August 20, 2019

May 31 - June 2, 2007

Lattices and Trajectories:
A Symposium of Mathematical Chemistry in honour of
Ray Kapral and Stu Whittington

Fields Institute, Toronto

The Connaught Fund


Richard Brak, The University of Melbourne
Directed path models of certain polymer phase transitions

I will review a range of directed path models of certain polymer phase transition problems, in particular, polymer collapse using interacting partially directed paths, polymer absorption onto a surface by Motzkin paths, and sensitized flocculation and steric stabilization using Dyck paths interacting with a pair of surfaces. I will also discuss some unexpected applications of these results to a non-equilibrium problem, that of the simple asymmetric exclusion processes, and to an equilibrium model, that of directed compact percolation. This talk will have a strong combinatorial flavour.

Hugues Chaté, CEA
Collective properties of active polar and nematic particles
Active or self-propelled particles are in fashion today in models for the collective motion of animals, bacteria, cells, molecular motors, as well as driven granular matter or even for the swarming behavior of robots. I will review recent results obtained on minimal microscopic models of interacting polar and apolar (nematic) active particles, stressing the (probably) universal properties of the emerging collective dynamics. If time allows, recent proposals for mesoscopic descriptions of these systems will be discussed.

Giovanni Ciccotti, Università degli Studi di Roma-"La Sapienza"
Minimum free energy paths and "isocommittor" surfaces
A computational technique is proposed which combines the string method with a sampling technique to determine minimum free energy paths. The technique only requires to compute the mean force and another conditional expectation locally along the string, and therefore can be applied even if the number of collective variables kept in the free energy calculation is large. This is in contrast with other free energy sampling techniques which aim at mapping the full free energy landscape and whose cost increases exponentially with the number of collective variables kept in the free energy. Provided that the number of collective variables is large enough, the new technique captures the mechanism of transition in that it allows to determine the committor function for the reaction and, in particular, the transition state region. The new technique is illustrated on the example of alanine dipeptide, in which we compute the minimum free energy path for the isomerization transition using either two or four dihedral angles as collective variables. It is shown that the mechanism of transition can be captured using the four dihedral angles, but not using only two of them.

David Coker, Boston University
Modeling electronic and vibrational pure dephasing and dissipation dynamics in condensed phase systems
A path integral approach using a discrete representation for the quantum subsystem is implemented by linearizing in the difference between forward and backward paths for the continuous solvent degrees of freedom, and employing a full mapping Hamiltonian description for the discrete quantum subsystem states. The approach is employed to study electronic and vibrational dephasing in a realistic model of complex many-body systems that can be probed in experiments exploring, for example, the microscopic mechanism of how superposition states of the quantum subsystem undergo decoherence due to entanglement with their environment. Understanding decoherence mechanisms, the factors that affect these quantum processes, and how they might be controlled by molecular level engineering is important if advances are to be made in applications of quantum information theory with molecular systems, for example. The studies will address vibrational quantum decoherence effects in the presence of both weak and strong dissipation, and also explore how such phenomena are influenced by avoided crossings and conical intersections in realistic models including decoherence of a vibrational superposition state of electronically excited halogen molecules in condensed rare gas environments where recent detailed experiments are available.

Joern Davidsen, University of Calgary
Filament-induced surface spiral turbulence

Surface defect-mediated turbulence in bounded three-dimensional (3D) excitable media is investigated in the regime of negative line tension. In this regime turbulence arises due to unstable filaments associated with scroll waves and is purely a 3D phenomenon. In this talk, I will show that the statistical properties of the turbulent defect dynamics can be used to distinguish surface defect-mediated turbulence from its 2D analog. Mechanisms for the creation and annihilation of surface defects will be discussed and generalizations of Markov rate equations are employed to model the results.

Frank den Hollander, Universiteit Leiden
Copolymers in solution
This talk describes a model for a random copolymer in a random emulsion that was introduced in a recent paper with Stu Whittington. The copolymer is a two-dimensional directed self-avoiding walk, carrying a random concatenation of monomers of two types A and B, each occurring with density ½ . The emulsion is a random mixture of liquids of two types, A and B, organized in large square blocks occurring with density p and 1-p, respectively where p (0,1). The polymer in the emulsion has an energy that is minus times the number of AA-matches minus times the number of BB-matches, where , R are interaction parameters that may be restricted to the cone {( , ) R2 : | |}.

