Dynamical Chaos and Non-equilibrium
Statistical Mechanics:
From Rigorous Results to Applications
in Nano-systems
(1 Aug - 30 Sep 2006)
~ Abstracts ~
Fractional calculus and physics on
fractal
Vasily Tarasov, Moscow State University
In these lectures, the introduction to fractional
calculus, fractal sets and fractional dynamics is
considered.
Fractional calculus is used to describe processes and
systems in noninteger-dimensional sets.
1. The brief review of Hausdorff measure, Hausdorff
dimension and integration on fractals are suggested.
The integration in noninteger-dimensional spaces and the
fractional integrals are defined. Simple examples are
suggested.
Fractional mass dimension of fractal are discussed.
2. The fractional generalization of the equations of balance
of mass density, momentum density, and internal energy are
derived.
The moments of inertia for homogeneous fractal rigid bodies
are calculated.
Possible simple experiments with fractal media are
discussed.
A fractional generalization of the Chapman-Kolmogorov
equation is obtained to describe the Markov processes on
fractals.
From the fractional Chapman-Kolmogorov equation, the
Fokker-Planck equation for fractals is derived.
The fractional generalization of the
integral Maxwell equations is considered to describe
electric and magnetic fields on fractal.
3. Elements of fractional statistical mechanics on fractals
are considered. Fractional generalizations of normalization
conditions, average values and reduced distribution
functions are defined to describe processes in noninteger
dimensional phase-space. Fractional systems and its simple
examples.
Liouville, Bogoliubov, and Vlasov equations for phase space
fractals are derived.
Pseudo-chaotic orbits of kicked
oscillators
John Lowenstein, New York University, USA
We consider a one-dimensional harmonic oscillator which
receives delta-function kicks in resonance with its natural
frequency, where the kick amplitude is a periodic function
of position. When this function is smooth, e.g. sinusoidal,
one obtains a planar Poincar\`{e} section with crystalline
or quasi-crystalline symmetry and chaotic orbits which
extend to infinity within a web-like layer. The case of a
piecewise linear kick amplitude (sawtooth function), on the
other hand, leads to a much more restricted web, whose
orbits, while not truly chaotic, nevertheless are capable,
given sufficient time, of generating considerable geometric
complexity in phase space. Such behavior is known as
pseudo-chaos, and it is the asymptotic long-time behavior of
the pseudo-chaotic kicked-oscillator orbits which is the
primary focus of these lectures.
Persistent fluctuations
George Zaslavsky, New York University
Persistent fluctuations (PF) are the deviations from an
equilibrium state of dynamical system with non-Gaussian
distribution of the fluctuations. The main feature of the
distribution function is the power type tail that makes mean
values of the fluctuations infinite beginning from some
power. We expose possible reasons for the appearance of PF.
The most important (or better known) reason is the
stickiness of orbits to some set of objects . There will be
an attempt to classify the types of stickiness. Particularly
we introduce a sticky set and demonstrate its appearance and
topological structure for different maps, billiards, and
others. The demonstration includes riddle orbits for the
saw-tooth map with zero Lyapunov exponent. After describing
the origin and character of sticky orbits , we are
discussing their role in the foundation of statistical
physics. Particularly we will demonstrate few dynamical
models that work similarly to the Maxwell's Demon. The
demonstration of the persistent fluctuations in such type of
systems will be presentsd and discussed in details.
Introduction to q-breathers:
existence, localization, scaling and applications
Sergej Flach, Max-Planck-Institut fuer Physik komplexer
Systeme
The dynamics of a spatially confined (finite) Hamiltonian
system disentangles into exact normal mode excitations in
the harmonic approximation. Each separate excitation at a
fixed energy is a uniquely defined periodic orbit (PO).
These POs are continued into the anharmonic domain when
normal modes interact. The resulting q-breathers are
time-periodic solutions which exponentially localize in the
normal mode space (Phys. Rev. Lett. 95, 064102 (2005)). The
properties of q-breathers and small perturbations of them
account for all major ingredients of the famous 51 year old
Fermi-Pasta-Ulam (FPU) problem of (non)equipartition in a
nonlinear atomic chain (Phys. Rev. E 73, 036618 (2006)). By
construction q-breathers are obtained in two- and
three-dimensional lattices as well (Phys. Rev. Lett. 97,
025505 (2006)) and may become relevant objects to study the
energy locking and flow between normal modes of finite
systems. Further scaling of q-breathers to infinitely large
system sizes (nlin.PS/0607019)) provides rigorous results in
the century old problem of anharmonic lattice vibrations of
crystals. We find that at any finite nonzero energy density
(temperature) and nonlinearity coefficient an effective
nonlinearity parameter defines a wave vector range around
the band edges of the harmonic phonon dispersion where the
phonon concept breaks down completely due to strong
interaction, resonances and complete delocalization in
normal mode space. In particular these findings explain
scaling properties of anomalous heat conductivity studies in
large FPU chains where one needs to overcome the ballistic
regime of long wavelength phonons. Additional discrete
symmetries induce degeneracies between harmonic normal mode
frequencies. That leads to even richer q-breather
structures, including states with a vortex like flow of
energy among a few normal modes of the system. Finally I
will discuss extensions of the concept to (discrete)
nonlinear Schroedinger equations and discuss quantization of
q-breathers.
