysis, we show that the limiting functional is a Gamma-limit of a sequence of Ginzburg-Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit. This talk is part of the Friday Focus Group: Soft Matter. April 1, 2005,
11:15 am, Lind Hall 409 Telephone-cord instabilities in thin smectic capillaries Abstract: Telephone-cord patterns have been recently observed in smectic liquid crystal capillaries. In this talk we analyse the effects that may induce them. As long as the capillary keeps its linear shape, we show that a nonzero chiral cholesteric pitch favors the SmA*-SmC* transition. However, neither the cholesteric pitch nor the presence of an intrinsic bending stress are able to give rise to a curved capillary shape. The key ingredient for the telephone-cord instability is spontaneous polarization. The free energy minimizer of a spontaneously polarized SmA* is attained on a planar capillary, characterized by a nonzero curvature. More interestingly, in the SmC* phase the combined effect of the molecular tilt and the spontaneous polarization pushes towards a helicoidal capillary shape, with nonzero curvature and torsion. This talk is part of the Friday Focus Group: Soft Matter. April 4, 2005, 11:15 am,
Lind Hall 409 Action minimization and sharp interface limits for the Allen-Cahn equation This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. April 18, 2005,
11:15 am, Lind Hall 409 Numerical simulation of high dimensional Schrödinger equations and applications to molecular simulation This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. April
20, 2005, 11:15 am, Lind Hall 409 Mathematical analysis of nonlocal and effective behavior in liquid crystals Abstract: I will address mathematical problems arising in studies of switching processes in liquid crystals, taking into account nonlocal electric field interaction. The identification of the switching states is carried out by energy minimization with physically relevant boundary conditions. The proposed model is appropriate to Smectic~C* liquid crystals, presenting chirality effects and layering pattern, at different temperature ranges. The free energy presents quadric terms in the second gradient of the fields as well as other nonlocal effects. Dynamical problems of switching will also be presenting with the aim of exploring increasing speed mechanisms. The last part of the presentation will deal with analysis of related flow problems by means of homogenization. This talk is part of the Wednesday Focus Group: Singularities. April 22, 2005, 11:15 am, Lind Hall 305
Math Awareness Month/Einstein
Annus Mirabilis Centenary Special Lecture
Math and the Cosmos: The New Mathematical Gravitational Astronomy. April
25, 2005, 11:15 am, Lind Hall 409 Title: TBA This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. April
28, 2005, 1:25 pm, Lind Hall 409 Continuum modelling of the mechanics of curved crystalline objects: applications to nanotubes Abstract: I will introduce a method to construct nonlinear elastic models for curved lattices from atomistic models, and analyze the performance of this and other continuum models for the simple example of a discrete worm-like chain. I will also present applications of this method to study the rich mechanics of carbon nanotubes. Multiscale scale simulations accessing scales relevant to devices, have revealed an anomalous elastic response of thick multi-walled nanotubes. April 29, 2005,
10:10 am, Lind Hall 409 Negative index materials Abstract: Materials in which the refractive index is negative were first considered by Veselago[1] in 1968. Such materials, which do not occur naturally, can give rise to remarkable optical phenomena. Recent experiments with man made meta-materials at microwave and terahertz frequencies have verified many of the theoretical predictions. I will give a brief overview of the area, consider some of the underlying physics, highlight challenges to realizing nondispersive negative index materials at optical frequencies, and discuss potential applications. 1. V.G. Veselago, Sov. Phys. Usp. 10, 509 (1968) This talk is part of the Friday Focus Group: Soft Matter. April 29, 2005, 11:15 am, Lind Hall
409 Nematodynamics of nematic polymers in general linear flows and imposed external field Abstract: I will discuss the nematodynamics of nematic polymers using the Doi-Hess kinetic theory and closure models for linear flows and imposed external field. I will show that a planar linear flow copuled with an imposed external field is equivalent to a simple shear coupled with an imposed external field in the direction transverse to the shearing plane thereby establishing a correspondence principle between the two types of flows. For simple shear, we have already obtained the full flow diagram. Therefore, we can infer the flow behavior for linear flows using the correspondence principle. This talk is part of the Friday Focus Group: Soft Matter. May 6, 2005,
2:00-3:00 pm, Lind Hall
409 Optimal sampling of a reaction coordinate in molecular dynamics Keywords: dynamics and sampling Abstract: Estimating how free energy changes with the state of a system is a central goal in applications of statistical mechanics to problems of chemical or biological interest. From these free energy changes it is possible, for example, to establish which states of the system are stable, what are their probabilities and how the equilibria between these states are influenced by external conditions. Free energies are also of great utility in determining kinetics of transitions between different states. A variety of methods have been developed to compute free energies of condensed phase systems. Here, I will focus on one class of methods - those that allow for calculating free energy changes along one or several generalized coordinates in the system, often called "reaction coordinates" or order parameters. Considering that in almost all cases of practical interest a significant computational effort is required to determine free energy changes along such coordinates it is hardly surprising that efficiencies of different methods are of great concern. In most cases, the main difficulty is associated with its shape along the reaction coordinate. If the free energy changes markedly along this coordinate Boltzmann sampling of its different values becomes highly non-uniform. This, in