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A Faster Implementation of the Pivot Algorithm for Self-Avoiding Walks
Abstract
The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the selfavoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N q ) with q < 1. We estimate that q is less than 0:57 in two dimensions, and less than 0:85 in three dimensions. Corrections to the O(N q ) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N q ) cannot be seen for walk lengths which can be simulated. In simulations the eective q is around 0:7 in two dimensions and 0:9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for o-lattice models. Key words: self-avoiding walk, pivot algorithm, polymer. 1 1
... The pivot algorithm [12,13] is a Markov chain Monte Carlo algorithm which is the most efficient known method for sampling SAWs of fixed length. Recent improvements [14][15][16] have reduced the CPU time necessary to attempt a pivot move to O(log N ), which makes it possible to reach the regime of truly large N , where the limit is due to computer memory (RAM) availability, rather than any constraint of available computer time. ...
... In the seminal work of Madras and Sokal [13], the pivot algorithm was proved to sample SAWs uniformly at random, and it was also shown to be remarkably efficient at sampling global observables, such as R 2 E , due to the fact that with each successful pivot move a large change is made to global observables. As mentioned in the introduction, recent improvements [14][15][16] have increased the efficiency of the pivot algorithm still further, to the point that it is now possible to sample SAWs with 10 9 steps. See [13] for many more details about the pivot algorithm. ...
... We show the estimates of κ from our fits in Fig. 3. There we observe that the estimates are much closer to 1/4, and if we extrapolate the trends we obtain an estimate of κ = 0.2516 (14) which is plotted on the y-axis. The value of 1/4 is only just outside our confidence interval, and we conclude that κ = 1/4 is consistent with these fits. ...
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is .
... A schematic illustration of an operation on square lattice is shown in Figure 2.5. An off-lattice pivot algorithm was implemented by my college K. Li on the basis of the algorithm by Kennedy [102]. In the off-lattice algorithm, the pivot move is achieved by two transformation matrices: the rotation matrix and the reflection matrix. ...
... Also, the check for self-avoiding after the pivot requires much time than the local moves. Kennedy [102,104] argued that one should emphasize on the time required for accepting a pivot, and proposed an implementation that can reduce this time to O(N q ), where q is about 0.7 for the 2D walk and 0.9 for the 3D walk. ...
... In this paper we use two models to perform the Monte Carlo simulation and study the questions defined in section 3.1. First, when simulating very long chains in order to investigate the key question on the definition of a contact we employ a pivot algorithm based on the original idea of Sokal and Kennedy [70,102] in continuous space. There have been several applications of the continuous pivot algorithm in different polymer models. ...
Various modeling and simulations have been done to study the organization of chromosomes under different circumstance and for different purposes. In this thesis, we would first study the influence of bending rigidity and spatial confinement on the organization of the chromatin. More concretely, the effect of heterogeneity and the definition of contacts will be addressed. We find that the definition of a contact does not change the asymptotic behavior of the contact probability. The heterogeneity of bending rigidity is shown to render the chain more flexible by comparing the persistent length and the contact probability of homogeneous and heterogeneous chains. In addition, we simulate semiflexible chains in rectangular confinements with different aspect ratios. An oscillation in the contact probability and the orientational correlation function is found due to the spiraling of polymer when the box size is small enough. The entanglement of chains is another important aspect when studying chromatins. The processes of disentanglement of two flexible chains are studied using the Monte Carlo simulation. Specifically, several measurements such as the inter-contact of chains, dynamic structure factor are analyzed in the process. When only the excluded volume interaction exists in the system, the average time required for segregation is barely influenced by the initial configurations of the two chains according to our results. However, the intertwinement of chains indeed could impede the segregation at a small time scale. The number of contacts inside a self-avoiding chain is also analyzed. It is found that the total number grows linearly with the length of a free chain, while in cubic confinement it grows quadratically. The distribution function of contacts number between two halves shows a power-law decay behavior and then an exponential decay for a free chain. In confinement, the function has a maximum. As the chain becomes longer, the percentage of inter-half contacts among the total contacts has a power-law decay behavior with an exponent close to -1, which supports that the number of contacts between two halves is finite even when the chain is infinitely long. Finally, we studied the fractality and the topology in the self-avoiding walks. Specifically, we calculate the fractal dimension and growth rates of the Betti numbers of the system. These growth rates can be viewed as a topological signature for different systems. The intra-contacts of the self-avoiding walk is a subset of the original walk, and we find that this subset may have a slight multifractal property. In addition, the topological exponents are also different from the self-avoiding walk. Further, each contact gives rise to the formation of a loop. To elucidate how these loops influence the structure of the self-avoiding walk, we delete the loops in a similar way to the loop-erased random walk, thus producing a new walk: loop-deleted self-avoiding walk (LDSAW). The critical exponent of LDSAW is approximated by studying the scaling behavior of mean end-to-end distance, and the dependence of the mean length of LDSAW on the length of the original self-avoiding walk. Afterward, the fractal dimension and growth rates of Betti numbers of this LDSAW are calculated. The same calculations are also performed on the projection and random subsets of self-avoiding walks.