Escape from a circle and Riemann hypotheses
Leonid Bunimovich, Georgia Institute of Technology
A comparison of escape rates in open systems with one and with many "holes" may provide an interesting information about dynamics of the corresponding closed systems. It seems to be a quite challenging problem though, as, seemingly the simplest problem of this type, a comparison of escape rates from a circular billiard with one and with two holes, turned out to be equivalent to the Riemann hypotheses.
Application of dynamical
systems theory to fluids in and out of thermodynamic
equilibrium
Harald Posch, University of Vienne
The phase-space trajectory of many-body systems such as
atomic fluids is dynamically unstable. This instability is
described by a set of rate constants, the Lyapunov
exponents. We demonstrate that perturbations associated with
the large exponents are localized in space. However,
perturbation vectors connected with the smallest positive
Lyapunov exponents exhibit coherent patterns in space, the
so-called "Lyapunov modes". They were first observed
for hard-ball fluids in one, two and three dimension. Using
Fourier-transformation methods, they are shown to exist also
in soft-particle systems. We discuss the symmetry properties
and remark on the dynamics of the modes.
In nonequilibrium stationary states the sum of all Lyapunov
exponents is negative, an indication of the fractal nature
of the phase-space probability density. We review some
recent results for such systems with dynamical or
stochastic temperature control.
Bose atoms in optical lattices:
Statistical mechanics with a few particles
Andrey Kolovsky, Max-Planck-Institute for Physics of
Complex Systems
Ultracold atoms in optical lattices constitute an intense
research activity both in experimental and theoretical
physics. Up to now this system has mostly been used for
modelling the fundamental Hamiltonians of solid state theory
[see, M.~Greiner {\em et. al.}, Nature \textbf{415}, 39
(2002), for example] where the number of particles is
macroscopically large. However, the recent progress with
manipulating a countable number of atoms makes it possible
to build a system of arbitrary size, ranging from
microscopic to macroscopic. We discuss the spectral and
dynamical properties of a few ($N\sim 10$) interacting Bose
atoms in 1D lattices with a few sites. We show that
interacting Bose atoms in a lattice is generally a quantum
chaotic system and indicate the region of chaos in the
system's parameter space. A special attention is paied to
physical manifestations of chaos. In patticular, we consider
the problem of conductivity with cold atoms, where the
atomic current is induced by a static (for example,
gravitational) force. Universal Isaki-Tsu current-voltage
dependence is obtained from the first principle.
Fundamentals of
geophysical fluid dynamics (GFD)
Vladimir Tseitline, Ecole Normale Superieure
In this introductory lecture I will introduce the basic
notions and models in GFD, and describe the fundamental
dynamical processes. Interaction and/or dynamical splitting
of fast (wave) and slow (vortex) motion will be discussed in
the context of nonlinear geostrophic adjustment process.
Equatorial
dynamics and geostrophic adjustment in the equatorial region
Vladimir Tseitline, Ecole Normale Superieure
In this lecture the fundamentals of GFD in the equatorial
region will be explained and families of equatorial waves
and their properies will be described. We will then address
the problem of nonlinear geostrophic adjustment in the
equatorial region, and associated nonlinear dynamics of
equatorial waves. Equatorial Rossby-wave solitons and modons
and Kelvin wave fronts and their interactions will be
discussed in this context. The anomalous transport
properties related to nonlinear dynamics of equatorial waves
will be presented.
Weak turbulence of
equatorial waves
Vladimir Tseitline, Ecole Normale Superieure
In this lecture, after reminding basic ideas and methods
of the so-called weak (or wave-) turbulence theory, we will
apply it to statistical ensembles of short equatorial waves.
Anisotropic equilibrium spectra of westward and eastward
moving ensembles of equatorial waves will be obtained.
