17,390 research outputs found
Simple deterministic dynamical systems with fractal diffusion coefficients
We analyze a simple model of deterministic diffusion. The model consists of a
one-dimensional periodic array of scatterers in which point particles move from
cell to cell as defined by a piecewise linear map. The microscopic chaotic
scattering process of the map can be changed by a control parameter. This
induces a parameter dependence for the macroscopic diffusion coefficient. We
calculate the diffusion coefficent and the largest eigenmodes of the system by
using Markov partitions and by solving the eigenvalue problems of respective
topological transition matrices. For different boundary conditions we find that
the largest eigenmodes of the map match to the ones of the simple
phenomenological diffusion equation. Our main result is that the difffusion
coefficient exhibits a fractal structure by varying the system parameter. To
understand the origin of this fractal structure, we give qualitative and
quantitative arguments. These arguments relate the sequence of oscillations in
the strength of the parameter-dependent diffusion coefficient to the
microscopic coupling of the single scatterers which changes by varying the
control parameter.Comment: 28 pages (revtex), 12 figures (postscript), submitted to Phys. Rev.
Memory effects in non-adiabatic molecular dynamics at metal surfaces
We study the effect of temporal correlation in a Langevin equation describing
non-adiabatic dynamics at metal surfaces. For a harmonic oscillator the
Langevin equation preserves the quantum dynamics exactly and it is demonstrated
that memory effects are needed in order to conserve the ground state energy of
the oscillator. We then compare the result of Langevin dynamics in a harmonic
potential with a perturbative master equation approach and show that the
Langevin equation gives a better description in the non-perturbative range of
high temperatures and large friction. Unlike the master equation, this approach
is readily extended to anharmonic potentials. Using density functional theory
we calculate representative Langevin trajectories for associative desorption of
N from Ru(0001) and find that memory effects lowers the dissipation of
energy. Finally, we propose an ab-initio scheme to calculate the temporal
correlation function and dynamical friction within density functional theory
Ten reasons why a thermalized system cannot be described by a many-particle wave function
It is widely believed that the underlying reality behind statistical
mechanics is a deterministic and unitary time evolution of a many-particle wave
function, even though this is in conflict with the irreversible, stochastic
nature of statistical mechanics. The usual attempts to resolve this conflict
for instance by appealing to decoherence or eigenstate thermalization are
riddled with problems. This paper considers theoretical physics of thermalized
systems as it is done in practise and shows that all approaches to thermalized
systems presuppose in some form limits to linear superposition and
deterministic time evolution. These considerations include, among others, the
classical limit, extensivity, the concepts of entropy and equilibrium, and
symmetry breaking in phase transitions and quantum measurement. As a
conclusion, the paper argues that the irreversibility and stochasticity of
statistical mechanics should be taken as a true property of nature. It follows
that a gas of a macroscopic number of atoms in thermal equilibrium is best
represented by a collection of wave packets of a size of the order of the
thermal de Broglie wave length, which behave quantum mechanically below this
scale but classically sufficiently far beyond this scale. In particular, these
wave packets must localize again after scattering events, which requires
stochasticity and indicates a connection to the measurement process.Comment: Drastically rewritten version, with more explanations, with three new
reasons added and three old ones merged with other parts of the tex
Chaotic properties of systems with Markov dynamics
We present a general approach for computing the dynamic partition function of
a continuous-time Markov process. The Ruelle topological pressure is identified
with the large deviation function of a physical observable. We construct for
the first time a corresponding finite Kolmogorov-Sinai entropy for these
processes. Then, as an example, the latter is computed for a symmetric
exclusion process. We further present the first exact calculation of the
topological pressure for an N-body stochastic interacting system, namely an
infinite-range Ising model endowed with spin-flip dynamics. Expressions for the
Kolmogorov-Sinai and the topological entropies follow.Comment: 4 pages, to appear in the Physical Review Letter
Nonequilibrium stationary states and equilibrium models with long range interactions
It was recently suggested by Blythe and Evans that a properly defined steady
state normalisation factor can be seen as a partition function of a fictitious
statistical ensemble in which the transition rates of the stochastic process
play the role of fugacities. In analogy with the Lee-Yang description of phase
transition of equilibrium systems, they studied the zeroes in the complex plane
of the normalisation factor in order to find phase transitions in
nonequilibrium steady states. We show that like for equilibrium systems, the
``densities'' associated to the rates are non-decreasing functions of the rates
and therefore one can obtain the location and nature of phase transitions
directly from the analytical properties of the ``densities''. We illustrate
this phenomenon for the asymmetric exclusion process. We actually show that its
normalisation factor coincides with an equilibrium partition function of a walk
model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure
Some Exact Results for the Exclusion Process
The asymmetric simple exclusion process (ASEP) is a paradigm for
non-equilibrium physics that appears as a building block to model various
low-dimensional transport phenomena, ranging from intracellular traffic to
quantum dots. We review some recent results obtained for the system on a
periodic ring by using the Bethe Ansatz. We show that this method allows to
derive analytically many properties of the dynamics of the model such as the
spectral gap and the generating function of the current. We also discuss the
solution of a generalized exclusion process with -species of particles and
explain how a geometric construction inspired from queuing theory sheds light
on the Matrix Product Representation technique that has been very fruitful to
derive exact results for the ASEP.Comment: 21 pages; Proceedings of STATPHYS24 (Cairns, Australia, July 2010
The symmetric heavy-light ansatz
The symmetric heavy-light ansatz is a method for finding the ground state of
any dilute unpolarized system of attractive two-component fermions.
