Interacting Self-avoiding Walks Research Articles

Exact Enumeration Approach to Estimate the Theta Temperature of Interacting Self-Avoiding Walks on the Simple Cubic Lattice.

We compute the exact root-mean-square end-to-end distance of the interacting self-avoiding walk (ISAW) up to 27 steps on the simple cubic lattice. These data are used to construct a fixed point equation to estimate the theta temperature of the collapse transition of the ISAW. With the Bulirsch-Stoer extrapolation method, we obtain accurate results that can be compared with large-scale long-chain simulations. The free parameter ω in extrapolation is precisely determined using a parity property of the ISAW. The systematic improvement of this approach is feasible by adopting the combination of exact enumeration and multicanonical simulations.

Energy of the interacting self-avoiding walk at the θ point

We perform a numerical study of a new microcanonical polymer model on a three-dimensional cubic lattice, consisting of ideal chains whose range and number of nearest-neighbor contacts are fixed to given values. Our simulations suggest an interesting exact relation concerning the internal energy per monomer of the interacting self-avoiding walk at the θ point.

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents**All series data generated for this paper are available with the preprint at arXiv:1911.05852.

We investigate, by series methods, the behaviour of interacting self-avoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the θ-point to be at uc = 2.767 ± 0.002. The honeycomb lattice is unique among the regular two-dimensional lattices in that the exact growth constant is known for non-interacting walks, and is (Duminil-Copin H and Smirnov S 2014 Ann. Math. 175 1653–65), while for half-plane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is (Beaton N R et al 2014 Commun. Math. Phys. 326 727–54). We could not help but notice that We discuss the difficulties of trying to prove, or disprove, this possibility. For square lattice ISAWs we find uc = 1.9474 ± 0.001, which is consistent with the best Monte Carlo analysis. We also study bridges and terminally-attached walks (TAWs) on the square lattice at the θ-point. We estimate the exponents to be γb = 0.00 ± 0.03, and γ1 = 0.55 ± 0.03 respectively. The latter result is consistent with the prediction (Duplantier B and Saleur H 1987 Phys. Rev. Lett. 59 539–42; Seno F and Stella A L 1988 Europhys. Lett. 7 605–10; Stella A L et al 1993 J. Stat. Phys. 73 21–46) , albeit for a modified version of the problem, while the former estimate is predicted in [Duplantier B and Guttmann A J 2019 Statistical mechanics of confined polymer networks (in preparation)] to be zero.

Limit theorems for random walk excursion conditioned to enclose a typical area

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at criticality which is the object of a companion paper \citep{CarPet17a}. This requires to derive a reinforced version of our result in the case of a random walk with Laplace symmetric increments.

Collapse transition of the interacting prudent walk

This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by β > 0 and the set of trajectories consists of those self-avoiding paths respecting the prudent condition, which means that they do not take a step towards a previously visited lattice site. The IPSAW interpolates between the interacting partially directed self-avoiding walk (IPDSAW) that was analyzed in details in, e.g., Zwanzig and Lauritzen (1968), Brak et al. (1992), Carmona et al. (2016) and Nguyen and Petrelis (2013), and the interacting self-avoiding walk (ISAW) for which the collapse transition was conjectured in Saleur (1986). Three main theorems are proven. We show first that IPSAW undergoes a collapse transition at finite temperature and, up to our knowledge, there was so far no proof in the literature of the existence of a collapse transition for a non-directed model built with self-avoiding path. We also prove that the free energy of IPSAW is equal to that of a restricted version of IPSAW, i.e., the interacting two-sided prudent walk. Such free energy is computed by considering only those prudent path with a general northeast orientation. As a by-product of this result we obtain that the exponential growth rate of generic prudent paths equals that of two-sided prudent paths and this answers an open problem raised in e.g., Bousquet-Melou (2010) or Dethridge and Guttmann (2008). Finally we show that, for every β > 0, the free energy of ISAW itself is always larger than β and this rules out a possible self-touching saturation of ISAW in its conjectured collapsed phase.