... The pivot algorithm is the most efficient method known for sampling self-avoiding walks [15,16], and recent improvements [17,13,12] have made it even more effective, especially in the large N limit. These improvements are highly beneficial as they allow one to obtain accurate data for large N , which reduces systematic errors due to corrections-to-scaling. ...
... Madras and Sokal [16] gave a hash table implementation of the pivot algorithm which resulted in mean CPU time of O(N 1−p ) = O(N 0.89 ) per pivot attempt (alternatively, CPU time O(N ) per successful pivot). This has since been improved by Kennedy [17] to roughly mean CPU time of O(N 0.74 ) per pivot attempt, and further still by the present author [12,12] to O(log N ). This makes the pivot algorithm extremely efficient for sampling global observables for self-avoiding walks, but it is not obvious how efficient it is for sampling our observable B. ...
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent for self-avoiding walks to unprecedented accuracy. Our final estimate for is 1.15695300(95).
... [16]-if one uses the Verdier-Stockmayer algorithm [5] in two dimensions, one does not sample 3.2%, 1.4%, 5% of the most compact configurations for N = 11, 13, 15. These ergodicity problems can be solved by using a different ensemble [17][18][19][20], chain-growth algorithms [21][22][23][24], or nonlocal algorithms [25][26][27][28][29][30]. However, in the presence of interactions, nonlocal algorithms become inefficient since nonlocal moves generate new walks with large energy differences and thus they will be rejected making the dynamics very slow. ...
... For the simulation of weakly interacting walks, i.e. for β < β θ , there exist powerful nonlocal algorithms [27,30]. However, these algorithms cannot be used in confined geometries-they are not ergodic-and are very inefficient in the presence of strong interactions. ...
We consider lattice self-avoiding walks and discuss the dynamic critical behavior of two dynamics that use local and bilocal moves and generalize the usual reptation dynamics. We determine the integrated and exponential autocorrelation times for several observables, perform a dynamic finite-size scaling study of the autocorrelation functions, and compute the associated dynamic critical exponents z. For the variables that describe the size of the walks, in the absence of interactions we find in two dimensions and in three dimensions. At the -point in two dimensions we have .
... The method involves generating a correlated sequence of SAWs by performing rotations and reflections on part of the walk. Recent development of fast implementations [15,[19][20][21] have increased the speed, and hence utility, of the method still further. ...
... This data structure allows successful pivot moves to be performed in mean CPU time O(log N), and builds on the earlier important work by Kennedy [19] who first had the insight that successful pivot moves could be performed in CPU time o(N) if certain bookkeeping operations were provided by an appropriate data structure. ...
We extend a binary tree implementation of the pivot algorithm to the face-centered cubic and body-centered cubic lattices, and use it to calculate the growth constant, , for self-avoiding walks on these lattices. We find that \mufcc =10.037\, 057\, 85 \pm 0.000\, 000\, 14 and \mubcc = 6.530\, 511\, 501 \pm 0.000\, 000\, 084. In addition, we estimate the amplitudes \Afcc = 1.17119 \pm 0.00003 and \Abcc = 1.17637 \pm 0.00003, and provide convincing numerical evidence in support of the hypothesis that the critical exponent is a universal quantity.
