153 research outputs found
The Adsorption and Collapse Transitions in a Linear Polymer Chain near an Attractive Wall
We deduce the qualitative phase diagram of a long flexible neutral polymer
chain immersed in a poor solvent near an attracting surface using
phenomenological arguments. The actual positions of the phase boundaries are
estimated numerically from series expansion up to 19 sites of a self-attracting
self avoiding walk in three dimensions. In two dimensions, we calculate
analytically phase boundaries in some cases for a partially directed model.
Both the numerical as well as analytical results corroborate the proposed
qualitative phase diagram.Comment: 8 pages, 8 figures, revte
Critical points and quenched disorder: From Harris criterion to rare regions and smearing
We consider the influence of quenched spatial disorder on phase transitions
in classical and quantum systems. We show that rare strong disorder
fluctuations can have dramatic effects on critical points. In classical systems
with sufficiently correlated disorder or in quantum systems with overdamped
dynamics they can completely destroy the sharp phase transition by smearing.
This is caused by effects similar to but stronger than Griffiths phenomena:
True static order can develop on a rare region while the bulk system is still
in the disordered phase. We discuss the thermodynamic behavior in the vicinity
of such a smeared transition using optimal fluctuation theory, and we present
numerical results for a two-dimensional model system.Comment: 10 pages, 5 eps figures, contribution to the Festschrift for Michael
Schreiber's 50th birthday, final version as publishe
Fluctuation-dissipation relationship in chaotic dynamics
We consider a general N-degree-of-freedom dissipative system which admits of
chaotic behaviour. Based on a Fokker-Planck description associated with the
dynamics we establish that the drift and the diffusion coefficients can be
related through a set of stochastic parameters which characterize the steady
state of the dynamical system in a way similar to fluctuation-dissipation
relation in non-equilibrium statistical mechanics. The proposed relationship is
verified by numerical experiments on a driven double well system.Comment: Revtex, 23 pages, 2 figure
Limit theorems for von Mises statistics of a measure preserving transformation
For a measure preserving transformation of a probability space
we investigate almost sure and distributional convergence
of random variables of the form where (called the \emph{kernel})
is a function from to and are appropriate normalizing
constants. We observe that the above random variables are well defined and
belong to provided that the kernel is chosen from the projective
tensor product with We establish a form of the individual ergodic theorem for such
sequences. Next, we give a martingale approximation argument to derive a
central limit theorem in the non-degenerate case (in the sense of the classical
Hoeffding's decomposition). Furthermore, for and a wide class of
canonical kernels we also show that the convergence holds in distribution
towards a quadratic form in independent
standard Gaussian variables . Our results on the
distributional convergence use a --\,invariant filtration as a prerequisite
and are derived from uni- and multivariate martingale approximations
Stochastic Renormalization Group in Percolation: I. Fluctuations and Crossover
A generalization of the Renormalization Group, which describes
order-parameter fluctuations in finite systems, is developed in the specific
context of percolation. This ``Stochastic Renormalization Group'' (SRG)
expresses statistical self-similarity through a non-stationary branching
process. The SRG provides a theoretical basis for analytical or numerical
approximations, both at and away from criticality, whenever the correlation
length is much larger than the lattice spacing (regardless of the system size).
For example, the SRG predicts order-parameter distributions and finite-size
scaling functions for the complete crossover between phases. For percolation,
the simplest SRG describes structural quantities conditional on spanning, such
as the total cluster mass or the minimum chemical distance between two
boundaries. In these cases, the Central Limit Theorem (for independent random
variables) holds at the stable, off-critical fixed points, while a ``Fractal
Central Limit Theorem'' (describing long-range correlations) holds at the
unstable, critical fixed point. This first part of a series of articles
explains these basic concepts and a general theory of crossover. Subsequent
parts will focus on limit theorems and comparisons of small-cell SRG
approximations with simulation results.Comment: 33 pages, 6 figures, to appear in Physica A; v2: some typos corrected
and Eqs. (26)-(27) cast in a simpler (but equivalent) for
Corrections to scaling in 2--dimensional polymer statistics
Writing for the mean
square end--to--end length of a self--avoiding polymer chain of
links, we have calculated for the two--dimensional {\em continuum}
case from a new {\em finite} perturbation method based on the ground state of
Edwards self consistent solution which predicts the (exact) exponent.
This calculation yields . A finite size scaling analysis of data
generated for the continuum using a biased sampling Monte Carlo algorithm
supports this value, as does a re--analysis of exact data for two--dimensional
lattices.Comment: 10 pages of RevTex, 5 Postscript figures. Accepted for publication in
Phys. Rev. B. Brief Reports. Also submitted to J. Phys.
