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E. J. Janse van Rensburg

Publications and source records attributed to E. J. Janse van Rensburg.

14 recordsLinked to original sources
arXiv2016

Forces and pressures in adsorbing partially directed walks

Polymers in confined spaces lose conformational entropy. This induces a net repulsive entropic force on the walls of the confining space. A model for this phenomenon is a lattice walk between confining walls, and in this paper a model of an adsorbing partially directed walk is used. The walk is placed in a half square lattice $L^2_+$ with boundary $\partial L^2_+$, and confined between two vertical parallel walls, which are vertical lines in the lattice, a distance $w$ apart. The free energy of the walk is determined, as a function of $w$, for walks with endpoints in the confining walls and adsorbing in $\partial L^2_+$. This gives the entropic force on the confining walls as a function of $w$. It is shown that there are zero force points in this model and the locations of these points are determined, in some cases exactly, and in other cases asymptotically.

cond-mat.stat-mech↗
arXiv2014

Phase diagram of inhomogeneous percolation with a defect plane

Let ${\mathbb{L}}$ be the $d$-dimensional hypercubic lattice and let ${\mathbb{L}}_0$ be an $s$-dimensional sublattice, with $2 \leq s < d$. We consider a model of inhomogeneous bond percolation on ${\mathbb{L}}$ at densities $p$ and $σ$, in which edges in ${\mathbb{L}}\setminus {\mathbb{L}}_0$ are open with probability $p$, and edges in ${\mathbb{L}}_0$ open with probability $σ$. We generalizee several classical results of (homogeneous) bond percolation to this inhomogeneous model. The phase diagram of the model is presented, and it is shown that there is a subcritical regime for $σ< σ^*(p)$ and $p p_c(d)$, and a surface supercritical regime for $p σ^*(p)$. We show that $σ^*(p)$ is a strictly decreasing function for $p\in[0,p_c(d)]$, with a jump discontinuity at $p_c(d)$. We extend the Aizenman-Barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase. We prove that the cluster size distribution decays exponentially in the subcritical phase, and sub-exponentially in the supercritical phases. For a model of lattice animals with a defect plane, the free energy is related to functions of the inhomogeneous percolation model, and we show how the percolation transition implies a non-analyticity in the free energy of the animal model. Finally, we present simulation estimates of the critical curve $σ^*(p)$.

math-ph↗
arXiv2013

On trivial words in finitely presented groups

We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of such groups by finding the cogrowth series for Baumslag-Solitar groups $\mathrm{BS}(N,N) = < a,b | a^N b = b a^N >$ and prove that their cogrowth rates are algebraic numbers.

math.GR↗
arXiv2013

Random sampling of trivials words in finitely presented groups

We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution \begin{align*}π(w) &= (|w|+1)^α β^{|w|} \cdot Z^{-1} \end{align*} where $|w|$ is the length of a word $w$, $α$ and $β$ are parameters of the algorithm, and $Z$ is a normalising constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which then allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the Baumslag-Solitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), and Richard Thompson's group $F$.

math.GR↗
arXiv2013

Adsorbed self-avoiding walks subject to a force

We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that the appropriate limiting free energy exists where there is an applied force and a surface potential term, and prove that this free energy is convex in appropriate variables. We then derive an expression for the limiting free energy in terms of the free energy without a force and the free energy with no surface interaction. Finally we show that there is a phase boundary between the adsorbed phase and the desorbed phase in the presence of a force, prove some qualitative properties of this boundary and derive bounds on the location of the boundary.

math-ph↗
arXiv2013

The entropic pressure of a lattice polygon

The entropic pressure in the vicinity of a two dimensional square lattice polygon is examined as a model of the entropic pressure near a planar ring polymer. The scaling of the pressure as a function of distance from the polygon and length of the polygon is determined and tested numerically.

cond-mat.stat-mech↗
arXiv2013

The pressure exerted by adsorbing directed lattice paths and staircase polygons

A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in the first quadrant. Our models show that the pressure vanishes in the limit of long paths in the desorbed phase, but there is a non-zero pressure in the adsorbed phase. We determine asymptotic approximations of the pressure for finite length Dyck paths and directed paths, as well as for a model of adsorbing staircase polygons with both ends grafted to the X-axis.