We consider the quenched free energy per monomer in the limit as the length n of the polymer tends to infinity and the blocks in the emulsion have size Ln such that Ln and Ln/n 0. To make the model mathematically tractable, we assume that the polymer can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. Although this is an unphysical restriction, it turns out that the model exhibits rich and physically relevant behaviour.

Let pc 0.64 be the critical probability for directed bond percolation on the square lattice. We show that for p pc there is a localization vs. delocalization phase transition along one (!) critical curve in the cone, which turns out to be independent of p, while for p < pc there are three (!) critical curves, all of which depend on p. We derive a number of qualitative and quantitative properties of these curves.
(This is joint work with Nicolas Pétrélis and Stu Whittington.)

Rashmi Desai, University of Toronto
Epitaxial Growth in coherent, strained, asymmetric alloy films
I shall report on our recent work on epitaxial growth and surface instabilities in coherent, strained, asymmetric alloy films. A nonequilibrium continuum model is used to explore coupling of alloy segregation instability and morphological instability which arises from lattice mismatch and other elastic effects. Even though the model has interesting nonlinearities, interesting effects occur even in linear approximation (valid for thin films). I shall also discuss application to some real materials.

Leon Glass, McGill University
Predicting and Preventing Sudden Cardiac Death
Sudden cardiac death kills hundreds of thousands of North Americans each year. This number could be reduced significantly if a medical device -- the implantable cardiac defibrillator -- had been implanted prior to the sudden death. However, since we do not have good ways of predicting who will suffer sudden cardiac death or when, physicians face a major problem in deciding in whom to implant a cardiac defibrillator. This problem is made more severe since implantable cardiac defibrillators are expensive, and complications, though rare, do add to the risk of using the devices in those who would not benefit. In this talk I will describe attempts to understand cardiac arrhythmias -- especially those responsible for sudden cardiac death. The methods include analysis of electrocardiographic records of patients who experienced sudden cardiac death, analysis of arrhythmias in German Shepherd dogs that experience sudden cardiac death, recording activity in tissue culture models of cardiac arrhythmias, and the formulation of mathematical models of cardiac arrhythmia employing a range of techniques from number theory to nonlinear dynamics.

Tony Guttman, University of Melbourne
Role of conformational entropy in force-induced bio-polymer unfolding
A statistical mechanical description of flexible and semi-flexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. The existence of many intermediate states at low temperatures stabilized by the force is found. A unified response of pulling and compressing force has been obtained in the constant distance ensemble. We show the signature of a cross-over length which increases linearly with the chain length. Below this cross-over length, the critical force of unfolding decreases with temperature, while above it increases with temperature. For stiff chains, we report for the first time a "saw-tooth" like behavior in the force-extension curves which has been seen earlier in the case of protein unfolding.
(Joint work with Sanjay Kumar, Iwan Jensen and Jesper L. Jacobsen.)

James T. Hynes, University of Colorado and Ecole Normale Supérieure, Paris
Solvation and photochemical funnels: Environmental effects on conical intersection structure and dynamics

Excited electronic state processes at conical intersections (CIs) have received intense scrutiny in photochemical experiment and theory in recent years. CIs often provide a "funnel" for (often ultrafast) passage from a photochemically accessed S1 state to the ground state S0, governing nonadiabatic transition rates; they have been referred to as "transition states" for photochemical processes.

Recent experiments on, e.g., photoactive proteins highlight the pronounced influence of a solvent or protein nanospace environment on CI dynamics. S1-S0 population transfer can be substantially modified, suggesting major changes in the underlying CI topology and dynamics. A central theoretical challenge is to select and describe the relevant features governing the complex chromophore-environment supermolecular systems.