Microscopic Aspects of Stretched Exponential Relaxation (SER) in Homogeneous Molecular and Network Glasses and Polymers
Because the theory of SER is still a work in progress, the phenomenon itself
can be said to be the oldest unsolved problem in science, as it started with
Kohlrausch in 1847. Many electrical and optical phenomena exhibit SER with
probe relaxation I(t) ~ exp[-(t/{\tau}){\beta}], with 0 < {\beta} < 1. Here
{\tau} is a material-sensitive parameter, useful for discussing chemical
trends. The "shape" parameter {\beta} is dimensionless and plays the role of a
non-equilibrium scaling exponent; its value, especially in glasses, is both
practically useful and theoretically significant. The mathematical complexity
of SER is such that rigorous derivations of this peculiar function were not
achieved until the 1970's. The focus of much of the 1970's pioneering work was
spatial relaxation of electronic charge, but SER is a universal phenomenon, and
today atomic and molecular relaxation of glasses and deeply supercooled liquids
provide the most reliable data. As the data base grew, the need for a
quantitative theory increased; this need was finally met by the
diffusion-to-traps topological model, which yields a remarkably simple
expression for the shape parameter {\beta}, given by d*/(d* + 2). At first
sight this expression appears to be identical to d/(d + 2), where d is the
actual spatial dimensionality, as originally derived. The original model,
however, failed to explain much of the data base. Here the theme of earlier
reviews, based on the observation that in the presence of short-range forces
only d* = d = 3 is the actual spatial dimensionality, while for mixed short-
and long-range forces, d* = fd = d/2, is applied to four new spectacular
examples, where it turns out that SER is useful not only for purposes of
quality control, but also for defining what is meant by a glass in novel
contexts. (Please see full abstract in main text
Field Theory Approaches to Nonequilibrium Dynamics
It is explained how field-theoretic methods and the dynamic renormalisation
group (RG) can be applied to study the universal scaling properties of systems
that either undergo a continuous phase transition or display generic scale
invariance, both near and far from thermal equilibrium. Part 1 introduces the
response functional field theory representation of (nonlinear) Langevin
equations. The RG is employed to compute the scaling exponents for several
universality classes governing the critical dynamics near second-order phase
transitions in equilibrium. The effects of reversible mode-coupling terms,
quenching from random initial conditions to the critical point, and violating
the detailed balance constraints are briefly discussed. It is shown how the
same formalism can be applied to nonequilibrium systems such as driven
diffusive lattice gases. Part 2 describes how the master equation for
stochastic particle reaction processes can be mapped onto a field theory
action. The RG is then used to analyse simple diffusion-limited annihilation
reactions as well as generic continuous transitions from active to inactive,
absorbing states, which are characterised by the power laws of (critical)
directed percolation. Certain other important universality classes are
mentioned, and some open issues are listed.Comment: 54 pages, 9 figures, Lecture Notes for Luxembourg Summer School
"Ageing and the Glass Transition", submitted to Springer Lecture Notes in
Physics (www.springeronline/com/series/5304/
Four-point renormalized coupling constant and Callan-Symanzik beta-function in O(N) models
We investigate some issues concerning the zero-momentum four-point
renormalized coupling constant g in the symmetric phase of O(N) models, and the
corresponding Callan-Symanzik beta-function. In the framework of the 1/N
expansion we show that the Callan- Symanzik beta-function is non-analytic at
its zero, i.e. at the fixed-point value g^* of g. This fact calls for a check
of the actual accuracy of the determination of g^* from the resummation of the
d=3 perturbative g-expansion, which is usually performed assuming analyticity
of the beta-function. Two alternative approaches are exploited. We extend the
\epsilon-expansion of g^* to O(\epsilon^4). Quite accurate estimates of g^* are
then obtained by an analysis exploiting the analytic behavior of g^* as
function of d and the known values of g^* for lower-dimensional O(N) models,
i.e. for d=2,1,0. Accurate estimates of g^* are also obtained by a reanalysis
of the strong-coupling expansion of lattice N-vector models allowing for the
leading confluent singularity. The agreement among the g-, \epsilon-, and
strong-coupling expansion results is good for all N. However, at N=0,1,
\epsilon- and strong-coupling expansion favor values of g^* which are sligthly
lower than those obtained by the resummation of the g-expansion assuming
analyticity in the Callan-Symanzik beta-function.Comment: 35 pages (3 figs), added Ref. for GRT, some estimates are revised,
other minor change
A statistical learning strategy for closed-loop control of fluid flows
This work discusses a closed-loop control strategy for complex systems utilizing scarce and streaming data. A discrete embedding space is first built using hash functions applied to the sensor measurements from which a Markov process model is derived, approximating the complex system’s dynamics. A control strategy is then learned using reinforcement learning once rewards relevant with respect to the control objective are identified. This method is designed for experimental configurations, requiring no computations nor prior knowledge of the system, and enjoys intrinsic robustness. It is illustrated on two systems: the control of the transitions of a Lorenz’63 dynamical system, and the control of the drag of a cylinder flow. The method is shown to perform well
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