Self-attracting polymers in two dimensions with three low-temperature phases

We study via Monte Carlo simulation a generalisation of the so-called vertex interacting self-avoiding walk (VISAW) model on the square lattice. The configurations are actually not self-avoiding walks but rather restricted self-avoiding trails (bond avoiding paths) which may visit a site of the lattice twice provided the path does not cross itself: to distinguish this subset of trails we shall call these configurations grooves. Three distinct interactions are added to the configurations: firstly the VISAW interaction, which is associated with doubly visited sites, secondly a nearest neighbour interaction in the same fashion as the canonical interacting self-avoiding walk (ISAW) and thirdly, a stiffness energy to enhance or decrease the probability of bends in the configuration.In addition to the normal high temperature phase we find three low temperature phases: (i) the usual amorphous liquid drop-like ‘globular’ phase, (ii) an anisotropic ‘β-sheet’ phase with dominant configurations consisting of aligned long straight segments, which has been found in semi-flexible nearest neighbour ISAW models, and (iii) a maximally dense phase, where the all sites of the path are associated with doubly visited sites (except those of the boundary of the configuration), previously observed in interacting self-avoiding trails.We construct a phase diagram using the fluctuations of the energy parameters and three order parameters. The β-sheet and maximally dense phases do not seem to meet in the phase space and are always separated by either the extended or globular phases. We focus attention on the transition between the extended and maximally dense phases, as that is the transition in the original VISAW model. We find that for the path lengths considered there is a range of parameters where the transition is first order and it is otherwise continuous.

Exact Partition Functions of Interacting Self-Avoiding Walks on Lattices

Ideas and methods of statistical physics have been shown to be useful for understanding some interesting problems in physical systems, e.g. universality and scaling in critical systems. The interacting self-avoiding walk (ISAW) on a lattice is the simplest model for homopolymers and serves as the framework of simple models for biopolymers, such as DNA, RNA, and protein, which are important components in complex systems in biology. In this paper, we briefly review our recent work on exact partition functions of ISAW. Based on zeros of these exact partition functions, we have developed a novel method in which both loci of zeros and thermodynamic functions associated with them are considered. With this method, the first zeros can be identified clearly without ambiguity. The critical point of a small system can then be defined as the peak position of the heat capacity component associated with the first zeros. For the system with two phase transitions, two pairs of first zeros corresponding to two phase transitions can be identified and overlapping C υ can be well separated. ISAW on the simple cubic lattice is such a system where in addition to a standard collapse transition, there is another freezing transition occurring at a lower temperature. Our approach can give a clear scenario for the collapse and the freezing transitions.

Monte Carlo simulations of polymers with nearest- and next nearest-neighbor interactions on square and cubic lattices

We study a generalized interacting self-avoiding walk (ISAW) model with nearest- and next nearest-neighbor (NN and NNN) interactions on square and cubic lattices. In both dimensions, the phase diagrams show coil and globule phases separated by continuous transition lines. Along these lines, we calculate the metric νt, crossover ϕt and entropic γt exponents, all of them in good agreement with the exact values of the Θ universality class. Therefore, the introduction of NNN interactions does not change the class of the ISAW model, which still exists even for repulsive forces. The growth parameters μt are shown to change monotonically with temperature along the Θ-lines. In the square lattice, the Θ-line has an almost linear behavior, which was not found in the cubic one. Although the region of repulsive NNN interactions, with attractive NN ones, leads to stiff polymers, no evidence of a transition to a crystalline phase was found.

Polymers in disordered environments

A brief review of our recent studies aiming at a better understanding of the scaling behaviour of polymers in disordered environments is given. The main emphasis is on a simple generic model where the polymers are represented by (interacting) self-avoiding walks and the disordered environment by critical percolation clusters. The scaling behaviour of the number of conformations and their average spatial extent as a function of the number of monomers and the associated critical exponents $\gamma$ and $\nu$ are examined with two complementary approaches: numerical chain-growth computer simulations using the PERM algorithm and complete enumerations of all possible polymer conformations employing a recently developed very efficient exact counting method.

Heat capacity decomposition by partition function zeros for interacting self-avoiding walks

A novel method based on partition function zeros is developed to demonstrate the additional advantages by considering both loci of partition function zeros and thermodynamical functions associated with them. With this method, the first pair of complex conjugate zeros (first zeros) can be defined without ambiguity and the critical point of a small system can be defined as the peak position of the heat capacity component associated with the first zeros. For the system with two phase transitions, two pairs of first zeros corresponding to two phase transitions can be identified and two overlapping phase transitions can be well separated. This method is applied to the interacting self-avoiding walk (ISAW) of homopolymer with N monomers on the simple cubic lattice, which has a collapse transition at a higher temperature and a freezing transition at a low temperature. The exact partition functions ZN with N up to 27 are calculated and our approach gives a clear scenario for the collapse and the freezing transitions.