Nonlinear wave
phenomena in the semi-transparent equatorial wave-guide
Vladimir Tseitline, Ecole Normale Superieure
In this lecture we will show how equatorial waves,
trapped in the vicinity of equator may be parametrically
excited by passing non-trapped waves (coming e.g. from
mid-latitudes). This process leads to Ginzburg - Landau type
dynamics for the envelopes of trapped waves which could
explain some slow-time (climatic) variability in the
equatorial region.
Transport and
equilibration within finite quantum systems
Jochen Gemmer, Universität Osnabrück
We essentially adress two different but connected
subjects:
- The (possible) approach to equilibrium of few level
quantum sytems which are coupled to quantum environments
that may only consist of a few particles.
- Transport within finite modular quantum systems, i.e.,
sytems that may be decomposed into few weakly
interacting subunits which are located on some kind of lattice.
In both cases, we concentrate on "design models" which
feature some crucial properties of real systems but
completely neglect others. Due to this abstraction the
models may be viewed as models for the transport of any
conserved quantity like e.g. energy, particles, etc.
To reconcile the irreversibility of equilibration with the
reversibilty of the underlying Schroedinger equation we
exploit the concept of Hilbert space averages.
Particles dynamics in regular and chaotic flows of vortices and Charney-Hasegawa-Mima model
Xavier Laurent Yvon Leoncini, Université de Provence -
CNRS
Part 1: Particles dynamics in regular flows
In this part we will consider the dynamics of passive
tracers in a system of three point vortices and in an array
of vortices. These dynamics are all particular examples of
chaotic advection. This phenomenon has fundamental
implication in various physical systems. On large scales one
may for instance think as plasma physics (confinement),
geophysical flows (pollution). But also on a much smaller
scale for micro-fluidic devices, for which chaotic advection
seems to be the best candidate in order to trigger mixing
without breaking anything. For the point vortex flow some
motivation of the choice of the flow and the dynamics of
vortices will be given. Then transport properties of the
tracers will be discussed and the anomalous effects induced
by the stickiness around regular islands presented. While
for the array of vortices I will present a way to perturb
the flow in order to enhance mixing while limiting
transport.
Part 2: Particle dynamics in chaotic flows
In this part we will consider the advection of passive
particles in flows with four point vortices and more, as
well as flows described by the Charney-Hasegawa-Mima
equation. In these settings the flows are not integrable,
thus characterizing the dynamics of tracers explaining
transport properties are not as intuitive as for integrable
periodic flows for which Poincaré section of the phase space
can be performed. The transport of particles will be
analyzed, and the anomalous behavior will be linked to the
presence in these flows of an "hidden" order namely chaotic
jets. Within these jets particles remain for large times in
each other's vicinity, which affects the mixing properties
of the flow and as a consequence transport as well. One will
also emphasize that looking for these jets can be thought of
as actually making measurements of space-time complexity.
Finally the similarity in transport properties of the
various flows will be emphazised as well as the possible use
of fractional kinetics as a possible useful tool, in order
to model and consider transport properties.
Heat conduction in
harmonic and anharmonic lattices
Raman Research Institute, Bangalore, India
I will discuss a formalism for heat transport in open
systems using generalized Langevin equations and
nonequilibrium Green's function. As applications I will
discuss heat conduction in harmonic lattices in the presence
of (i) disorder and (ii) self-consistent reservoirs. I will
also discuss some recent surprising results on heat
conduction in anharmonic lattices.
Low dimensional
Hamiltonian systems
Vered Rom-Kedar, Weizmann Institute, Israel
Lecture 1: From near integrable Hamiltonians to steep
billiard potentials.
The emergence of the two limits (integrable Hamiltonians and
billiards) as skeletons for studying nearby systems will be
introduced with some motivating examples. The connection
between these two limits will be discussed. Then, we will
start with the study of the integrable limit by reviewing
some basic notions regarding the structure of the level
sets.
Lecture 2: Two degrees of freedom near integrable
Hamiltonians
The framework of hierarchy of bifurcations for studying near
integarble Hamiltonian systems will be developed and
explained. It will be applied to several prototype examples,
and the implications of it on the perturbed orbit structure
will be shown. In particular the emergence of parabolic
resonances will be discussed.
(based on joint works with E. Shlizerman and A.
Litvak-Hinenzon)
Lecture 3: Classifications of instabilities in near
integrable Hamiltonians.
The main ideas leading to the complete classification of the
foliations of non-degenerate integrable 2 d.o.f. Hamiltonian
systems will be intoduced. Partial results regarding
representations of the integrable systems and the
instability mechanisms in higher dimensional systems will be
discussed.
