Self-avoiding Walks Research Articles

From loop probability of self-avoiding walk to constitutive models of hydrogels for thickness, strain-softening and hardening effects

A novel self-avoiding walk (SAW) chain with loop probability model is proposed, incorporating parameters with clear physical meanings at the microscopic scale. By contrasting it with the phenomenally described Gent model, this research not only discovers the physical foundations of the Gent model assumed purely empirical previously but also provides insights into the stiffness-toughness conflict as well as the thickness effect of hydrogels. Coupled with the physically based Arruda-Boyce model, the proposed model empowers the Kuhn length with the ability to shorten or elongate, allowing for modelling strain-softening and hardening effects for both soft regular and highly entangled hydrogels. The proposed model is validated through comparisons with published experiments on stress-stretch curves of ultrathin hydrogel films and both soft regular and highly entangled hydrogels.

Physical properties of a generalized model of multilayer adsorption of dimers.

We investigate the transport properties of a complex porous structure with branched fractal architectures formed due to the gradual deposition of dimers in a model of multilayer adsorption. We thoroughly study the interplay between the orientational anisotropy parameter p_{0} of deposited dimers and the formation of porous structures, as well as its impact on the conductivity of the system, through extensive numerical simulations. By systematically varying the value of p_{0}, several critical and off-critical scaling relations characterizing the behavior of the system are examined. The results demonstrate that the degree of orientational anisotropy of dimers plays a significant role in determining the structural and physical characteristics of the system. We find that the Einstein relation relating to the size scaling of the electrical conductance holds true only in the limiting case of p_{0}→1. Monitoring the fractal dimension of the interface of the multilayer formation for various p_{0} values, we reveal that in a wide range of p_{0}>0.2 interface shows the characteristic of a self-avoiding random walk, compared to the limiting case of p_{0}→0 where it is characterized by the fractal dimension of the backbone of ordinary percolation cluster at criticality. Our results thus can provide useful information about the fundamental mechanisms underlying the formation and behavior of wide varieties of amorphous and disordered systems that are of paramount importance both in science and technology as well as in environmental studies.

Alignment interaction and band formation in assemblies of autochemorepulsive walkers.

Chemotaxis refers to the motion of an organism induced by chemical stimuli and is a motility mode shared by many living species that has been developed by evolution to optimize certain biological processes such as foraging or immune response. In particular, autochemotaxis refers to chemotaxis mediated by a cue produced by the chemotactic particle itself. Here, we investigate the collective behavior of autochemotactic particles that are repelled by the cue and therefore migrate preferentially towards low-concentration regions. To this end, we introduce a lattice model inspired by the true self-avoiding walk which reduces to the Keller-Segel model in the continuous limit, for which we describe the rich phase behavior. We first rationalize the chemically mediated alignment interaction between walkers in the limit of stationary concentration fields, and then describe the various large-scale structures that can spontaneously form and the conditions for them to emerge, among which we find stable bands traveling at constant speed in the direction transverse to the band.

Assessing the evolution of educational accessibility with self-avoiding random walk: insights from Helsinki

Rapid urbanization has posed challenges to accessibility to critical services that require in-depth analysis. Complex networks theory has been used to evaluate the evolution of network topologies or the overall accessibility of transportation systems. However, topological metrics to explain the temporal changes in accessibility levels do not fully capture the dynamics and implications of accessibility to specific critical services. In this study, we address this gap and investigate the opportunities of using a self-avoiding random walk (SARW) algorithm to evaluate and explain the evolution of spatial accessibility to education facilities. We used hotspot analysis to understand the temporal changes and investigated changes in hot and cold spots over time. Furthermore, we explored the relationship between the network indicators and the SARW-based accessibility metric. We illustrated this method in a case study from Helsinki, where large-scale open data spanning from 1991 to 2016 is available. Our findings indicate that the SARW-based metric delivers more detailed node-level results than the traditional isochrone-based metric. The latter generates accessibility zones where accessibility is assumed to be uniform, while the SARW metric captures the dynamic nature of educational facility accessibility more accurately. The developed methodology helps to identify the impacts on the historical development of accessibility and can be applied to investigate accessibility to other critical services.