turn, may have considerable, detrimental effect on the performance of many methods for calculating free energies. Several approaches have been proposed to overcome this difficulty. Recently two methods have been developed that allow for recovering the free energy profile from trajectories, in which sampling along the reaction coordinate is exactly or approximately uniform. One method is based on Jarzynski's identity and requires generating a series of appropriate non-equilibrium trajectories that describe evolution of the system along the reaction coordinate. The other method, called Adaptive Biasing Force (ABF), devcloped by Darve and Pohorille, relies on thermodynamic integration of the average force acting on the reaction coordinate during unconstrained molecular dynamics simulations. I will outline the derivation of the general formula for thermodynamic force in such simulations, discuss how this formula can be used to obtain both uniform sampling of the reaction coordinate and unbiased estimates of the accompanied free energy changes, and derive error estimate for the method. It will be also demonstrated that the efficiency of ABF is considerably better than the efficiency of non-equilibrium simulations. Recently, ABF was successfully applied by several groups to calculate free energies of conformational changes in flexible molecules, transfer of solutes across phase interfaces, stretching helical peptides, association of proteins in membranes and ion transport through transmembrane channels. I will discuss the results of these simulations. This talk is part of the Focus Group: Molecular Dynamics and Sampling. May 9, 2005, 11:15 am, Lind Hall 409 A mathematical description of the invasion of Bacteriophage T4 Abstract: Bacteriophage T4 is a virus that attacks bacteria by a unique mechanism. It lands on the surface of the bacterium and attaches its baseplate to the cell wall. Aided by Brownian motion and chemical bonding, its tail fibers stick to the cell wall, producing a large moment on the baseplate. This triggers an amazing phase transformation in the tail sheath, of martensitic type, that causes it to shorten and fatten. The transformation strain is about 50%. With a thrusting and twisting motion, this transformation drives the stiff inner tail core through the cell wall of the bacterium. The DNA of the virus then enters the cell through the hollow tail core, leading to the invasion of the host. This is a natural machine. As we ponder the possibility of making man-made machines that can have intimate interactions with natural ones, on the scale of biochemical processes, it is an interesting prototype. We present a mathematical theory of the martensitic transformation that occurs in T4 tail sheath. Following a suggestion of Pauling, we propose a theory of an active protein sheet with certain local interactions between molecules. The free energy is found to have a double-well structure. Using the explicit geometry of T4 tail sheath we introduce constraints to simplify the theory. Configurations corresponding to the two phases are found and a formula for the force generated by contraction is given. The predicted behavior of the sheet is completely unlike macroscopic sheets. To understand the position of this bioactuator relative to nonbiological actuators, the forces and energies are compared with those generated by inorganic actuators, including nonbiological martensitic transformations. Joint work with WF@ddt.biochem.umn.edu ,Wayne Falk and R. D. James, An elasticity theory for self-assembled protein lattices with application to the martensitic transformation in Bacteriophage T4 tail sheath, preprint. K. Bhattacharya and R. D. James, The material is the machine, Science 307 (2005), pp. 53-54. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. May 11,
2005, 11:15-12:15 pm, Lind Hall 409 Vortex patterns in Bose Einstein condensates This talk is part of the Wednesday Focus Group: Singularities. May 20, 2005, 11:15, Lind Hall 409 Molecular dynamics simulations: stability, multiscale approaches and the art of trajectory analysis Abstract: Molecular dynamics (MD) is a venerable computer simulation technique in biomolecular modeling. MD is also known to be very compute-intensive. Using multiple time stepping (MTS) (quasi-)multiscale integrators is one of the key methods for speeding up MD simulations. In this talk, I will revisit the stability issues of MTS MD simulations and show that MTS integrators are really limited by nonlinear instabilities. Then I will present a family of MTS quasi-mutiscale integrators based on targeted Langevin stabilization of stiff modes. Such schemes would become more powerful when they are developed under a general mathematical framework termed as Projective Thermostatting Dynamics. Ideas of new development will be presented. I will also present a case study in which we apply MTS MD to an enzyme system, the soluble guanylyl cyclase (sGC). While our aim is to reveal the science behind the phenomena, the MD trajectory analysis is more an art than anything else. I will present how we go about making analysis to infer the mechanism of allosteric activation of sGC. May 23, 2005, 11:15 am, Lind Hall 305
Retrieving useful statistics and closure approximations in multiscale simulations Abstract: With colleagues at Penn State, we have been working on a number of interdisciplinary projects related to multiscale simulations of multicomponent alloys, complex fluids, and vesicle bio-membranes. In this talk, we will discuss the needs and techniques for retrieving useful statistics and closure approximations. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. May 26, 2005, 11:15 am, Lind Hall 305 The many facets of film growth modeling The growth of epitaxial thin films is studied on an enormous range of length and time scales using a number of distinct computational tools. This talk will provide an overview of the film growth process and a survey of some of the most popular models, including the solid-on-solid (SOS) model and the Burton-Cabrera-Frank (BCF) model. The SOS model is typically simulated via kinetic Monte Carlo, where as the BCF model is formulated as a free boundary problem coupling diffusion equations on adjacent domains. I will discuss some work that couples these two approaches in a multiscale simulation.