... Two walks representative of different parts of the probability distribution are shown. The distributions and walks were generated using Monte Carlo using [33]. ...
... Inserting this into (B.1), and approximating the integral using the saddle point approximation, we find f (y) = e y 2 /12 y 2 + O(y −2 Table 1: Data from a Monte Carlo simulation of a self-avoiding walk with endpoints pulled apart with force F , so the weighting of a given walk is e rF . For each value we used approximately 5 million walks, using a slightly modified version of Tom Kennedy's code [33]. The coefficient c 2 was fit assuming the form (C.3). ...
We study quantum JT gravity at finite cutoff using a mapping to the statistical mechanics of a self-avoiding loop in hyperbolic space, with positive pressure and fixed length. The semiclassical limit (small ) corresponds to large pressure, and we solve the problem in that limit in three overlapping regimes that apply for different loop sizes. For intermediate loop sizes, a semiclassical effective description is valid, but for very large or very small loops, fluctuations dominate. For large loops, this quantum regime is controlled by the Schwarzian theory. For small loops, the effective description fails altogether, but the problem is controlled using a conjecture from the theory of self-avoiding walks.
... A few years ago [1,2], one of the present authors demonstrated significant progress in calculating universal properties of self-avoiding walks (SAWs) [3] on a lattice, which is the standard model to describe the static equilibrium properties of isolated polymer chains in good solvent. These advances were made possible through the use of a recursive data structure called the SAW-tree, which allows for very fast checking of self-overlaps in Monte Carlo (MC) simulations based upon the pivot algorithm [4][5][6][7], such that chains with up to N ≈ 34 × 10 6 monomers (repeat units) could be studied. Universal quantities that are accessible include critical exponents such as the Flory exponent ν = 0.587597(7) [1], which connects the mean polymer size R with the degree of polymerization N via the scaling law R ∝ bN ν , where b is the typical monomer size, and universal amplitude ratios such as the ratio of two different ways to define the size of the coil. ...
... The standard hash table implementation [5] then requires mean CPU time O(N) to generate an essentially new configuration with respect to global observables such as R 2 E . Recent algorithmic improvements [1,2,7] have further increased the relative advantage of the pivot algorithm over other methods. ...
The universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self-avoiding walks is estimated by high-resolution Monte Carlo simulations. By studying chains of length of up to monomers, we find that the ratio takes the value , which is several orders of magnitude more accurate than the previous state of the art. This is facilitated by a sampling scheme which is quite general, and which allows for the efficient estimation of averages of a large class of observables. The competing corrections to scaling for the hydrodynamic radius are clearly discernible. We also find improved estimates for other universal properties that measure the chain dimension. In particular, a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the Flory exponent of .
... For the cubic lattice, g is selected from the octahedral group, which has 48 elements. Generally, one disregards the trivial movement, identity operation, because it does not change the path, turning the algorithm slightly inefficient [40] (more pivot attempts are required to change the path). A natural question to ask about the pivot algorithm concerns the efficiency to generate SAWs. ...
... and match i with some i after the lattice symmetry operation. Such a procedure is the same as the one used by Kennedy [40]. For instance, the 120 • rotation of 1 in the hexagonal lattice (4) with fixed Φ 1 = 2ν 0 − 2 and Φ 2 = 2ν 0 − ∆ 1 − 1. ...
In this study, we show the convergence and new properties of persistence length, λN, for the self-avoiding random Walk model (SAW) using Monte Carlo data. We generate high precision estimates of several conformational quantities with pivot algorithm for the square, hexagonal, triangular, cubic and diamond lattices with path lengths of 103 steps. For each lattice, we accurately estimate the asymptotic limit λ∞, which corroborates the convergence of λN to a constant value, and allows us to check the universality on the λN/λ∞ curves. Based on the λ∞ estimates we make an ansatz for λ∞ dependency with lattice cell and spatial dimension, we also find a new geometric interpretation for the persistence length.