Enumeration of Self-Avoiding Random Walks on Lattices as Model Chains in Polymer Crystals

Recent simulation studies have revealed a wealth of distinct crystal polymorphs encountered in the self-organization of polymer systems driven by entropy or free energy. The present analysis, based on the concept of self-avoiding random walks (SAWs) on crystal lattices, is useful to calculate upper bounds for the entropy difference of the crystals that are formed during polymer crystallization and thus to predict the thermodynamic stability of distinct polymorphs. Here, we compare two pairs of crystals sharing the same coordination number, ncoord: hexagonal close-packed (HCP) and face centered cubic (FCC), both having ncoord = 12 and the same packing density, and the less dense simple hexagonal (HEX) and body centered cubic (BCC) lattices, with ncoord = 8. In both cases, once a critical number of steps is reached, one of the crystals shows a higher number of SAWs compatible with its geometry. We explain the observed trends in terms of the bending and torsion angles as imposed by the geometric constraints of the crystal lattice.

Comparison of Linear and Ring Polymers Partitioning into Pores Using Random Walks and Self‐Avoiding Walks Models

AbstractMonte Carlo simulations that examine the partitioning of dilute solutions of ring or linear chains into a slit pore using two chain models, the random‐walk (RW) model and the self‐avoiding walk (SAW) model are presented. The partitioning coefficients K for the ring and the linear chains at the surface interaction both under the critical adsorption point (CAP) and above the CAP are compared. In both chain models, K for the ring remains larger than K for the linear chains. The ring chain crosses over the point K = 1 at a weaker surface interaction than the linear chain. When extrapolated to infinite chain length, the cross‐over point for the ring and linear chain becomes the same (within statistical errors) for the RW model but remains different for the SAW model. The density profiles of the ring within the pore reveal the development of humps near the wall as the surface interaction crosses over the CAP. The excluded volume interaction in the SAW model additionally impacts the partitioning of chains in a solution consisting of a binary mixture of ring and linear chains and makes the K values in a binary mixture differ from the monodispersed solutions.

The escape transition in a self-avoiding walk model of linear polymers

A linear polymer grafted to a hard wall and underneath an atomic force microscopy tip can be modeled in a lattice as a grafted lattice polymer (self-avoiding walk) compressed underneath a piston near the wall. As the piston approaches the wall the increasingly confined polymer escapes from the confined region to explore conformations beside the piston. This conformational change is believed to be a phase transition in the thermodynamic limit, and has been argued to be first order, based on numerical results in the literature. In this paper a lattice self-avoiding walk model of the escape transition is constructed. It is proven that this model has a critical point in the thermodynamic limit corresponding to the escape transition of compressed grafted linear polymers. This result relies on the analysis of self-avoiding walks in slits and slabs in the square and cubic lattices. Additionally, numerical estimates of the location of the escape transition critical point is reported based on Monte Carlo simulations of self-avoiding walks in slits and in slabs.

Percolation in random sequential adsorption of polydisperse mixtures of extended objects on a triangular lattice

Percolation properties of an adsorbed polydisperse mixture of extended objects on a triangular lattice are studied by Monte Carlo simulations. The depositing objects of various shapes are formed by self-avoiding walks on the lattice. We study polydisperse mixtures in which the size of the shape making the mixture increases gradually with the number of components. This study examines the influence of the shape of the primary object defining a polydisperse mixture on its percolation and jamming properties. The dependence of the jamming density and percolation threshold on the number of components n making the mixture is analyzed. Determining the contribution of the individual components in the lattice covering allowed a better insight into the deposit structure of the n-component mixture at the percolation threshold. In addition, we studied mixtures of objects of various shapes but the same size.

Self-avoiding walks of specified lengths on rectangular grid graphs

AbstractThe investigation of self-avoiding walks on graphs has an extensive literature. We study the notion of wrong steps of self-avoiding walks on rectangular shape $$n\times m$$ n × m grids of square cells (Manhattan graphs) and examine some general and special cases. We determine the number of self-avoiding walks with one and with two wrong steps in general. We also establish some properties, like unimodality and sum of the rows of the Pascal-like triangles corresponding to the walks. We also present particular recurrence relations on the number of self-avoiding walks on the $$n\times 2$$ n × 2 grids with any specified number of wrong steps.

Peptide conformational sampling using the Quantum Approximate Optimization Algorithm

Protein folding has attracted considerable research effort in biochemistry in recent decades. In this work, we explore the potential of quantum computing to solve a simplified version of protein folding. More precisely, we numerically investigate the performance of the Quantum Approximate Optimization Algorithm (QAOA) in sampling low-energy conformations of short peptides. We start by benchmarking the algorithm on an even simpler problem: sampling self-avoiding walks. Motivated by promising results, we then apply the algorithm to a more complete version of protein folding, including a simplified physical potential. In this case, we find less promising results: deep quantum circuits are required to achieve accurate results, and the performance of QAOA can be matched by random sampling up to a small overhead. Overall, these results cast serious doubt on the ability of QAOA to address the protein folding problem in the near term, even in an extremely simplified setting.

