122 research outputs found
Collapsing lattice animals and lattice trees in two dimensions
We present high statistics simulations of weighted lattice bond animals and
lattice trees on the square lattice, with fugacities for each non-bonded
contact and for each bond between two neighbouring monomers. The simulations
are performed using a newly developed sequential sampling method with
resampling, very similar to the pruned-enriched Rosenbluth method (PERM) used
for linear chain polymers. We determine with high precision the line of second
order transitions from an extended to a collapsed phase in the resulting
2-dimensional phase diagram. This line includes critical bond percolation as a
multicritical point, and we verify that this point divides the line into two
different universality classes. One of them corresponds to the collapse driven
by contacts and includes the collapse of (weakly embeddable) trees, but the
other is {\it not yet} bond driven and does not contain the Derrida-Herrmann
model as special point. Instead it ends at a multicritical point where a
transition line between two collapsed phases (one bond-driven and the other
contact-driven) sparks off. The Derrida-Herrmann model is representative for
the bond driven collapse, which then forms the fourth universality class on the
transition line (collapsing trees, critical percolation, intermediate regime,
and Derrida-Herrmann). We obtain very precise estimates for all critical
exponents for collapsing trees. It is already harder to estimate the critical
exponents for the intermediate regime. Finally, it is very difficult to obtain
with our method good estimates of the critical parameters of the
Derrida-Herrmann universality class. As regards the bond-driven to
contact-driven transition in the collapsed phase, we have some evidence for its
existence and rough location, but no precise estimates of critical exponents.Comment: 11 pages, 16 figures, 1 tabl
Polar decompositions of quaternion matrices in indefinite inner product spaces
Polar decompositions of quaternion matrices with respect to a given
indefinite inner product are studied. Necessary and sufficient conditions for
the existence of an -polar decomposition are found. In the process an
equivalent to Witt's theorem on extending -isometries to -unitary
matrices is given for quaternion matrices
Knot localization in adsorbing polymer rings
We study by Monte Carlo simulations a model of knotted polymer ring adsorbing
onto an impenetrable, attractive wall. The polymer is described by a
self-avoiding polygon (SAP) on the cubic lattice. We find that the adsorption
transition temperature, the crossover exponent and the metric exponent
, are the same as in the model where the topology of the ring is
unrestricted. By measuring the average length of the knotted portion of the
ring we are able to show that adsorbed knots are localized. This knot
localization transition is triggered by the adsorption transition but is
accompanied by a less sharp variation of the exponent related to the degree of
localization. Indeed, for a whole interval below the adsorption transition, one
can not exclude a contiuous variation with temperature of this exponent. Deep
into the adsorbed phase we are able to verify that knot localization is strong
and well described in terms of the flat knot model.Comment: 27 pages, 10 figures. Submitter to Phys. Rev.
Knotting probabilities after a local strand passage in unknotted self-avoiding polygons
We investigate the knotting probability after a local strand passage is
performed in an unknotted self-avoiding polygon on the simple cubic lattice. We
assume that two polygon segments have already been brought close together for
the purpose of performing a strand passage, and model this using Theta-SAPs,
polygons that contain the pattern Theta at a fixed location. It is proved that
the number of n-edge Theta-SAPs grows exponentially (with n) at the same rate
as the total number of n-edge unknotted self-avoiding polygons, and that the
same holds for subsets of n-edge Theta-SAPs that yield a specific
after-strand-passage knot-type. Thus the probability of a given
after-strand-passage knot-type does not grow (or decay) exponentially with n,
and we conjecture that instead it approaches a knot-type dependent amplitude
ratio lying strictly between 0 and 1. This is supported by critical exponent
estimates obtained from a new maximum likelihood method for Theta-SAPs that are
generated by a composite (aka multiple) Markov Chain Monte Carlo BFACF
algorithm. We also give strong numerical evidence that the after-strand-passage
knotting probability depends on the local structure around the strand passage
site. Considering both the local structure and the crossing-sign at the strand
passage site, we observe that the more "compact" the local structure, the less
likely the after-strand-passage polygon is to be knotted. This trend is
consistent with results from other strand-passage models, however, we are the
first to note the influence of the crossing-sign information. Two measures of
"compactness" are used: the size of a smallest polygon that contains the
structure and the structure's "opening" angle. The opening angle definition is
consistent with one that is measurable from single molecule DNA experiments.Comment: 31 pages, 12 figures, submitted to Journal of Physics
Punctured polygons and polyominoes on the square lattice
