### ABSTRACTS

**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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