Using deterministic self-avoiding walks as a small-world metric on Watts–Strogatz networks

Using deterministic self-avoiding walks as a small-world metric on Watts–Strogatz networks

ACA: Automatic search strategy for radioactive source

Nowadays, mobile robots have been used to search for uncontrolled radioactive source in indoor environments to avoid radiation exposure for technicians. However, in the indoor environments, especially in the presence of obstacles, how to make the robots with limited sensing capabilities automatically search for the radioactive source remains a major challenge. Also, the source search efficiency of robots needs to be further improved to meet practical scenarios such as limited exploration time. This paper proposes an automatic source search strategy, abbreviated as ACA: the location of source is estimated by a convolutional neural network (CNN), and the path is planned by the A-star algorithm. First, the search area is represented as an occupancy grid map. Then, the radiation dose distribution of the radioactive source in the occupancy grid map is obtained by Monte Carlo (MC) method simulation, and multiple sets of radiation data are collected through the eight neighborhood self-avoiding random walk (ENSAW) algorithm as the radiation data set. Further, the radiation data set is fed into the designed CNN architecture to train the network model in advance. When the searcher enters the search area where the radioactive source exists, the location of source is estimated by the network model and the search path is planned by the A-star algorithm, and this process is iterated continuously until the searcher reaches the location of radioactive source. The experimental results show that the average number of radiometric measurements and the average number of moving steps of the ACA algorithm are only 2.1% and 33.2% of those of the gradient search (GS) algorithm in the indoor environment without obstacles. In the indoor environment shielded by concrete walls, the GS algorithm fails to search for the source, while the ACA algorithm successfully searches for the source with fewer moving steps and sparse radiometric data.

Critical behaviour of the extended-ballistic transition for pulled self-avoiding walks

In order to study long chain polymers many lattice models accommodate a pulling force applied to a particular part of the chain, often a free endpoint. This is in addition to well-studied features such as energetic interaction between the lattice polymer and a surface. However, the critical behaviour of the pulling force alone is less well studied, such as characterising the nature of the phase transition and particularly the values of the associated exponents. We investigate a simple model of lattice polymers subject to forced extension, namely self-avoiding walks (SAWs) on the square and simple cubic lattices with one endpoint attached to an impermeable surface and a force applied to the other endpoint acting perpendicular to the surface. In the thermodynamic limit the system undergoes a transition to a ballistic phase as the force is varied and it is known that this transition occurs whenever the magnitude of the force is positive, i.e. f>fc=0. Using well established scaling arguments we show that the crossover exponent ϕ for the finite-size model is identical to the well-known exponent νd, which controls the scaling of the size of the polymer in d-dimensions. With extensive Monte Carlo simulations we test this conjecture and show that the value of ϕ is indeed consistent with the known values of ν2=3/4 and ν3=0.587597(7). Scaling arguments, in turn, imply the specific heat exponent α is 2/3 in two dimensions and 0.29815(2) in three dimensions.

Random sequential adsorption of self-avoiding chains on two-dimensional lattices.

Random sequential adsorption of extended objects deposited on two-dimensional regular lattices is studied. The depositing objects are chains formed by occupying adsorption sites on the substrate through a self-avoiding walk of k lattice steps; these objects are also called "tortuous k-mers." We study how the jamming coverage, θ_{j,k}, depends on k for lattices with different connectivity (honeycomb, square, and triangular). The dependence can be fitted by the function θ_{j,k}=θ_{j,k→∞}+B/k+C/k^{2}, where B and C are found to be shared parameters by the three lattices and θ_{j,k→∞} (>0) is the jamming coverage for infinitely long k-mers for each of them. The jamming coverage is found to have a growing behavior with the connectivity of the lattice. In addition, θ_{j,k} is found to be higher for tortuous k-mers than for the previously reported for linear k-mers in each lattice. The results were obtained by means of numerical simulation through an efficient algorithm whose characteristics are discussed in detail. The computational method introduced here also allows us to investigate the full-time kinetics of the surface coverage θ_{k}(t) [θ_{j,k}≡θ_{k}(t→∞)]. Along this line, different time regimes are identified and characterized.