(based on joint works with E. Shlizerman, A. Litvak-Hinenzon
and M. Radnovic)
Lecture 4: Steep billiard potentials
Various billiard models serve as approximations to the
classical and semi-classical motion of particles in some
steep potentials (e.g. for studying classical molecular
dynamics, cold atom's motion in dark optical traps and
microwave dynamics). Here we develop methodologies for
examining the validity and accuracy of this approximation.
These methodologies are used to establish that on one hand
separatrix splittings which appear in the billiard map
persist for the smooth flow, and on the other, that even
when the billiard is ergodic and hyperbolic (Sinai billiard)
a steep smooth flow which is arbitrarily close to this
billiard has stability islands. The former results are
established for any dimension and geometry. The latter
results are established analytically in 2 dimensions and
numerically in n dimensions.
(Based on joint works with A. Rapoport and D. Turaev.)
« Back...
Dynamical Tunneling in
a Mixed Phase Space Roland Ketzmerick TU Dresden
Roland Ketzmerik,
Institut für Theoretische Physik, Germany
The phase space of mixed systems consists of regular islands which in two dimensions are dynamically separated from the chaotic sea. Quantum mechanically these phase space regions are connected by dynamical tunneling. We derive a simple formula for the tunneling rates that incorporates the properties of the regular island and the chaotic dynamics. It applies to the case when Plancks constant is a few times smaller than the size of the island, where resonance assisted tunneling can still be neglected. We demonstrate for some kicked systems that it gives excellent agreement with numerically determined tunneling rates.
It will be discussed why it is difficult to evaluate the formula in general.
Large deviation
theory and its many applications in statistical physics
Hugo Touchette, Queen Mary, University of London,
United Kingdom
My goal in this talk is to give a basic introduction to
the theory of large deviations, and discuss a few of its
many applications in statistical physics. The talk will be
divided into four sections covering four important classes
of mathematical and physical systems:
- sums of random variables
- equilibrium systems
- non-equilibrium systems
- multifractals and chaotic maps.
The case of sums of random variables will be presented to
illustrate in a very simple way the basic principles and
results of large deviation theory. These results will then
be used to re-formulate in the language of large deviations
many known results of statistical physics related to the
other topics listed above. In going through this exercise of
're-transcribing physics in a different language', I hope to
convey the feeling that many results of statistical physics
are large deviation results in disguise, and that large
deviation theory provides, in the end, a useful, unified
framework in which to formulate and solve problems of
statistical physics.
Complexity of 2D
microcavity lasers
Takahisa Harayama, ATR Wave Engineering Lab, Japan
Various kinds of devices such as lasers and musical
instruments utilize stationary wave oscillations in resonant
cavities. In order to maintain the stationary oscillation in
these devices, nonlinearity is essential in the mechanism
for balancing the pumping of the external energy and the
decay of the wave of the quasi-stable resonance in the
resonant cavity. Besides, the interaction between
nonlinearities and the morphology of the boundary condition
imposed on a resonating wave system by the shape of the
cavity is also very important for determining the modes of
oscillation.
One-dimensional simple shapes have been used for laser
cavities because they are suitable for fabrication as well
as application of directional emission. However, recent
advances in processing technology of dry-etching for
semiconductor laser diodes have made it possible to
fabricate two-dimensional (2D) microcavity lasers of
arbitrary 2D shapes with potential applications of 2D
emission of laser light in optical communications, optical
integrated circuits, and optical sensing.
We will review the theory of 2D mcirocavity lasers, and
discuss their complex dynamics and applications.
Coupled heat and matter flow in
a simple hamiltonian system
Francois Leyvraz, Cenrto de Ciencias Fisicas
We study heat and matter transport in a quasi-one
dimensional Lorentz gas with freely rotating circular
scatterers which scatter point particles via perfectly rough
collisions. Upon imposing either any combination of
temperature and chemical potential gradients, we find that a
stationary state is attained for which local thermal
equilibrium holds. We find that transport in this system is
normal, that is, the transport coefficients which
characterize the flow of heat and matter are finite in the
thermodynamic limit. Furthermore, the two flows are
non-trivially coupled and the Green-Kubo relations, and
hence Onsager repriprocity, are found to hold within
numerical accuracy. This system is therefore one of the
simplest possible to show fully realistic transport
behaviour while still being exactly solvable at the level of
equilibrium statistical mechanics.
Non-Equilibrium Properties of
Random Matrix Driven Systems
Dimitri Kusnezov, Yale University
Random Matrix ensembles, as a quantum realization of
classically chaotic Hamiltonians, provide an approach to
understanding the properties of non-equilibium quantum
systems. Using diagrammatic methods in the large N limit of
the theory, it is possible to set up non-equilibrium steady
state boundary conditions on arbitrary systems and evaluate
the resulting behavior.