The present contribution focuses on excited electronic state processes at CIs where a charge transfer is involved. We describe the key features of a theoretical formulation recently developed to describe the chromophore-environment interaction and its consequences. This generalizes considerably an early important treatment by Bonacic-Koutecky, Koutecky and Michl to include important molecular coordinates, e.g., isomerization twisting motions, and the polar/polarizable environment's influence. The environment's electrostatic effects are accounted for by a dielectric continuum model. Applications to a model for the S1-S0 CI in protonated Schiff bases provide a free energy surfaces description for the coupled system represented by molecular coordinates (e.g., twisting and bond stretching/contracting) plus a solvent coordinate. The environment's significant impact on the CI is investigated, as are "reaction paths" and the dynamics leading to and through the CI. Nonequilibrium "solvation" effects are shown to be critical.
(This work has been carried out in collaboration with Irene Burghardt (ENS, Paris), Riccardo Spezia (ENS, Paris), Joao Malhado (ENS, Paris) and L. Cederbaum (Heidelberg).)

Jennifer Lee, University of Toronto
Collapse transition in the presence of an applied force
This talk will focus on a collapse transition of a linear homopolymer in dilute solution in the presence of an applied force. An interacting partially self-avoiding walk (IPDSAW) model was used to describe the system conditions, with energy and applied force variables associated with the near-neighbour contacts in the walk. Exact expressions were generated, where the analytic structure of such expressions will be presented. Theoretical results were then used as a comparison model for investigating a collapse transition in single molecule experiments. Force spectroscopy obtained using AFM generated single molecule force-extension profiles and will be presented.
(This work is carried out in collaboration with S. G. Whittington, R. Brak, A. J. Guttmann and G. C. Walker.)

Neal Madras, York University
Polymers on hyperbolic lattices
This talk discusses traditional lattice models of polymers (self-avoiding walks, lattice trees, and lattice animals) on "non-Euclidean lattices", specifically graphs that correspond to regular tilings of the hyperbolic plane (or 3-space). One example is the infinite planar graph in which every face is a triangle and eight triangles meet at every vertex. On such lattices, these models should exhibit mean field behaviour, as they would in high-dimensional Euclidean space, or, more simply, on an infinite regular tree. We have made progress towards rigorous understanding of these issues, as well as analogous ones for percolation, but some open questions remain.
(This talk is based on joint work with C. Chris Wu.)

G. Nicolis and C. Nicolis, Université Libre de Bruxelles
Nonlinear dynamics and self-organization in the presence of metastable phases
There is increasing evidence that self-organization phenomena in a variety of nanosize materials occur in the presence of metastable phases. This switches on non-standard nucleation mechanisms with combined structural and density fluctuations, entailing that kinetic effects and nonequilibrium states are playing an important role. In this presentation the effect of metastable phases on the free energy landscape is determined for a class of materials in which the attractive part of interparticle interactions is weak and short-ranged. The kinetics of the fluctuation-induced transitions between the different states, stable as well as metastable, is subsequently analyzed using a generic model involving two order parameters. Conditions are identified under which the transition rate towards the most stable state can be enhanced and the relevance of the results in the crystallization of protein solutions is discussed.

Steven Nielsen, University of Texas at Dallas
Quantifying the surfactant coverage of nanoparticles by molecular dynamics simulation: The physisorbed versus chemisorbed cases
Potential energy terms are derived for the interactions between surfactants and solvent, and a spherical nanoparticle, which depend parametrically on the nanoparticle radius. The gradient of these potentials with respect to the nanoparticle radius allows the mean force of constraint on the radius to be calculated during a molecular dynamics simulation. This free energy method allows the optimal, or saturated, surfactant coverage to be found. The effects of curvature, surfactant geometry, and chemisorbed versus physisorbed conditions are explored.

Gian-Luca Oppo, University of Strathclyde
Spatio-temporal structures in photonics and chemistry
We compare nonlinear spatio-temporal structures such as patterns, spatial solitons, spirals, defect-mediated turbulence, etc. in prototype models of photonic and chemical systems. Analogies and differences are drawn between systems driven by either diffraction (photonics) or diffusion (chemistry). It is found that while the investigated structures often have very similar nature, their names differ between these two research fields. Typical examples are chemical spirals and optical vortices, chemical spots and cavity solitons. As Gershwin said: "you like potato and I like potahto, you like tomato and I like tomahto".