The Θ points of interacting self-avoiding walks and rings on a 2D square lattice

We propose an order parameter to locate theΘ points of interacting self-avoiding walks (ISAWs) and self-avoiding rings (SARs) on atwo dimensional square lattice. Using exact enumeration results for ISAWs offinite size we find that the order parameter as a function of temperature showsa discontinuous jump at the transition temperature. The computed transitiontemperature fluctuates with the size of the walk and appears to converge well to theΘ point as the size increases. The value for theΘ point of the linear homopolymer phase transition obtained from our methodagrees with recent high precision Monte Carlo estimates. We tested thereliability of our method for longer walk lengths by using the density of statesobtained from the recently proposed flat energy histogram method. TheΘ point is also computed for SARs on a square lattice using exact enumeration results availablein the literature. Using two other independent methods, namely the partition function zeroson the complex temperature plane and the inflection point of microcanonical entropy, theΘ point of an SAR is computed and is found to be in agreement with the result obtained withthe proposed order parameter.

Versatile Approach to Access the Low Temperature Thermodynamics of Lattice Polymers and Proteins

We show that Wang-Landau sampling, combined with suitable Monte Carlo trial moves, provides a powerful method for both the ground state search and the determination of the density of states for the hydrophobic-polar (HP) protein model and the interacting self-avoiding walk (ISAW) model for homopolymers. We obtain accurate estimates of thermodynamic quantities for HP sequences with >100 monomers and for ISAWs up to >500 monomers. Our procedure possesses an intrinsic simplicity and overcomes the limitations inherent in more tailored approaches making it interesting for a broad range of protein and polymer models.

PULL-INDUCED PHASE TRANSITION IN COMPACT CHAINS

The study presents the low temperature phase transition of compact chains with one end fixed based on the interacting self-avoiding walks(ISAWS) on the square lattice.The simulation is realized by transfer matrix(TM) method.During the tensile process,the average pulling force F has an obviously oscillations at T0.1,and the oscillations vanishes with the temperature increasing.Since the compact chain has "frozen conformations" like β-sheets,and when about half a layer has been opened,the average pulling force F decreases obviously.The results agree well with the conclusions by Sanjay Kumar et al.The average pulling force F,the average free energy G and the phase transition for compact chains adsorbed on the neutral surface and the attractive surface are investigated respectively.During the initial tensile process,in order to overcome the adsorbed surface,the average pulling force F has a peak value,and this phenomenon is also explained by the free energy.The adsorbed surface can make the available average pulling force F decrease,so that the area of collapsed phase decreases.On the other hand,the average pulling force F is also influenced by temperature.At the beginning,when the end-to-end distance x12,with the influence of excluded volume effects,the average pulling force F decreases with the temperature.When end-to-end distance 12≤x≤16,the average pulling force F keeps a plateau of a constant force around 0.9 with different temperatures.But with the further extension of the compact chain,when the end-to-end distance x16,the average pulling force F increases with the temperature.This result has already been found by Kumar et al.for the biopolymers without considering the attractive surface.With considering the influence of excluded volume effects,when the end-to-end distance x is less than the average size of the coil(without force),the compact chain needs a compressing force instead of a pulling force,i.e.when T=1.0,the end-to-end distance x=8,the pull-induced force is compressing force,and comparing with the average compressing forces of compact chains adsorbed on the neutral surface and the attractive surface,the adsorbed surface makes the available average compressing force decreases.These investigations may provide some insights into the phase transition in the force-induced compact chains.

A flat histogram method for interacting self avoiding walks

We present a Monte Carlo method to estimate microcanonical entropy of interacting self avoiding walks (ISAWs) employing blind, myopic and discerning ant algorithms. We also propose a flat energy histogram model for discerning ant algorithm for studying coil-globule transition of linear homopolymers.

A growth walk model for estimating the canonical partition function of interacting self-avoiding walk

We have explained in detail why the canonical partition function of interacting self-avoiding walk (ISAW) is exactly equivalent to the configurational average of the weights associated with growth walks, such as the interacting growth walk (IGW), if the average is taken over the entire genealogical tree of the walk. In this context, we have shown that it is not always possible to factor the density of states out of the canonical partition function if the local growth rule is temperature dependent. We have presented Monte Carlo results for IGWs on a diamond lattice in order to demonstrate that the actual set of IGW configurations available for study is temperature dependent even though the weighted averages lead to the expected thermodynamic behavior of ISAW.