May 27, 2005, 10:00 am, Lind Hall 409
New balances that steer phase evolution Phase changes are ubiquitous. This is especially true here at IMA: a naive Google search in its web site gave me 3680 occurrences of the term "phase" , 469 of them containing the phrase "phase change" - not to count "phase transformation" (239), "phase field" (552) "two-phase" (154), etcetera. However, "phase" is mostly used as a descriptive, non-quantitative term. Phase can change in time, but there are no kinematic variables to describe its evolution - with the notable exception of phase field theories, where nonetheless the phase-related order parameter is more a contrivance than a physical quantity. I am dissatisfied with this state of affairs. During the last five years or so, I have been developing a format for continuum mechanics - which I like to call "material remodelling" - that has explicit phase descriptors and new balance laws to govern their time evolution. In this talk I try to give the flavor of the theory and of (some of) its applications, presenting the basics as straightforwardly as I can. May 27,
2005, 11:15 am, Lind Hall 409 Thoughts about diffusion mediated transport: can we study motion in small systems? Abstract: Diffusion mediated transport is implicated in the operation of many molecular level systems. These include some liquid crystal and lipid bilayer systems, and, especially, the motor proteins responsible for eukaryotic cellular traffic. All of these systems are extremely complex and involve subtle interactions on varying scales, as exemplified by the talks in the last workshop. The chemical/mechanical transduction in motor proteins is, by contrast to many materials microstructure situations, quite distant from equilibrium. These systems function in a dynamically metastable range. Our plan is to look at the relationship of the Monge-Kantorovich mass transfer problem to models for conventional kinesin type motors and their relatives. These concepts permit us to establish consistent thermodynamical dissipation principles from which evolution equations follow. What properties are necessary for transport? What is the role of diffusion? What is the role of other elements of the system and how can dissipation be exploited to understand this? How successful are we? June 1, 2005, 11:15 am, Lind Hall 409
From discrete to continuum systems: crystallization in two dimensions Abstract: While the analysis of contiuum models of spatially extended systems is a well studied mathematical discipline, much less is know about the relation between discrete systems and their continuum limits. I will highlight some of the challenges which arise when passing from discrete to continuum scales and give an overview of recent mathematical developments. A fundamental challenge in this area is the crystallization problem where one attempts to characterize the asymptotic behavior of the ground state of N particles that interact via Lennard-Jones type potentials with each other. I will report on new mathematical results which partically solve this problem. This talk is part of the Monday Focus Group: Multiscale Modeling and Computing. June 3, 2005, 2:30 pm, Lind Hall 409
Exploring complicated energetic landscapes: from atomistic to continuum and back The talk will start with an exploration of the energetic landscape of martensitic at the atomistic (lattice) level. Steel is the most prominent example of an irreversible martensitic transformation, while shape-memory alloys are reversible martensitic transformations. Why is steel soft compared to shape-memory materials? One reason is the distinctive nature of their atomistic landscapes. They turn out to be dictated by the change in symmetry groups that are involved in the transformation. This leads us to a discussion of the consequences for continuum models, where we introduce the concept of relaxation and some problems in the actual computation of a relaxed (continuum) energetic landscape. The talk concludes with a few remarks on how to travel in a complicated non-convex energetic landscape from one well to another. |
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