Adsorption of melting deoxyribonucleic acid

The melting of a homopolymer double-stranded (ds) deoxyribonucleic acid (DNA) in the dilute limit is studied numerically in the presence of an attractive and impenetrable surface on a simple cubic lattice. The two strands of the DNA are modeled using two self-avoiding walks, capable of interacting at complementary sites, thereby mimicking the base pairing. The impenetrable surface is modeled by restricting the DNA configurations at the z≥0 plane, with attractive interactions for monomers at z = 0. Further, we consider two variants for z = 0 occupations by ds segments, where one or two surface interactions are counted. This consideration has significant consequences, to the extent of changing the stability of the bound phase in the adsorbed state. Interestingly, adsorption changes from critical to first-order with a modified exponent on coinciding with the melting transition. For simulations, we use the pruned and enriched Rosenbluth algorithm.

Weakly self-avoiding walk on a high-dimensional torus

Weakly self-avoiding walk on a high-dimensional torus

Counting independent sets in amenable groups

AbstractGiven a locally finite graph $\Gamma $ , an amenable subgroup G of graph automorphisms acting freely and almost transitively on its vertices, and a G-invariant activity function $\unicode{x3bb} $ , consider the free energy $f_G(\Gamma ,\unicode{x3bb} )$ of the hardcore model defined on the set of independent sets in $\Gamma $ weighted by $\unicode{x3bb} $ . Under the assumption that G is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|\unicode{x3bb} \|_\infty < \unicode{x3bb} _c(\Delta )$ , there exists a randomized $\epsilon $ -additive approximation scheme for $f_G(\Gamma ,\unicode{x3bb} )$ that runs in time $\mathrm {poly}((1+\epsilon ^{-1})\lvert \Gamma /G \rvert )$ , where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $ -regular tree. In addition, if G has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. However, we observe that if $\|\unicode{x3bb} \|_\infty> \unicode{x3bb} _c(\Delta )$ , there is no efficient approximation scheme, unless $\mathrm {NP} = \mathrm {RP}$ . This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

Densest packing of flexible polymers in 2D films.

How dense objects, particles, atoms, and molecules can be packed is intimately related to the properties of the corresponding hosts and macrosystems. We present results from extensive Monte Carlo simulations on maximally compressed packings of linear, freely jointed chains of tangent hard spheres of uniform size in films whose thickness is equal to the monomer diameter. We demonstrate that fully flexible chains of hard spheres can be packed as efficiently as monomeric analogs, within a statistical tolerance of less than 1%. The resulting ordered polymer morphology corresponds to an almost perfect hexagonal triangular (TRI) crystal of the p6m wallpaper group, whose sites are occupied by the chain monomers. The Flory scaling exponent, which corresponds to the maximally dense polymer packing in 2D, has a value of ν = 0.62, which lies between the limits of 0.50 (compact and collapsed state) and 0.75 (self-avoiding random walk).

Self-avoiding walks and polygons confined to a square

We prove several rigorous results about the asymptotic behaviour of the numbers of polygons and self-avoiding walks confined to a square on the square lattice. Specifically we prove that the dominant asymptotic behaviour of polygons confined to an L × L square is identical to that of self-avoiding walks that cross an L × L square from one corner vertex to the opposite corner vertex. We prove results about the sub-dominant asymptotic behaviour of self-avoiding walks crossing a square and polygons confined to a square and extend some results to self-avoiding walks and polygons in a hypercube in .

Solvent-Induced Negative Energetic Elasticity in a Lattice Polymer Chain.

The negative internal energetic contribution to the elastic modulus (negative energetic elasticity) has been recently observed in polymer gels. This finding challenges the conventional notion that the elastic moduli of rubberlike materials are determined mainly by entropic elasticity. However, the microscopic origin of negative energetic elasticity has not yet been clarified. Here, we consider the n-step interacting self-avoiding walk on a cubic lattice as a model of a single polymer chain (a subchain of a network in a polymer gel) in a solvent. We theoretically demonstrate the emergence of negative energetic elasticity based on an exact enumeration up to n=20 and analytic expressions for arbitrary n in special cases. Furthermore, we demonstrate that the negative energetic elasticity of this model originates from the attractive polymer-solvent interaction, which locally stiffens the chain and conversely softens the stiffness of the entire chain. This model qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the polymer-gel experiments, indicating that the analysis of a single chain can explain the properties of negative energetic elasticity in polymer gels.