Garnett Ord, Ryerson University
Counting oriented rectangles and the propagation of waves
We propose a simple counting problem involving chains of rectangles on a planar lattice. The boundaries of the chains form a type of random walk with a finite inner scale. With orientation neglected, the continuum limit of the walk densities obeys the Telegraph equation, a form of diffusion equation with a finite signal velocity. Taking into account the orientation of the rectangles, the same continuum limit yields the Dirac equation. This provides an interesting context in which the Dirac equation is phenomenological rather than fundamental.

E. Orlandini, Universita degli Studi di Padova
Directed walk models of polymers stretched by a force
In recent years, the mechanical properties of individual polymers and filaments have been thoroughly investigated experimentally, thanks to the rapid development of micromanipulation techniques such as optical tweezers and atomic force microscopy (AFM). Experiments such as the stretching of single DNA polymers or the force-induced desorption from an attractive surface enhance the possibility of understanding the physical properties of the single molecule. In order to interpret experiments quantitatively several theoretical models which allow one to calculate the response of a polymer to external forces have recently been introduced and studied by several authors. In this respect, Stu Whittington has been a pioneer in the field through the introduction of simple directed-walk models of polymers, subjected to an elongational force, that are either adsorbed on the surface or localized between two different solvents.

In this talk I will review some of these models, showing that, although they are simple enough to be solved analytically, they can catch much of the underlying physics of the problem. Moreover, they can be extended to describe other interesting phenomena such as the mechanical unzipping of double-stranded DNA or the stretching of compact polymers.

Aleksander Owczarek, University of Melbourne
Polymers in a slab with attractive walls: Scaling and numerical results

We summarize the latest results concerning models of polymers in a slab with sticky walls. We present a conjectured scaling theory and numerical confirmation.

Antonio Politi, CNR
Chaos without exponential instability
Since several years, it is known that coupled map models can exhibit pseudochaotic behaviour, even in the absence of a strictly positive maximum Lyapunov exponent. Quite recently some more realistic systems have been identified, where this behaviour can be generated. In particular, I refer to a chain of hard-point particles and to a network of globally coupled leaky integrate-and-fire neurons. The peculiarity of this type of dynamical behaviour and the conditions for its generation will be discussed.

Andrew Rechnitzer, University of British Columbia
Mean unknotting times of random knots and embeddings
We study mean unknotting times of knots and knot embeddings by crossing reversals, in a problem motivated by DNA entanglement. Using self-avoiding polygons (SAPs) and self-avoiding polygon trails (SAPTs) we prove that the mean unknotting time grows exponentially in the length of the SAPT and at least exponentially with the length of the SAP. The proof uses Kesten's pattern theorem, together with results for mean first-passage times in the two-parameter Ehrenfest urn model. We use the pivot algorithm to generate random SAPTs and calculate the corresponding unknotting times, and find that the mean unknotting time grows very slowly even at moderate lengths. Our methods are quite general -- for example the lower bound on the mean unknotting time applies also to Gaussian random polygons.
(This is work together with Aleks Owcarek and Yao-ban Chan at the University of Melbourne, and Gord Slade at the University of British Columbia.)

Katrin Rohlf, Ryerson University
From excitable media on the large scale to reaction-diffusion mechanisms on the small scale
In honour of Raymond Kapral, this talk will highlight some of our past and current work concerning simulations for reactive media both on the large scale, as well as on the small scale.

The first part of the talk will be devoted to recent results concerning the self-organizational properties of spiral waves in a FitzHugh-Nagumo system. Such systems have a wide range of physical, chemical and biological applications, and -- in particular --have often been used to describe the electrical activity of heart tissue. Our results have important implications on the proper assessment of drug treatment options for cardiac arrhythmias.

The talk will conclude with an overview of current work concerning the time evolution of a chemically reacting medium using a particle-based approach. In particular, some results will be presented for a Selkov reactive mechanism in a spatially extended system, and we will show its connection to a stochastic phase-space description for which the total number of particles in the system is not conserved.