We use the finite lattice method to count the number of punctured staircase
and self-avoiding polygons with up to three holes on the square lattice. New or
radically extended series have been derived for both the perimeter and area
generating functions. We show that the critical point is unchanged by a finite
number of punctures, and that the critical exponent increases by a fixed amount
for each puncture. The increase is 1.5 per puncture when enumerating by
perimeter and 1.0 when enumerating by area. A refined estimate of the
connective constant for polygons by area is given. A similar set of results is
obtained for finitely punctured polyominoes. The exponent increase is proved to
be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure
Simulations of lattice animals and trees
The scaling behaviour of randomly branched polymers in a good solvent is
studied in two to nine dimensions, using as microscopic models lattice animals
and lattice trees on simple hypercubic lattices. As a stochastic sampling
method we use a biased sequential sampling algorithm with re-sampling, similar
to the pruned-enriched Rosenbluth method (PERM) used extensively for linear
polymers. Essentially we start simulating percolation clusters (either site or
bond), re-weigh them according to the animal (tree) ensemble, and prune or
branch the further growth according to a heuristic fitness function. In
contrast to previous applications of PERM, this fitness function is {\it not}
the weight with which the actual configuration would contribute to the
partition sum, but is closely related to it. We obtain high statistics of
animals with up to several thousand sites in all dimension 2 <= d <= 9. In
addition to the partition sum (number of different animals) we estimate
gyration radii and numbers of perimeter sites. In all dimensions we verify the
Parisi-Sourlas prediction, and we verify all exactly known critical exponents
in dimensions 2, 3, 4, and >= 8. In addition, we present the hitherto most
precise estimates for growth constants in d >= 3. For clusters with one site
attached to an attractive surface, we verify the superuniversality of the
cross-over exponent at the adsorption transition predicted by Janssen and
Lyssy. Finally, we discuss the collapse of animals and trees, arguing that our
present version of the algorithm is also efficient for some of the models
studied in this context, but showing that it is {\it not} very efficient for
the `classical' model for collapsing animals.Comment: 17 pages RevTeX, 29 figures include
Equilibrium size of large ring molecules
The equilibrium properties of isolated ring molecules were investigated using
an off-lattice model with no excluded volume but with dynamics that preserve
the topological class. Using an efficient set of long range moves, chains of
more than 2000 monomers were studied. Despite the lack of any excluded volume
interaction, the radius of gyration scaled like that of a self avoiding walk,
as had been previously conjectured. However this scaling was only seen for
chains greater than 500 monomers.Comment: 11 pages, 3 eps figures, latex, psfi
Simulations of grafted polymers in a good solvent
We present improved simulations of three-dimensional self avoiding walks with
one end attached to an impenetrable surface on the simple cubic lattice. This
surface can either be a-thermal, having thus only an entropic effect, or
attractive. In the latter case we concentrate on the adsorption transition, We
find clear evidence for the cross-over exponent to be smaller than 1/2, in
contrast to all previous simulations but in agreement with a re-summed field
theoretic -expansion. Since we use the pruned-enriched Rosenbluth
method (PERM) which allows very precise estimates of the partition sum itself,
we also obtain improved estimates for all entropic critical exponents.Comment: 5 pages with 9 figures included; minor change
Average Structures of a Single Knotted Ring Polymer
Two types of average structures of a single knotted ring polymer are studied
by Brownian dynamics simulations. For a ring polymer with N segments, its
structure is represented by a 3N -dimensional conformation vector consisting of
the Cartesian coordinates of the segment positions relative to the center of
mass of the ring polymer. The average structure is given by the average
conformation vector, which is self-consistently defined as the average of the
conformation vectors obtained from a simulation each of which is rotated to
minimize its distance from the average conformation vector. From each
conformation vector sampled in a simulation, 2N conformation vectors are
generated by changing the numbering of the segments. Among the 2N conformation
vectors, the one closest to the average conformation vector is used for one
type of the average structure. The other type of the averages structure uses
all the conformation vectors generated from those sampled in a simulation. In
thecase of the former average structure, the knotted part of the average
structure is delocalized for small N and becomes localized as N is increased.
In the case of the latter average structure, the average structure changes from
a double loop structure for small N to a single loop structure for large N,
which indicates the localization-delocalization transition of the knotted part.Comment: 15 pages, 19 figures, uses jpsj2.cl
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