Operationally it can be viewed as a generalization of the Kohn-Sham equations
in density functional theory applied to N-body density correlations. While the
original Hamiltonian has an exact Z_2 symmetry, the heavy-light ansatz breaks
this symmetry by skewing the mass ratio of the two components. In the limit
where one component is infinitely heavy, the many-body problem can be solved in
terms of single-particle orbitals. The original Z_2 symmetry is recovered by
enforcing Z_2 symmetry as a constraint on N-body density correlations for the
two components. For the 1D, 2D, and 3D attractive Hubbard models the method is
in very good agreement with exact Lanczos calculations for few-body systems at
arbitrary coupling. For the 3D attractive Hubbard model there is very good
agreement with lattice Monte Carlo results for many-body systems in the limit
of infinite scattering length.Comment: 38 pages, 13 figures, revised manuscript includes results for 1D, 2D,
and 3
Chaotic and fractal properties of deterministic diffusion-reaction processes
We study the consequences of deterministic chaos for diffusion-controlled
reaction. As an example, we analyze a diffusive-reactive deterministic
multibaker and a parameter-dependent variation of it. We construct the
diffusive and the reactive modes of the models as eigenstates of the
Frobenius-Perron operator. The associated eigenvalues provide the dispersion
relations of diffusion and reaction and, hence, they determine the reaction
rate. For the simplest model we show explicitly that the reaction rate behaves
as phenomenologically expected for one-dimensional diffusion-controlled
reaction. Under parametric variation, we find that both the diffusion
coefficient and the reaction rate have fractal-like dependences on the system
parameter.Comment: 14 pages (revtex), 12 figures (postscript), to appear in CHAO
Quantum field theory of cooperative atom response: Low light intensity
We study the interactions of a possibly dense and/or quantum degenerate gas
with driving light. Both the atoms and the electromagnetic fields are
represented by quantum fields throughout the analysis. We introduce a field
theory version of Markov and Born approximations for the interactions of light
with matter, and devise a procedure whereby certain types of products of atom
and light fields may be put to a desired, essentially normal, order. In the
limit of low light intensity we find a hierarchy of equations of motion for
correlation functions that contain one excited-atom field and one, two, three,
etc., ground state atom fields. It is conjectured that the entire linear
hierarchy may be solved by solving numerically the classical equations for the
coupled system of electromagnetic fields and charged harmonic oscillators. We
discuss the emergence of resonant dipole-dipole interactions and collective
linewidths, and delineate the limits of validity of the column density approach
in terms of non-cooperative atoms by presenting a mathematical example in which
this approach is exact.Comment: 35 pages, RevTe
Thermodynamic formalism for systems with Markov dynamics
The thermodynamic formalism allows one to access the chaotic properties of
equilibrium and out-of-equilibrium systems, by deriving those from a dynamical
partition function. The definition that has been given for this partition
function within the framework of discrete time Markov chains was not suitable
for continuous time Markov dynamics. Here we propose another interpretation of
the definition that allows us to apply the thermodynamic formalism to
continuous time.
We also generalize the formalism --a dynamical Gibbs ensemble construction--
to a whole family of observables and their associated large deviation
functions. This allows us to make the connection between the thermodynamic
formalism and the observable involved in the much-studied fluctuation theorem.
We illustrate our approach on various physical systems: random walks,
exclusion processes, an Ising model and the contact process. In the latter
cases, we identify a signature of the occurrence of dynamical phase
transitions. We show that this signature can already be unravelled using the
simplest dynamical ensemble one could define, based on the number of
configuration changes a system has undergone over an asymptotically large time
window.Comment: 64 pages, LaTeX; version accepted for publication in Journal of
Statistical Physic
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