Perturbative methods for
dynamical systems theory
Federico Bonetto, Georgia Institute of Technology,
USA
In recent years we have developed some perturbative tools
to analyze Dynamical Systems. The goal is to obtain
constructive and explicit expression for the quantity of
relevance having in mind mainly possible application to
Nonequilibrium Statistical Mechanics. Among the results are
the construction of the SRB measure for Coupled Anosov Maps
and for Anosov Flows. I'll review some of this results
together with possible application to open problems in
Dynamical Systems Theory.
Weighted Technological Networks
Driven By Traffic Flow
Bing-Hong Wang, Institute of Theoretical Physics,
Department of Modern Physics,
University of Science and Technology of China, China
For most technical networks, the interplay of dynamics,
traffic and topology is assumed crucial to their evolution.
We propose a traffic flow driven evolution model of weighted
technological networks. By introducing a general
strength-coupling mechanism under which the traffic and
topology mutually interact, the model gives power-law
distributions of degree, weight and strength, as confirmed
in many real networks. Particularly, depending on a
parameter W that controls the total weight growth of the
system, the nontrivial clustering coefficient C, degree
assortativity coefficient r and degree-strength
correlation are all in consistence with empirical evidences.
Weighted Technological Networks Driven By Traffic Flow
Suggestion of New Routing Strategy
for the Future Communication Networks
Bing-Hong Wang, Institute of Theoretical Physics,
Department of Modern Physics,
University of Science and Technology of China, China
To improve the transportation efficiency on the
communication networks, we suggest several new routing
strategies. Instead of using the routing strategy for
shortest path, we give a generalized routing algorithm to
find the so-called efficient path, which considers the
possible congestion in the nodes along actual paths. Since
the nodes with largest degree are very susceptible to
traffic congestion, an effective way to improve traffic and
control congestion, as our new strategy, can be as
redistributing traffic load in central nodes to other
non-central nodes. Simulation results indicate that the
network capability in processing traffic is improved more
than 10 times by optimizing the efficient path, which is in
good agreement with the analysis.
Furthermore, a packet routing strategy with a tunable
parameter based on the local structural information of a
scale-free network was proposed and investigated. As free
traffic flow on the communication networks is a key to their
normal and efficient functioning, we focus on the network
capacity that can be measured by the critical point of phase
transition from free flow to congestion. Simulations show
that the maximal capacity corresponds to a parameter with
negative value opposite with our usual intuition. To explain
this, we investigate the number of packets of each node
depending on its degree in the free flow state and observe
the power law behavior. Other dynamic properties including
average packet traveling time and traffic load are also
studied. Inspiringly, our results indicate that some
fundamental relationships exist between the dynamics of
synchronization and traffic on the scale-free networks.
The above local routing protocol can also be generalized to
the next-nearest-neighbor search strategy case. It is found
that by tuning a strategy parameter, the scale-free network
capacity measured by the order parameter is considerably
enhanced compared to the normal next-nearest-neighbor
strategy. Due to the low cost of acquiring
next-nearest-neighbor information and the strongly improved
network capacity, our strategy may be useful for the
protocol designing of future communication networks.
The efficiency of traffic routing on communication networks
should be reflected by two key measurements, i.e., the
network capacity and the average travel time of data
packets. We propose furthermore a mixing routing strategy by
integrating local static and dynamic information for
enhancing the efficiency of traffic on scale-free networks.
The strategy is governed by a single parameter. Simulation
results show that maximizing the network capacity and
reducing the packet travel time can generate an optimal
parameter value. Compared with the strategy of adopting
exclusive local static information, the new strategy shows
its advantages in improving the efficiency of the system.
The detailed analysis of the mixing strategy is provided for
explaining its effects on traffic routing. The work
indicates that effectively utilizing the larger degree nodes
plays a key role for speeding up the information
transmission speed in the communication network and other
scale-free traffic systems.
« Back...
Measures and dynamics of
entanglement
Andreas Buchleitner, Max Planck Institute for the
Physics of Complex Systems, Germany
I'll review some recent progress on mixed state
entanglement measures, as well as of their use for assessing
the time evolution of entanglement under environment
coupling. Furthermore, an experimental approach for the
direct measurement of entanglement will be discussed.
Condensation on weighted
scale-free networks
Zonghua Liu, Institute of Theoretical Physics and
Department of Physics, East China Normal University, China
We study the condensation phenomenon in a zero range
process on
weighted scale-free networks in order to show how the
weighted
transport influences the particle condensation. Instead of
the
approach of grand canonical ensemble which is generally used
in a
zero range process, we introduce an alternate approach of
the mean
field equations to study the dynamics of particle transport.
