Self-avoiding walks on three-dimensional lattices are flexible linear objects which can be self-entangled. The authors discuss several ways to measure entanglement complexity for n-step walks, and prove that these measures tend infinity with n. For small n, they use Monte Carlo methods estimate compare the n-dependence of two measures.

The incidence of knots in lattice polygons the face-centred cubic is investigated numerically. authors generate a sample using pivot algorithm and detect knotted by calculating Alexander polynomial. If p0n( phi ) probability that polygon with n edges unknot, then it known lim supn to infinity )1n/=e- varies as (0)<1. They find ( )=(7.6+or-0.9)*10-6. effect solvent quality on considered. data show being increases rapidly deteriorates.

We use Monte Carlo methods to investigate the asymptotic behaviour of number and mean-square radius gyration polygons in simple cubic lattice with fixed knot type. Let be n-edge a type lattice, let mean square all counted by . If we assume that , where is growth constant entropic exponent then our numerical data are consistent relation unknot prime factors both being independent These results support claims made Janse van Rensburg Whittington (1991a 24 3935) Orlandini et al (1996 29 L299,...

The sixth, seventh and eighth virial coefficients of hard discs spheres are evaluated numerically (Monte Carlo integration). I improve on the best previous estimates for coefficients, integration coefficient is new. these B7 /b6 = 0.114 86(7) B8/b7 0.065 14(8); B7/b6 0.01307(7) 0.00432(10). b second in each case. Pade approximations to excess pressure free energy computed from results compared data otherwise obtained.

The BFACF algorithm applied to polygons involves sampling on a Markov chain whose state space is the set of all polygons. In three dimensions, for simple cubic lattice. authors prove that ergodic classes this are knot

The numerical simulation of self-avoiding walks remains a significant component in the study random objects lattices. In this review, I give comprehensive overview current state Monte Carlo simulations models walks. walk model is revisited, and motivations for are discussed. Efficient sampling an elusive objective, but progress has been made over last three decades. still poses challenging questions however, review specific methods improved including general techniques such as Metropolis...

Self-interacting walks and polygons on the simple cubic lattice undergo a collapse transition at -point. We consider self-avoiding with an additional interaction between pairs of vertices which are unit distance apart but not joined by edge walk or polygon. prove that these have same limiting free energy if interactions nearest-neighbour repulsive. The attractive regime is investigated using Monte Carlo methods, we find evidence energies also equal here. In particular, this means models...

A self-avoiding walk adsorbing on a line in the square lattice, and plane cubic is studied numerically as model of an polymer dilute solution. The simulated by multiple Markov chain Monte Carlo implementation pivot algorithm for walks. Vertices that are visits or weighted eβ. critical value β, where adsorbs plane, determined considering energy ratios approximations to free energy. We determine values β In addition, crossover exponent determined: Metric quantities, including mean radius...

The authors study the dimensions (mean-square radius of gyration and mean span) self-avoiding polygons on simple cubic lattice with fixed knot type. approach used is a Monte Carlo algorithm which combination BFACF pivot algorithm, so that are studied in grand canonical ensemble, but autocorrelation time not too large. They show that, although sensitive to type, critical exponent (v) leading amplitude independent type polygon. influences confluent correction scaling term hence rate limiting behaviour.

The writhe of a self-avoiding walk in three-dimensional space is the average over all projections onto plane sum signed crossings. We compute this number using Monte Carlo simulation. Our results suggest that absolute value walks increases as nalpha , where n length and alpha approximately=0.5. mean crossing also computed found to have power-law dependence on walks. In addition, we consider effects solvent quality

A new nonlocal algorithm for the simulation of trees on lattice Zd is proposed. The authors study implementation and properties algorithm, show that it decisively better than an which performs only local moves. They use to investigate in two, three, four, eight nine dimensions.

We consider a model of circular polyelectrolyte, such as DNA, in which the molecule is represented by polygon three-dimensional simple cubic lattice. A short-range attractive force between nonbonded monomers included (to account for solvent quality) together with screened Coulomb potential effect added salt). compute probability that ring knotted function number ring, and ionic strength solution. The results show same general behavior recent experimental Shaw Wang [Science 260, 533 (1993)]...

How many edges are necessary and sufficient to construct a knot of type K in the cubic lattice? Define minimal edge number be this edges. To what extend does measure complexity knot? What is behaviour under connected sum knots, its limiting behaviour? We consider these questions show that may computed using simulated annealing.

The writhe of a knot in the simple cubic lattice [Formula: see text] can be computed as average linking number with its pushoffs into four non-antipodal octants. We use Monte Carlo algorithm to generate sample knots specified type, and estimate distribution function length knots. If expected value is not zero, then chiral. prove that additive under concatenation observe mean appears connected sum operation. In addition we linear crossing certain families.

We study the linking probability of polygons on simple cubic lattice. In particular, we consider two each having n edges, confined to a cube side L, and ask for as function L. also other situations in which are restricted be not too far apart, but necessarily cube. prove several rigorous results, use Monte Carlo methods address some questions unable answer rigorously. An interesting feature is that L/nv, where v exponent characterizing radius gyration polygon.

A polymer in the confined spaces between colloid particles loses entropy and exerts a repulsive entropic force on confining particles. This situation can be modelled by self-avoiding walk slab two parallel planes lattice. In this paper, we prove existence of limiting free energy for general case that is interacting with bounding planes. We also strictly increasing distance some regions phase diagram. These results demonstrate presence non-zero model. Finally, examine relation model walks...

We show that the classical Rosenbluth method for sampling self-avoiding walks (Hammersley and Morton 1954 J. R. Stat. Soc. B 16 23, 1955 Chem. Phys. 23 356) can be extended to a general algorithm many families of objects, including polygons. The implementation relies on an elementary move which is generalization kinetic growth; rather than only appending edges endpoint, may inserted at any vertex provided resulting objects still lie within same family. This gives, first time, growth...

In this paper we examine the relative knotting probabilities in a lattice model of ring polymers confined cavity. The is knot size $n$ cubic lattice, to cube side-length $L$ and with volume $V=(L{+}1)^3$ sites. We use Monte Carlo algorithms approximately enumerate number conformations knots confining cube. If $p_{n,L}(K)$ polygon length type $K$ $L^3$, then probability have $K$, that unknot (the trivial knot, denoted by $0_1$), $\rho_{n,L}(K/0_1) = p_{n,L}(K)/p_{n,L}(0_1)$. determine...

We define a statistic an(w), the size of atmosphere self-avoiding walk, w, length n, with property that ⟨an(w)⟩ → μ as n ∞, where is growth constant lattice walks. Both and entropic exponent γ may be estimated to high precision from ⟨a(w)⟩ using canonical Monte Carlo simulations Previous measurements have used grand simulations. Our indicate 2.63816 ± 0.00006 = 1.345 0.002. These results, based on modest computer run, are comparable best estimates for (grand canonical) simulations, at most...

We study the knot probability of polygons confined to slabs or prisms, considered as subsets simple cubic lattice. show rigorously that almost all sufficiently long in a slab are knotted and we use Monte Carlo methods investigate behaviour function width prism number edges polygon. In addition consider effect solvent quality on these geometries.