... The pivot algorithm [18,21] is a Markov chain Monte Carlo algorithm which is the most efficient known method for sampling SAWs of fixed length. Recent improvements [7,8,17] have reduced the CPU time necessary to attempt a pivot move to O(log N ), which makes it possible to reach the regime of truly large N , where the limit is due to computer memory (RAM) availability, rather than any constraint of available computer time. ...
... In the seminal work of Madras and Sokal [21], the pivot algorithm was proved to sample SAWs uniformly at random, and it was also shown to be remarkably efficient at sampling global observables, such as R 2 E , due to the fact that with each successful pivot move a large change is made to global observables. As mentioned in the introduction, recent improvements [7,8,17] have increased the efficiency of the pivot algorithm still further, to the point that it is now possible to sample SAWs with 10 9 steps. See [21] for many more details about the pivot algorithm. ...
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is .
... The pivot algorithm is the most efficient method known for sampling self-avoiding walks [15,16], and recent improvements [12,13,17] have made it even more effective, especially in the large N limit. These improvements are highly beneficial as they allow one to obtain accurate data for large N, which reduces systematic errors due to corrections-to-scaling. ...
... Madras and Sokal [16] gave a hash table implementation of the pivot algorithm which resulted in mean CPU time of O(N 1−p ) = O(N 0.89 ) per pivot attempt (alternatively, CPU time O(N) per successful pivot). This has since been improved by Kennedy [17] to roughly mean CPU time of O(N 0.74 ) per pivot attempt, and further still by the present author [12] to O(log N). This makes the pivot algorithm extremely efficient for sampling global observables for self-avoiding walks, but it is not obvious how efficient it is for sampling our observable B. ...
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent for self-avoiding walks to unprecedented accuracy. Our final estimate for is 1.15695300(95).
... On one side, new algorithms have been developed which, together with the availability of increasingly powerful computers, have allowed the generation of very long asymptotic series and the simulation of very large systems (for lattice polymer systems, see, e.g. [4][5][6][7]). This has significantly reduced the hard-to-control systematic errors due to the confluent singularities. ...
... If ⩾ / i N 2, we check whether + r i 1 intersects the subchain { } … − r r , , i 1 1 , then we consider + r i 2 , and so on. To verify whether r j , ⩾ + j i 1, intersects { } … − r r , , i 1 1 or not, we implement the strategy proposed in [6]. At the beginning we set k = i − 1. ...
Extensive Monte Carlo simulations are used to investigate how model systems of mixtures of polymers and hard spheres approach the scaling limit. We represent polymers as lattice random walks of length L with an energy penalty w for each intersection (Domb-Joyce model), interacting with hard spheres of radius Rc via a hard-core pair potential of range Rmon + Rc, where R mon is identified as the monomer radius. We show that the mixed polymer-colloid interaction gives rise to new confluent corrections. The leading ones scale as L-v, V≈ where is the usual Flory exponent. Finally, we determine optimal values of the model parameters w and R mon that guarantee the absence of the two leading confluent corrections. This improved model shows a significantly faster convergence to the asymptotic limit L→ ∞ and is amenable for extensive and accurate numerical simulations at finite density, with only a limited computational effort.
... The following simulations can be seen as a new way of probing the adequation between SAW's and SLE 8/3 : In order to check that the statistics of flights over a self avoiding walk follow the expected law with d = 4 3 , we have performed extended computer simulations. We have generated a set of self-avoiding walks on a square lattice using an implementation of the pivot algorithm described by Kennedy [Ken02] (see also [Ken05]). The number of steps of the self avoiding walk is fixed at 10 5 . ...
... It was shown in [GKL + 06] that for a boundary of fractal dimension d embedded in an Euclidian space of Euclidean dimension d e , we should have Figure 3: Two examples of self avoiding walk in 2D generated by the pivot algorithm [Ken02]. The number of steps in each SAW is fixed at 10 5 . ...