We
find that the condensation on scale-free network is easier
to
occur in the case of weighted transport than in the case of
weight-free. In the weighted transport, especially, a
dynamical
condensation is even possible for the case of no interaction
among
particles, which is impossible in the case of weight-free.
Visual Chaos
Leonid Bunimovich, Georgia Institute of Technology
I'll explain on very simple examples what strongly
chaotic, weakly chaotic, mixed, intermittent and integrable
motion mean. The talk will be accessible for undegraduates.
Fractal asymptotics
Carl Dettmann, University of Bristol
I will discuss exact expansions and discrete scale invariance (aka log- periodic oscillations, complex dimensions) first for exactly self- similar Cantor sets, then a more difficult problem of the escape rate of a repeller in the presence of small stochastic perturbations. In the process I will give an introduction to a number of widely applicable methods of physical asymptotics, including Mellin transforms, periodic orbit theory and Borel resummation.
Measure Synchronization in
Coupled Hamiltonian Systems
Xingang Wang, National University of Singapore
Chaos synchronization so far has been exclusively studied
for coupled dissipative systems, while for Hamiltonian
systems, due to the conservation of phase space,
trajectories are not allowed to be converged. However, based
on the definition of system measurement, synchronization can
still be established in coupled Hamiltonian systems. In
specific, for two coupled Hamiltonian oscillators, there
exists a transition from a state where the two oscillators
visit different phase space domains to a state where the two
oscillators share the same domain. For the latter, we say
measure synchronization is reached iff that, given the time
evolution longer enough, any state of one oscillator could
be arbitrarily closed by the obit of another oscillator. One
direct observation is that all the macroscopic quantities,
e.g. the energy, of the synchronized systems should be the
same. Since synchronization, despite of the forms, always
reflects some degree of coherence between coupled systems,
it is of great interest to explore its roles played in
fundamental statistics problems, e.g. the FPU problem, and
in practical applications, e.g. heat conductivity.
In this study I shall begin by introducing the basic
concepts in measure synchronization, including the
phenomenon, the definition, and the role of the critical
coupling; then I shall go on to characterize the transition
from non-synchronization to synchronization via the
variations of some macroscopic quantities such as the bare
energy, the phase coherence, and the bifurcations; when
oscillators are coupled in a chain, I shall show you the
phenomenon of partial synchronization and the process of
pattern formation via cluster integrations; Finally, I shall
take another approach to the FPU problem by measure
synchronization and show you some new findings which might
provide some solutions from a very different angle.
Quantum Brownian motion and the
Third law of Thermodynamics
Peter Hänggi, Universität Augsburg
The quantum thermodynamic behavior of small systems is
investigated in presence of finite quantum dissipation. We
consider the archetype cases of a damped harmonic oscillator
and a free quantum Brownian particle. A main finding is that
quantum dissipation helps to ensure the validity of the
Third Law. For the quantum oscillator, finite damping
replaces the zero-coupling result of an exponential
suppression of the specific heat at low temperatures by a
power-law behavior. Rather intriguing is the behavior of the
free quantum Brownian particle. In this case, quantum
dissipation is able to restore the Third Law: Instead of
being constant down to zero temperature, the specific heat
now vanishes proportional to temperature with an amplitude
that is inversely proportional to the ohmic dissipation
strength. A distinct subtlety of finite quantum dissipation
is the result that the various thermodynamic functions of
the sub-system do not only depend on the dissipation
strength but depend as well on the prescription employed in
their definition [1].
[1] P. Hänggi and G.L. Ingold, Acta Physica Polonica B 37,
1537–1550 (2006).
Stationary Hamiltonian dynamics
under dc-bias
Peter Hänggi, Universität Augsburg
We obtain stationary transport in a Hamiltonian system
with ac driving in the presence of a dc bias. A particle in
a periodic potential under the influence of a time-periodic
field possesses a mixed phase space with regular and chaotic
components. An additional external dc bias allows to
separate effectively these structures. We show the existence
of a stationary current which originates from the persisting
invariant manifolds (regular islands, periodic orbits, and
cantori). The transient dynamics of the accelerated chaotic
domain separates fast chaotic motion from ballistic type
trajectories which stick to the vicinity of the invariant
submanifold. Experimental studies with cold atoms in
laser-induced optical lattices are ideal candidates for the
observation of these unexpected findings.
(Based on joint works with Sergey Denisov and Sergej Flach)
Microscopic physics of systems
under thermal gradients
Kenichiro Aoki, Keio University, Japan
Some microscopic aspects of systems with heat flow are
investigated. We quantitatively analyze the behavior of
systems close and far from equilibrium. Local and global
properties such as validity of local equilibrium and linear
response are studied. The relation between heat flow and
dimensional loss in the phase space is also discussed.
