372 research outputs found
Exact Solution of the Discrete (1+1)-dimensional RSOS Model in a Slit with Field and Wall Interactions
We present the solution of a linear Restricted Solid--on--Solid (RSOS) model
confined to a slit. We include a field-like energy, which equivalently weights
the area under the interface, and also include independent interaction terms
with both walls. This model can also be mapped to a lattice polymer model of
Motzkin paths in a slit interacting with both walls and including an osmotic
pressure. This work generalises previous work on the RSOS model in the
half-plane which has a solution that was shown recently to exhibit a novel
mathematical structure involving basic hypergeometric functions .
Because of the mathematical relationship between half-plane and slit this work
hence effectively explores the underlying -orthogonal polynomial structure
to that solution. It also generalises two other recent works: one on Dyck paths
weighted with an osmotic pressure in a slit and another concerning Motzkin
paths without an osmotic pressure term in a slit
A self-interacting partially directed walk subject to a force
We consider a directed walk model of a homopolymer (in two dimensions) which
is self-interacting and can undergo a collapse transition, subject to an
applied tensile force. We review and interpret all the results already in the
literature concerning the case where this force is in the preferred direction
of the walk. We consider the force extension curves at different temperatures
as well as the critical-force temperature curve. We demonstrate that this model
can be analysed rigorously for all key quantities of interest even when there
may not be explicit expressions for these quantities available. We show which
of the techniques available can be extended to the full model, where the force
has components in the preferred direction and the direction perpendicular to
this. Whilst the solution of the generating function is available, its analysis
is far more complicated and not all the rigorous techniques are available.
However, many results can be extracted including the location of the critical
point which gives the general critical-force temperature curve. Lastly, we
generalise the model to a three-dimensional analogue and show that several key
properties can be analysed if the force is restricted to the plane of preferred
directions.Comment: 35 pages, 14 figure
Geometrical Properties of Two-Dimensional Interacting Self-Avoiding Walks at the Theta-Point
We perform a Monte Carlo simulation of two-dimensional N-step interacting
self-avoiding walks at the theta point, with lengths up to N=3200. We compute
the critical exponents, verifying the Coulomb-gas predictions, the theta-point
temperature T_theta = 1.4986(11), and several invariant size ratios. Then, we
focus on the geometrical features of the walks, computing the instantaneous
shape ratios, the average asphericity, and the end-to-end distribution
function. For the latter quantity, we verify in detail the theoretical
predictions for its small- and large-distance behavior.Comment: 23 pages, 4 figure
Four-dimensional polymer collapse II: Pseudo-First-Order Transition in Interacting Self-avoiding Walks
In earlier work we provided the first evidence that the collapse, or
coil-globule, transition of an isolated polymer in solution can be seen in a
four-dimensional model. Here we investigate, via Monte Carlo simulations, the
canonical lattice model of polymer collapse, namely interacting self-avoiding
walks, to show that it not only has a distinct collapse transition at finite
temperature but that for any finite polymer length this collapse has many
characteristics of a rounded first-order phase transition. However, we also
show that there exists a `-point' where the polymer behaves in a simple
Gaussian manner (which is a critical state), to which these finite-size
transition temperatures approach as the polymer length is increased. The
resolution of these seemingly incompatible conclusions involves the argument
that the first-order-like rounded transition is scaled away in the
thermodynamic limit to leave a mean-field second-order transition. Essentially
this happens because the finite-size \emph{shift} of the transition is
asymptotically much larger than the \emph{width} of the pseudo-transition and
the latent heat decays to zero (algebraically) with polymer length. This
scenario can be inferred from the application of the theory of Lifshitz,
Grosberg and Khokhlov (based upon the framework of Lifshitz) to four
dimensions: the conclusions of which were written down some time ago by
Khokhlov. In fact it is precisely above the upper critical dimension, which is
3 for this problem, that the theory of Lifshitz may be quantitatively
applicable to polymer collapse.Comment: 30 pages, 14 figures included in tex
Critical Percolation in High Dimensions
We present Monte Carlo estimates for site and bond percolation thresholds in
simple hypercubic lattices with 4 to 13 dimensions. For d<6 they are
preliminary, for d >= 6 they are between 20 to 10^4 times more precise than the
best previous estimates. This was achieved by three ingredients: (i) simple and
fast hashing which allowed us to simulate clusters of millions of sites on
computers with less than 500 MB memory; (ii) a histogram method which allowed
us to obtain information for several p values from a single simulation; and
(iii) a new variance reduction technique which is especially efficient at high
dimensions where it reduces error bars by a factor up to approximately 30 and
more. Based on these data we propose a new scaling law for finite cluster size
corrections.Comment: 5 pages including figures, RevTe
Exact Solution of the Discrete (1+1)-dimensional RSOS Model with Field and Surface Interactions
We present the solution of a linear Restricted Solid--on--Solid (RSOS) model
in a field. Aside from the origins of this model in the context of describing
the phase boundary in a magnet, interest also comes from more recent work on
the steady state of non-equilibrium models of molecular motors. While similar
to a previously solved (non-restricted) SOS model in its physical behaviour,
mathematically the solution is more complex. Involving basic hypergeometric
functions , it introduces a new form of solution to the lexicon of
directed lattice path generating functions.Comment: 10 pages, 2 figure
Pulling self-interacting polymers in two-dimensions
We investigate a two-dimensional problem of an isolated self-interacting
end-grafted polymer, pulled by one end. In the thermodynamic limit, we find
that the model has only two different phases, namely a collapsed phase and a
stretched phase. We show that the phase diagram obtained by Kumar {\it at al.\}
[Phys. Rev. Lett. {\bf 98}, 128101 (2007)] for small systems, where differences
between various statistical ensembles play an important role, differ from the
phase diagram obtained here in the thermodynamic limit.Comment: 20 pages, 22 figure
A simple model of a vesicle drop in a confined geometry
We present the exact solution of a two-dimensional directed walk model of a
drop, or half vesicle, confined between two walls, and attached to one wall.
This model is also a generalisation of a polymer model of steric stabilisation
recently investigated. We explore the competition between a sticky potential on
the two walls and the effect of a pressure-like term in the system. We show
that a negative pressure ensures the drop/polymer is unaffected by confinement
when the walls are a macroscopic distance apart
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