Tom Shiokawa, National Center for Theoretical Sciences
Non-Markovian dynamics and quantum Brownian motion

Kenneth Showalter, West Virginia University
Spatiotemporal dynamics of networks of excitable nodes
A network of excitable nodes based on the photosensitive Belousov-Zhabotinsky reaction is studied in experiments and simulations. The addressable medium allows both local and nonlocal links between the nodes. The initial spread of excitation across the network as well as the asymptotic oscillatory behavior are described. Synchronization of the spatiotemporal dynamics occurs by entrainment to high-frequency network pacemakers formed by excitation loops. Analysis of the asymptotic behavior reveals that the dynamics of the network is governed by a subnetwork selected during the initial transient period.
(In collaboration with Aaron J. Steele and Mark Tinsley.)

Christine Soteros, University of Saskatchewan
Random copolymer models
Self-avoiding walk models have been used for about 50 years to study linear polymers (long chain molecules) in dilute solution. For such models, the vertices of the walk represent the monomer units which compose the polymer and an edge of the walk joins two monomer units which are chemically bonded together in the polymer chain. Distinct self-avoiding walks on a lattice, such as the square or simple cubic lattice, represent distinct conformations of the polymer chain. Recently there has been much interest in extending the standard self-avoiding walk model of homopolymers (all monomer units considered identical) to the study of random copolymers. A random copolymer is a polymer composed of several types of comonomers where the specific distribution of these comonomers along the polymer chain is determined by a random process. The comonomer sequence can be thought of as being determined in or by the polymerization process (assumed to involve a random process) but then once determined the sequence of comonomers is fixed; this is an example of what is known as quenched randomness. In the simplest self-avoiding walk model of a random copolymer, one assumes that there are two types of comonomers and that they are distributed independently along the polymer chain.

Based on a series of seminal papers by Stu Whittington and others, I will review the progress that has been made using self-avoiding walk models to study phase transitions, such as the absorption phase transition and the localization phase transition, in random copolymer systems. Special emphasis will be placed on recent progress made by us, in collaboration with Stu Whittington, studying bounds on the limiting quenched average free energy for directed walk models such as Dyck and Motzkin paths.

De Witt Sumners, Florida State University
Random thoughts about random knotting
At the interface between statistical mechanics and geometry/topology, one encounters the very interesting problem of length dependence of the spectrum of geometric/topological properties (writhing, knotting, linking, etc.) of randomly embedded graphs. Stu Whittington has been at the forefront of research in this area, and this talk will discuss the proof of the Frisch-Wasserman-Delbruck conjecture (the longer a random circle, the more likely it is to be knotted), with some of its generalizations and scientific applications.

E. J. Janse van Rensburg, York University
Knotted lattice polygons
Let pn(K) be the number of lattice polygons of length n and knot type K. It is known that limn [log pn( )]/n = log exists, where K = is the unknot, and is the growth constant of unknotted polygons in the cubic lattice. In addition, < , where is the growth constant of lattice polygons. This result implies that almost all lattice polygons are knotted in the large n limit. In this talk I shall review the statistical and scaling properties of lattice polygons of fixed knot types in the lattice and also in a slab geometry by using rigorous and scaling arguments and by presenting numerical results from Monte Carlo simulations using the BFACF algorithm.

Xiao-Guang (Charles) Wu, Revionics, Inc.
Ion dynamics in non-perfect quadrupole traps
Ion dynamics in non-perfect quadrupole traps differ from those in a pure quadrupole field. We obtain an analytic expression for a quadrupole field superimposed with weak, higher-order multipole fields. Single ion dynamics in such trapping fields close to the instability point are investigated. We show that for an in-phase octopole field, oscillating envelopes of the axial displacement grow exponentially with the parameter deviation; whereas for an out-of-phase octopole field the growth of the oscillating envelopes follows a square-root law. A hard-sphere scattering model is assumed to incorporate collisions with buffer-gas molecules. The collision frequency and cross-section are defined. A simulation algorithm for many-ion dynamics is developed based on the Verlet algorithm and Monte Carlo techniques. We show how a weak octopole field affects the mass resolution in a significant way.

Royce Zia, Virginia Tech
Percolation of a collection of finite random walks: A model for gas permeation through thin polymeric membranes
Bond percolation on a square lattice is well known. What if the bonds are not randomly distributed, but are correlated somehow? In particular, consider placing a fixed density of (non-self-avoiding) random walks of l bonds on the lattice. How does the critical density depend on l? This problem is motivated by a model for gas transport through thin polymeric films.


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