Heat transport in quantum systems
Keiji Saito, Tokyo University, Japan
Recently heat transport in quantum magnetic systems is
intensively studied in experiments.
As well known, isotropic 1/2-Heisenberg chain shows
ballistic heat transport.
Recent experiments show such ballistic nature in heat
transport. In many other systems, magnon mainly conveys heat
at low temperature. Magnetic systems are very useful to to
study fundamental properties in quantum heat transport both
theoretically and experimentally.
I first systematically study how the Fourier heat law is
realized in a one-dimensional quantum magnetic system using
the transverse Ising system. When the system is integrable
and the energy statistics is well described by the Poisson
statistics, ballistic transport is observed. On the other
hand, when the systems is nonintegrable and the energy
statistics is described by the random matrices, the Fourier
heat law is observed. I will show the local equilibrium
properties and energy profiles using the quantum master
equation and also convergence of current-current correlation
function in the Green-Kubo formula.
I next consider controlling of heat in magnetic systems. The
characteristics of heat flow in a coupled magnetic system
(Heisenberg chains) are studied. The coupled system is
composed of a gapped chain and a gapless chain. The system
size is assumed to be quite small so that the mean free path
is comparable to it. When the parameter set of the
temperatures of reservoirs is exchanged, the characteristics
of heat flow are studied with the Keldysh Green function
technique. The asymmetry of current is found in the presence
of local equilibrium process at the contact between the
magnetic systems.
Field electron emission from
nanotubes and nanowires: Large-scale simulations and
theoretical models
Zhibing Li, Zhong Shan University, China
Field electron emission (FE) is a quantum tunneling
process, in which electrons are injected from materials into
the vacuum under the influence of an applied electric field.
The excellent FE properties of quasi-one-dimensional
materials, such as carbon nanotubes (CNT) and various
nanowires, have provided a great opportunity for
popularization of the electron sources based on FE
technique. The applications of FE have included flat panel
displays, high-power vacuum electronic devices,
microwave-generation devices, and vacuum microelectronic
devices, etc.
On the side of theoretical interest, it is a challenge to
simulate and understand these systems that consists of huge
number of freedoms yet still feasible for computer
simulation. The quasi-one-dimensional system for the purpose
of FE is a typical multi-scale system in which the electric
field, electron current, and the detail atomic and
electronic structures play role together, in the
length-scale covered angstroms and micrometers. New
simulation algorithms should be developed for the
multi-scale systems. To guide the simulations and
experiments, theoretical models are always important.
The following contents will be addressed:
(1) An introduction to a quick quantum mechanical simulation
algorithm for CNTs;
(2) The field emission mechanism responsible for the
superior FE properties of CNTs;
(3) The apex structure effects on the FE of CNTs;
(4) A model for FE of CNTs;
(5) A model for FE of nanowires made in wide band-gap
semiconductors.
Partial Synchronization on Complex
Networks with Applications
Zhi Gang Zheng, Beijing Normal University, China
Synchronization, as a universal cooperative behavior and
a fundamental mechanism in nature, has been extensively
studied in relating to numerous phenomena in physics,
chemistry, and biology. In recent years, there has been a
growing interest in the synchronization of spatiotemporal
systems, especially in synchronous dynamics on networks.
Network topology plays an important role in governing the
collective dynamics. Synchronizations on typical complex
networks, e.g., on small-world networks or scale-free
networks, have been investigated recently.
In spite of these efforts, much less was explored for
partial synchronization prior to the global case. Moreover,
the mechanism for synchronization on complex networks is
still not clear. A good understanding of this issue should
be relevant to many collective behaviors in spatiotemporal
systems, especially in complex networks.
In this talk, partial synchronization (PaS) on regular
networks with a few non-local links is explored. Different
PaS patterns out of the symmetry breaking are observed for
different ways of non-local couplings. The criterion for the
emergence of PaS is studied. The emergence of PaS is related
to the loss of degeneration in Lyapunov exponent spectrum.
Theoretical and numerical analysis indicate that non-local
coupling may drastically change the dynamical feature of the
network, emphasizing the important topological dependence of
collective dynamics on complex networks. Furthermore, the
criterion we proposed above can be well applied to the
studies of synchronizations between spatiotemporal systems
with sparse couplings.