We characterize probability measures whose Hausdorff dimension or packing di- mension can be calculated by an entropy formula. In particular, we prove that such measures are unidimensional. We also construct examples of unidimensional measures for which entropy does not calculate the dimension. Let D be an integer greater than 1 and m be a probability measure in (0,1)D. Fix ℓ ≥ 1 and denote by Fn the family of ℓ-adic cubes of the nth generation, that is
... Another important question is whether the pivot algorithm [48,49] is ergodic for eSAW. One reason for the importance of this question is that recent improvements to the implementation of the pivot algorithm [50,44,51] have made it even more powerful. In addition, it is now clear that the pivot algorithm can be used for extremely accurate calculations of μ, γ , and ν for SAW. ...
... The enumeration results and analysis confirmed our expectations that γ = 1 for eSAW, but more serious numerical work should be done to provide stronger confirmation. In future we will apply Monte Carlo techniques [49,50,44,51] to accurately estimate critical exponents. It would be desirable to improve the rigor of our heuristic arguments that γ e = 1 and ν e = ν. ...
We introduce a self-avoiding walk model for which end-effects are completely
eliminated. We enumerate the number of these walks for various lattices in
dimensions two and three, and use these enumerations to study the properties of
this model. We find that endless self-avoiding walks have the same connective
constant as self-avoiding walks, and the same Flory exponent . However,
there is no power law correction to the exponential number growth for this new
model, i.e. the critical exponent exactly. In addition, we have
convincing numerical evidence to support the hypothesis that the amplitude for
the number growth is a universal quantity, and somewhat weaker evidence which
suggests that the number growth has no analytic corrections to scaling. The
technique by which end-effects are eliminated may be generalised to other
models of polymers such as interacting self-avoiding walks.
... To study the crossover, we perform Monte Carlo simulations of isolated chains for values of d/b ranging from 0.0125 to 0.7 and polymer lengths L from 250 to 8000. We use a continuum generalization of the pivot algorithm [65], implementing the Kennedy algorithm [66] to speed up the self-intersection check. We compute the second virial coefficient [90] B 2 and the radius of gyration R g . ...
We consider three different continuum polymer models, that all depend on a tunable parameter r that determines the strength of the excluded-volume interactions. In the first model chains are obtained by concatenating hard spherocylinders of height b and diameter rb (we call them thick self- avoiding chains). The other two models are generalizations of the tangent hard-sphere and of the Kremer-Grest models. We show that, for a specific value r*, all models show an optimal behavior: asymptotic long-chain behavior is observed for relatively short chains. For r < r*, instead, the behavior can be parametrized by using the two-parameter model that also describes the thermal crossover close to the {\theta} point. The bonds of thick self-avoiding chains cannot cross each other and, therefore, the model is suited for the investigation of topological properties and for dynamical studies. Such a model also provides a coarse-grained description of double-stranded DNA, so that we can use our results to discuss under which conditions DNA can be considered as a model good-solvent polymer.
... Madras and Sokal argued that O(N ) is best possible because it takes time of order N to merely write down an N -step SAW. However, Kennedy [9] recognized that it is not necessary to write down the SAW for each successful pivot, and from clever use of geometric constraints developed an algorithm that broke the O(N ) barrier. The CPU time for this implementation grows as a dimension-dependent fractional power of N (see Table I). ...
We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we use to obtain large samples of walks on the cubic lattice of up to steps. Consequently the critical exponent for three-dimensional self-avoiding walks is determined to great accuracy; the final estimate is . The method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum.
... For example, additional possible moves useful in modeling polymers are discussed in [32] and include "end flips," "corner flips," and "crankshaft moves." Other modifications to speed up the algorithm have been described, for example, in [38] [39]. Algorithms for random walks with fixed endpoints have also been proposed [40]. ...
... The growth constant (or connective constant) µ is lattice-dependent, and believed to be 2 + √ 2 for the honeycomb lattice, and another bi-quadratic number (approximately 2.64) for the square lattice [16]. 1365-8050 c 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France Fig. 1: A self-avoiding walk on the square lattice, and a (quasi-)random SAW of length 1,000,000, constructed by Kennedy using a pivot algorithm [17]. ...
