A series test of the scaling limit of self-avoiding walks

It is widely believed that the scaling limit of self-avoiding walks (SAWs) at the critical temperature is (i) conformally invariant, and (ii) describable by Schramm-Loewner Evolution (SLE) with parameter $\kappa = 8/3.$ We consider SAWs in a rectangle, which originate at its centre and end when they reach the boundary. We assume that the scaling limit of SAWs is describable by ${\rm SLE}_\kappa,$ with the value of $\kappa$ to be determined. It has previously been shown by Guttmann and Kennedy \cite{GK13} that, in the scaling limit, the ratio of the probability that a SAW hits the side of the rectangle to the probability that it hits the end of the rectangle, depends on $\kappa.$ By considering rectangles of fixed aspect ratio 2, and also rectangles of aspect ratio 10, we calculate the probabilities exactly for larger and larger rectangles. By extrapolating this data to infinite rectangle size, we obtain the estimate $\kappa = 2.66664 \pm 0.00007$ for rectangles of aspect ratio 2 and $\kappa = 2.66675 \pm 0.00015$ for rectangles of aspect ratio 10. We also provide numerical evidence supporting the conjectured distribution of SAWs striking the boundary at various points in the case of rectangles with aspect ratio 2.Comment: 19 pages, 4 figures (colour

Role of conformational entropy in force-induced bio-polymer unfolding

A statistical mechanical description of flexible and semi-flexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. We predict the existence of many intermediate states at low temperatures stabilized by the force. A unified response to pulling and compressing forces has been obtained in the constant distance ensemble. We show the signature of a cross-over length which increases linearly with the chain length. Below this cross-over length, the critical force of unfolding decreases with temperature, while above, it increases with temperature. For stiff chains, we report for the first time "saw-tooth" like behavior in the force-extension curves which has been seen earlier in the case of protein unfolding.Comment: 4 pages, 5 figures, ReVTeX4 style. Accepted in Phys. Rev. Let

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

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

Self-avoiding walks and polygons on the triangular lattice

We use new algorithms, based on the finite lattice method of series expansion, to extend the enumeration of self-avoiding walks and polygons on the triangular lattice to length 40 and 60, respectively. For self-avoiding walks to length 40 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. For self-avoiding polygons to length 58 we calculate series for the mean-square radius of gyration and the first 10 moments of the area. Analysis of the series yields accurate estimates for the connective constant of triangular self-avoiding walks, $\mu=4.150797226(26)$, and confirms to a high degree of accuracy several theoretical predictions for universal critical exponents and amplitude combinations.Comment: 24 pages, 6 figure

Information-theoretic determination of ponderomotive forces

From the equilibrium condition $\delta S=0$ applied to an isolated thermodynamic system of electrically charged particles and the fundamental equation of thermodynamics ($dU = T dS-(\mathbf{f}\cdot d\mathbf{r})$) subject to a new procedure, it is obtained the Lorentz's force together with non-inertial terms of mechanical nature. Other well known ponderomotive forces, like the Stern-Gerlach's force and a force term related to the Einstein-de Haas's effect are also obtained. In addition, a new force term appears, possibly related to a change in weight when a system of charged particles is accelerated.Comment: 10 page

High-temperature expansions through order 24 for the two-dimensional classical XY model on the square lattice

The high-temperature expansion of the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model on the square lattice is extended by three terms, from order 21 through order 24, and analyzed to improve the estimates of the critical parameters.Comment: 7 pages, 2 figure

Size and area of square lattice polygons

We use the finite lattice method to calculate the radius of gyration, the first and second area-weighted moments of self-avoiding polygons on the square lattice. The series have been calculated for polygons up to perimeter 82. Analysis of the series yields high accuracy estimates confirming theoretical predictions for the value of the size exponent, $\nu=3/4$, and certain universal amplitude combinations. Furthermore, a detailed analysis of the asymptotic form of the series coefficients provide the firmest evidence to date for the existence of a correction-to-scaling exponent, $\Delta = 3/2$.Comment: 12 pages 3 figure

Enumeration of self-avoiding walks on the square lattice

We describe a new algorithm for the enumeration of self-avoiding walks on the square lattice. Using up to 128 processors on a HP Alpha server cluster we have enumerated the number of self-avoiding walks on the square lattice to length 71. Series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and mean-square distance of monomers from the end points have been derived to length 59. Analysis of the resulting series yields accurate estimates of the critical exponents $\gamma$ and $\nu$ confirming predictions of their exact values. Likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations. Finally we report on an analysis giving compelling evidence that the leading non-analytic correction-to-scaling exponent $\Delta_1=3/2$.Comment: 24 pages, 6 figure