math-ph↗
arXiv2012

Lattice Knots in a Slab

In this paper the number and lengths of minimal length lattice knots confined to slabs of width $L$, is determined. Our data on minimal length verify the results by Sharein et.al. (2011) for the similar problem, expect in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when it is squeezed to a minimal state is determined.

cond-mat.soft↗
arXiv2011

Minimal knotted polygons in cubic lattices

An implementation of BFACF-style algorithms on knotted polygons in the simple cubic, face centered cubic and body centered cubic lattice is used to estimate the statistics and writhe of minimal length knotted polygons in each of the lattices. Data are collected and analysed on minimal length knotted polygons, their entropy, and their lattice curvature and writhe.

cond-mat.stat-mech↗
arXiv2011

On the universality of knot probability ratios

Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is $p_n(K)/p_n$ and is known to decay exponentially with length. Little is known rigorously about the asymptotics of $p_n(K)$, but there is substantial numerical evidence that $p_n(K)$ grows as $p_n(K) \simeq \, C_K \, μ_\emptyset^n \, n^{α-3+N_K}$, as $n \to \infty$, where $N_K$ is the number of prime components of the knot type $K$. It is believed that the entropic exponent, $α$, is universal, while the exponential growth rate, $μ_\emptyset$, is independent of the knot type but varies with the lattice. The amplitude, $C_K$, depends on both the lattice and the knot type. The above asymptotic form implies that the relative probability of a random polygon of length $n$ having prime knot type $K$ over prime knot type $L$ is $\frac{p_n(K)/p_n}{p_n(L)/p_n} = \frac{p_n(K)}{p_n(L)} \simeq [ \frac{C_K}{C_L} ]$. In the thermodynamic limit this probability ratio becomes an amplitude ratio; it should be universal and depend only on the knot types $K$ and $L$. In this letter we examine the universality of these probability ratios for polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices. Our results support the hypothesis that these are universal quantities. For example, we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure-eight, independent of the underlying lattice, giving an estimate of the intrinsic entropy associated with knot types in closed curves.

cond-mat.stat-mech↗
arXiv2010

BFACF-style algorithms for polygons in the body-centered and face-centered cubic lattices

In this paper the elementary moves of the BFACF-algorithm for lattice polygons are generalised to elementary moves of BFACF-style algorithms for lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic lattices. We prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the BCC and FCC lattices and so expand a similar result for the cubic lattice. Implementations of these algorithms for knotted polygons using the GAS algorithm produce estimates of the minimal length of knotted polygons in the BCC and FCC lattices.

cond-mat.stat-mech↗
arXiv2008

Generalised Atmospheric Rosenbluth Methods (GARM)

We show that the classical Rosenbluth method for sampling self-avoiding walks can be extended to a general algorithm for sampling many families of objects, including self-avoiding polygons. The implementation relies on an elementary move which is a generalisation of kinetic growth; rather than only appending edges to the endpoint, edges may be inserted at any vertex providing the resulting objects still lie within the same family. We implement this method using pruning and enrichment to sample self-avoiding walks and polygons. The algorithm can be further extended by mixing it with length-preserving moves, such pivots and crank-shaft moves.

cond-mat.stat-mech↗
arXiv2006

Partially directed paths in a wedge

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by $Y = \pm pX$, and an asymmetric wedge defined by the lines $Y= pX$ and Y=0, where $p > 0$ is an integer. We prove that the growth constant for all these models is equal to $1+\sqrt{2}$, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when $p=1$. From these we find asymptotic formulas for the number of partially directed paths of length $n$ in a wedge when $p=1$. The functional recurrences are solved by a variation of the kernel method, which we call the ``iterated kernel method''. This method appears to be similar to the obstinate kernel method used by Bousquet-Melou. This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi $θ$-functions, and have natural boundaries on the unit circle.

math.CO↗