415 research outputs found
What is the maximum rate at which entropy of a string can increase?
According to Susskind, a string falling toward a black hole spreads
exponentially over the stretched horizon due to repulsive interactions of the
string bits. In this paper such a string is modeled as a self-avoiding walk and
the string entropy is found. It is shown that the rate at which
information/entropy contained in the string spreads is the maximum rate allowed
by quantum theory. The maximum rate at which the black hole entropy can
increase when a string falls into a black hole is also discussed.Comment: 11 pages, no figures; formulas (18), (20) are corrected (the quantum
constant is added), a point concerning a relation between the Hawking and
Hagedorn temperatures is corrected, conclusions unchanged; accepted by
Physical Review D for publicatio
Lattice Green Function (at 0) for the 4d Hypercubic Lattice
The generating function for recurrent Polya walks on the four dimensional
hypercubic lattice is expressed as a Kampe-de-Feriet function. Various
properties of the associated walks are enumerated.Comment: latex, 5 pages, Res. Report 1
Existence of positive representations for complex weights
The necessity of computing integrals with complex weights over manifolds with
a large number of dimensions, e.g., in some field theoretical settings, poses a
problem for the use of Monte Carlo techniques. Here it is shown that very
general complex weight functions P(x) on R^d can be represented by real and
positive weights p(z) on C^d, in the sense that for any observable f, _P
= _p, f(z) being the analytical extension of f(x). The construction is
extended to arbitrary compact Lie groups.Comment: 9 pages, no figures. To appear in J.Phys.
Scaling of the atmosphere of self-avoiding walks
The number of free sites next to the end of a self-avoiding walk is known as
the atmosphere. The average atmosphere can be related to the number of
configurations. Here we study the distribution of atmospheres as a function of
length and how the number of walks of fixed atmosphere scale. Certain bounds on
these numbers can be proved. We use Monte Carlo estimates to verify our
conjectures. Of particular interest are walks that have zero atmosphere, which
are known as trapped. We demonstrate that these walks scale in the same way as
the full set of self-avoiding walks, barring an overall constant factor
Self-avoiding walks crossing a square
We study a restricted class of self-avoiding walks (SAW) which start at the
origin (0, 0), end at , and are entirely contained in the square on the square lattice . The number of distinct
walks is known to grow as . We estimate as well as obtaining strict upper and lower bounds,
We give exact results for the number of SAW of
length for and asymptotic results for .
We also consider the model in which a weight or {\em fugacity} is
associated with each step of the walk. This gives rise to a canonical model of
a phase transition. For the average length of a SAW grows as ,
while for it grows as
. Here is the growth constant of unconstrained SAW in . For we provide numerical evidence, but no proof, that the
average walk length grows as .
We also consider Hamiltonian walks under the same restriction. They are known
to grow as on the same lattice. We give
precise estimates for as well as upper and lower bounds, and prove that
Comment: 27 pages, 9 figures. Paper updated and reorganised following
refereein
Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution
We present an algorithm, based on the iteration of conformal maps, that
produces independent samples of self-avoiding paths in the plane. It is a
discrete process approximating radial Schramm-Loewner evolution growing to
infinity. We focus on the problem of reproducing the parametrization
corresponding to that of lattice models, namely self-avoiding walks on the
lattice, and we propose a strategy that gives rise to discrete paths where
consecutive points lie an approximately constant distance apart from each
other. This new method allows us to tackle two non-trivial features of
self-avoiding walks that critically depend on the parametrization: the
asphericity of a portion of chain and the correction-to-scaling exponent.Comment: 18 pages, 4 figures. Some sections rewritten (including title and
abstract), numerical results added, references added. Accepted for
publication in J. Stat. Phy
Honeycomb lattice polygons and walks as a test of series analysis techniques
We have calculated long series expansions for self-avoiding walks and
polygons on the honeycomb lattice, including series for metric properties such
as mean-squared radius of gyration as well as series for moments of the
area-distribution for polygons. Analysis of the series yields accurate
estimates for the connective constant, critical exponents and amplitudes of
honeycomb self-avoiding walks and polygons. The results from the numerical
analysis agree to a high degree of accuracy with theoretical predictions for
these quantities.Comment: 16 pages, 9 figures, jpconf style files. Presented at the conference
"Counting Complexity: An international workshop on statistical mechanics and
combinatorics." In celebration of Prof. Tony Guttmann's 60th birthda
Polymer-mediated entropic forces between scale-free objects
The number of configurations of a polymer is reduced in the presence of a
barrier or an obstacle. The resulting loss of entropy adds a repulsive
component to other forces generated by interaction potentials. When the
obstructions are scale invariant shapes (such as cones, wedges, lines or
planes) the only relevant length scales are the polymer size R_0 and
characteristic separations, severely constraining the functional form of
entropic forces. Specifically, we consider a polymer (single strand or star)
attached to the tip of a cone, at a separation h from a surface (or another
cone). At close proximity, such that h<<R_0, separation is the only remaining
relevant scale and the entropic force must take the form F=AkT/h. The amplitude
A is universal, and can be related to exponents \eta governing the anomalous
scaling of polymer correlations in the presence of obstacles. We use
analytical, numerical and epsilon-expansion techniques to compute the exponent
\eta for a polymer attached to the tip of the cone (with or without an
additional plate or cone) for ideal and self-avoiding polymers. The entropic
force is of the order of 0.1 pN at 0.1 micron for a single polymer, and can be
increased for a star polymer.Comment: LaTeX, 15 pages, 4 eps figure
Persistence length of a polyelectrolyte in salty water: a Monte-Carlo study
We address the long standing problem of the dependence of the electrostatic
persistence length of a flexible polyelectrolyte (PE) on the screening
length of the solution within the linear Debye-Huckel theory. The
standard Odijk, Skolnick and Fixman (OSF) theory suggests ,
while some variational theories and computer simulations suggest . In this paper, we use Monte-Carlo simulations to study the conformation
of a simple polyelectrolyte. Using four times longer PEs than in previous
simulations and refined methods for the treatment of the simulation data, we
show that the results are consistent with the OSF dependence . The linear charge density of the PE which enters in the coefficient of
this dependence is properly renormalized to take into account local
fluctuations.Comment: 7 pages, 6 figures. Various corrections in text and reference
Information Loss in Coarse Graining of Polymer Configurations via Contact Matrices
Contact matrices provide a coarse grained description of the configuration
omega of a linear chain (polymer or random walk) on Z^n: C_{ij}(omega)=1 when
the distance between the position of the i-th and j-th step are less than or
equal to some distance "a" and C_{ij}(omega)=0 otherwise. We consider models in
which polymers of length N have weights corresponding to simple and
self-avoiding random walks, SRW and SAW, with "a" the minimal permissible
distance. We prove that to leading order in N, the number of matrices equals
the number of walks for SRW, but not for SAW. The coarse grained Shannon
entropies for SRW agree with the fine grained ones for n <= 2, but differs for
n >= 3.Comment: 18 pages, 2 figures, latex2e Main change: the introduction is
rewritten in a less formal way with the main results explained in simple
term
- …