181 research outputs found
First-order scaling near a second-order phase transition: Tricritical polymer collapse
The coil-globule transition of an isolated polymer has been well established
to be a second-order phase transition described by a standard tricritical O(0)
field theory. We provide compelling evidence from Monte Carlo simulations in
four dimensions, where mean-field theory should apply, that the approach to
this (tri)critical point is dominated by the build-up of first-order-like
singularities masking the second-order nature of the coil-globule transition:
the distribution of the internal energy having two clear peaks that become more
distinct and sharp as the tricritical point is approached. However, the
distance between the peaks slowly decays to zero. The evidence shows that the
position of this (pseudo) first-order transition is shifted by an amount from
the tricritical point that is asymptotically much larger than the width of the
transition region. We suggest an explanation for the apparently contradictory
scaling predictions in the literature.Comment: 4 pages, 2 figures included in tex
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
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
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
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
IUPAC-NIST solubility data series. 81. Hydrocarbons with water and seawater - Revised and updated. Part 8. C9 hydrocarbons with water
The mutual solubility and related liquid-liquid equilibria of C9 hydrocarbons with water are exhaustively and critically reviewed. Reports of the experimental determination of solubility in 18 chemically distinct binary systems that appeared in the primary literature prior to the end of 2002 are compiled. For 8 systems, sufficient data are available to allow critical evaluation. All data are expressed as mass percent and mole fraction, as well as the originally reported units. In addition to the standard evaluation criteria used throughout the Solubility Date Series, a new method based on the evaluation of the all experimental data for a given homologous series of aliphatic and aromatic hydrocarbons was used
IUPAC-NIST solubility data series. 81. Hydrocarbons with water and seawater-revised and updated. Part 5. C7 hydrocarbons with water and heavy water
The mutual solubility and related liquid-liquid equilibria of C7 hydrocarbons with water and heavy water are exhaustively and critically reviewed. Reports of experimental determination of solubility in 23 chemically distinct binary systems that appeared in the primary literature prior to end of 2002 are compiled. For 9 systems sufficient data are available to allow critical evaluation. All data are expressed as mass percent and mole fraction as well as the originally reported units. In addition to the standard evaluation criteria used throughout the Solubility Data Series, a new method based on the evaluation of the all experimental data for a given homologous series of aliphatic and aromatic hydrocarbons was used
A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model
The free fermion condition of the six-vertex model provides a 5 parameter
sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter
into the eigenfunctions of the transfer matrices of the model decouple, hence
allowing explicit solutions. Such conditions arose originally in early
field-theoretic S-matrix approaches. Here we provide a combinatorial
explanation for the condition in terms of a generalised Gessel-Viennot
involution. By doing so we extend the use of the Gessel-Viennot theorem,
originally devised for non-intersecting walks only, to a special weighted type
of \emph{intersecting} walk, and hence express the partition function of
such walks starting and finishing at fixed endpoints in terms of the single
walk partition functions
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
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