MONTE CARLO INVESTIGATION OF LATTICE MODELS OF POLYMER COLLAPSE IN FIVE DIMENSIONS

Monte Carlo simulations, using the PERM algorithm, of interacting self-avoiding walks (ISAW) and interacting self-avoiding trails (ISAT) in five dimensions are presented which locate the collapse phase transition in those models. It is argued that the appearance of a transition (at least) as strong as a pseudo-first-order transition occurs in both models. The values of various theoretically-conjectured dimension-dependent exponents are shown to be consistent with the data obtained. Indeed the first-order nature of the transition is even stronger in five dimensions than four. The agreement with the theory is better for ISAW than ISAT and it cannot be ruled out that ISAT have a true first-order transition in dimension five. This latter difference would be intriguing if true. On the other hand, since simulations are more difficult for ISAT than ISAW at this transition in high dimensions, any discrepancy may well be due to the inability of the simulations to reach the true asymptotic regime.

Four-dimensional polymer collapse II: interacting self-avoiding trails

We have simulated four-dimensional interacting self-avoiding trails (ISAT) on the hypercubic lattice with standard interactions at a wide range of temperatures up to length 4096 and at some temperatures up to length 16384. The results confirm the earlier prediction (using data from a non-standard model at a single temperature) of a collapse phase transition occurring at finite temperature. Moreover they are in accord with the phenomenological theory originally proposed by Lifshitz, Grosberg and Khokhlov in three dimensions and recently given new impetus by its use in the description of simulational results for four-dimensional interacting self-avoiding walks (ISAW). In fact, we argue that the available data is consistent with the conclusion that the collapse transitions of ISAT and ISAW lie in the same universality class, in contradiction with long-standing predictions. We deduce that there exists a pseudo-first order transition for ISAT in four dimensions at finite lengths while the thermodynamic limit is described by the standard polymer mean-field theory (giving a second-order transition), in contradiction to the prediction that the upper critical dimension for ISAT is d u =4.

Exact enumeration study of free energies of interacting polygons and walks in two dimensions

We present analyses of substantially extended series for both interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the theta-temperature, extending the application of a theorem of Tesi et. al. to two dimensions. Below the $\theta$-temperature the conditions of this theorem break down, in contradistinction to three dimensions, but an analysis of the ratio of the partition functions for ISAP and ISAW indicates that the free energies are in fact equal at all temperatures to at least within 1%. Any perceived difference can be interpreted as the difference in the size of corrections-to-scaling in both problems. This may be used to explain the vastly different values of the cross-over exponent previously estimated for ISAP to that predicted theoretically, and numerically confirmed, for ISAW. An analysis of newly extended neighbour-avoiding SAW series is also given.

Grand canonical simulations of the interacting self-avoiding walk model

We report the results obtained from simulations of the interacting self-avoiding walk (ISAW) model on the square and simple cubic lattices. In particular we verify a new scaling form for the canonical partition function in the low temperature phase as proposed by Owczarek et al. We also obtain estimates of the critical exponent in two dimensions. Finally, we obtain good agreement with previous studies on the value of a correction to scaling amplitude at the point on the simple cubic lattice which is in disagreement with a field theoretic prediction.

Beyond Flory's Theory: A Computer Aided Phenomenology for Polymers

Abstract Extending the histogram method of Ferrenberg and Swendsen to the problem of a SAW in a bad solvent, we obtain a new expression for the free energy of such a system, which fits very properly the numerical results of our Monte Carlo simulations. The basic difference with Flory's theory lies in the reference system: instead of considering random walks (RW) we start with self-avoiding walks (SAW) for which we already proposed a model expression for the distribution of the radius of gyration. This distribution is universal for any class of homo- or heteropolymers and contains all the information concerning the excluded volume problem. For homopolymers we consider the standard case where attractive nearest neighbours interactions simulate a bad solvent. At a given radius of gyration r we compute the density of states P(r, m) for an interacting self avoiding walk (ISAW) as a function of its number of contacts m. We build a microscopic phenomenology for P(r, m) based on a factorisation procedure: P(r, m) is split into an a priori probability P(r) of finding a SAW with a given r, and a conditional probability P(m/r) of finding such a conformation with m contacts. A complete scaling is given for P(r) whereas P(m/r) is found to be well approximated by a gaussian distribution. Scaling laws for the first two cumulants of P(m/r) are exhibited from a finite-size scaling analysis. The theta-point temperature is recovered along with its dependence on the polymer length and the free energy of the globule phase is well reproduced. This approach is shown to be generalizable to heteropolymers by merely replacing m by u, the energy per monomer, and computing the density of states P(r, u) at a given r. The case of sequenced and random copolymers is examined with special attention to polyampholytes.