Cooperative Directional Transport in
Two- Dimensional Ratchet Potential Fields
Zhi Gang Zheng, Beijing Normal University, China
In recent years, much effort has been devoted to
understanding the nonequilibrium mechanism of generating net
currents by the rectification of thermal fluctuations in the
presence of certain drivings with temporally, spatially, and
statistically zero mean. directed transport (DT) has been
observed in the absence of any macroscopic gradient of
forces, if only the substrate potential exhibits the spatial
asymmetry and detailed balance is broken. These explorations
helped us to get a deeper understanding of the mechanism of
many phenomena in molecular motors, flux dynamics in
superconductors, Josephson junctions arrays, ladders, and
lines, transport in quantum dots, nano-device design,
particle separator, and solid surfaces treatment.
In recent years there have been a number of explorations on
directed transport in coupled systems, e.g., the rocking
overdamped ratchet lattice with harmonic couplings, ratchet
motion of particles with hard-core interactions,
asymmetrically coupled lattice in symmetric potentials
without external forces, ratchet motion by breaking the
spatiotemporal symmetries, and so on. However, till now very
few explorations have been focused on cooperative transport
in two-dimensional substrates. In this talk, we propose a
mechanism that cooperative directional transport can be
achieved in one direction through the zero-mean drivings in
the other direction. The energy inputted by the external
drivings can be translated to the work for directional
motion only by mutual couplings of individual elements. This
mechanism can be well applied to understand the transport of
polymers in microtubules.
Fidelity Decay: Theory and
experiment
Thomas Seligman, Centro Internacional de Ciencias A.C
Following a recent review with T. Gorin, T. Prosen and M.
Znidaric (Phys rep (in press)). I shall give a discussion of
fidelity decay in the correlation function approach. After
discussing some general features of integrable and chaotic
systems we shall compare random matrix results with
experiments elastic bodies and microwave cavities.
Long wave length phonons of carbon
nanotubes
Shengli Zhang, Xian Jiaotong University
Carbon nanotubes have attracted a great deal of interest
because of unique properties and a great potential in the
development of nanodevices and nanomaterials. Their thermal
properties, such as specific heat and thermal conductivity,
are important research fields. Making use of the continuum
model, phonon dispersion relations are derived numerically
and analytically. Neglecting the interaction between
twisting and longitudinal stress, the phonon dispersion
relations are determined analytically. The frequency of the
breathing mode is found to be inversely proportional to the
radius of SWNT. For a (10, 10) SWNT, its resonance spectral
wave number is found to be 170.76 (1/cm); which is
consistent with experimental observation (Science 275, 187
(1997)). Four acoustic modes have been analyzed and these
results are found to be in good agreement with the numerical
calculation of Saito (Phys. Rev. B 57 (1998) 4145). Our
results for the PDOS agree well with experimental data
(Phys. Rev. Lett. 85, 5222 (2000)). The expressions for
specific heat versus temperature are derived theoretically,
and we find that the lattice wave propagating along the
length of the SWNT plays the principle role in deciding the
value of the specific heat. Our theoretical results agree
well with experimental data (Science 289,1730 (2000)). The
vibrational spectra of double-wall carbon nanotubes (DWNTs)
are studied theoretically within a continuum model. The
phonon dispersion relations are derived numerically and
analytically, and two radial breathing modes (RBMs) are
calculated analytically, which agree with experimental data
[Phys. Rev. B 66, 075416 (2002)]. By comparison of the RBM
frequencies with the ones of isolated SWNT, we find that
there is a systematic upward shift for DWNTs RBM frequencies
due to the interlayer van der Waals interactions. In case of
counterphase modes, the upshift magnitude of RBM frequencies
increases with an increase of the outer-layer radius R.
However, for inphase RBMs, the upshift magnitude of RBM
frequencies may increase or decrease with an increase of R.
We discuss four acoustic phonons and find that there is no
transverse acoustic phonon in DWNTs. The general phonon
dispersion relations are calculated. According to
experimental data, we assign the specific chiralities to the
inner and outer layers by use of two RBM frequencies of the
DWNTs, and then calculate the diameters of the two layers
and even the interlayer distance. The interlayer spacing
between two layers b is found to be not a constant, but is
in the range 0.321 to 0.402 nm, which agrees well with
experimental data [W. Ren et al., Chem. Phys. Lett. 359, 196
(2002); R. Pfeiffer et al., Phys. Rev. Lett. 90, 225501
(2003)]. The values of both the upshift and downshift of
DWNTs’ RBM frequencies are not constant, but vary with the
interlayer spacings b and their chiralities. By calculating
and comparing the case of the chiralities being taken into
account with that of the chiralities being neglected in the
van der Waals forces, we find that the effect of chiralities
should not be neglected in order to assign the chiralities
of DWNTs though the difference between the two RBM
frequencies caused by the different chiralities of inner
and outer tubes is small.