International audience
We define a new family of self-avoiding walks (SAW) on the square lattice, called . These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.
Nous définissons une nouvelle famille de chemins auto-évitants (CAE) sur le réseau carré, appelés . Ces chemins ont une caractérisation simple en termes des ponts irréductibles qui les composent. Nous déterminons leur série génératrice. Cette série a une structure de singularité complexe et n'est en particulier pas D-finie. La constante de croissance est environ 2,54, ce qui est supérieur à toutes les familles naturelles de SAW étudiées jusqu'à présent, mais inférieur aux CAE généraux (dont la constante est environ 2,64). Nous prouvons également que la distance moyenne entre les extrémités du chemin croît linéairement. Enfin, nous étudions une variante diagonale du modèle.
... For example, additional possible moves useful in modeling polymers are discussed in [1] and include "end flips," "corner flips," and "crankshaft moves." Other modifications to speed up the algorithm have been described, for example, in [11,16]. Algorithms for random walks with fixed endpoints have also been proposed [24]. ...
In this work we propose a new algorithm for the computation of statistical equilibrium quantities on a cubic lattice when both an energy and a statistical temperature are involved. We demonstrate that the pivot algorithm used in situations such as protein folding works well for a small range of temperatures near the polymeric case, but it fails in other situations. The new algorithm, using localized transformations, seems to perform well for all possible temperature values. Having reliably approximated the values of equilibrium energy, we also propose an efficient way to compute equilibrium entropy for all temperature values. We apply the algorithms in the context of suction or supercritical vortices in a tornadic flow, which are approximated by vortex filaments on a cubic lattice. We confirm that supercritical (smooth, "straight") vortices have the highest energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and "balled up" to a great extent. The results support A. Chorin's findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold.
... To study the crossover, we perform Monte Carlo simulations of isolated chains for values of d/b ranging from 0.0125 to 0.7 and polymer lengths L from 250 to 8000. We use a continuum generalization of the pivot algorithm [65], implementing the Kennedy algorithm [66] to speed up the self-intersection check. We compute the second virial coefficient [90] B 2 and the radius of gyration R g . ...
We consider three different continuum polymer models, that all depend on a tunable parameter r that determines the strength of the excluded-volume interactions. In the first model chains are obtained by concatenating hard spherocylinders of height b and diameter rb (we call them thick self-avoiding chains).
The other two models are generalizations of the tangent hard-sphere and of the Kremer-Grest models. We show that, for a specific value r^*, all models show an optimal behavior: asymptotic long-chain behavior is observed for relatively short chains. For r < r^*, instead, the behavior can be parametrized by using the two-parameter model that also describes the thermal crossover close to the theta point. The bonds of thick self-avoiding chains cannot cross each other and, therefore, the model is suited for
the investigation of topological properties and for dynamical studies.
Such a model also provides a coarse-grained description of double-stranded DNA, so that we can use our results to discuss under which conditions DNA can be considered as a model good-solvent polymer.
... In addition, generating random self avoiding walks yields an approximation to the critical exponents and the connective constant. There are many algorithms to generate random self-avoiding walks; among them the pivot algorithm [14,12], the Berretti-Sokal algorithm [1], the Rosenbluth algorithm [19,9,18], flat-PERM [5]. All these algorithms are Monte Carlo algorithms [17]. ...
A self-avoiding walk (SAW) is extendable [Grimmett, G. R., A. E. Holroyd and Y. Peres, Extendable Self-Avoiding Walks, Ann. Inst. Henri Poincaré Comb. Phys. Interact., 1 (2014), pp. 61–75, Kremer, K. and J. W. Lyklema, Indefinitely growing self-avoiding walk, Phys. Rev. Lett. 54 (1985), pp. 267–269] if it can be extended into an infinite SAW. We give a simple proof that, for every lattice, extendable SAWs admit the same connective constant as the general SAWs and we give an optimal linear algorithm to generate random extendable SAWs. Our algorithm can generate every extendable SAW in dimension 2. For dimension , it generates only a subset of the extendable SAWs. We conjecture that this subset is “large” and has the same connective constant as the extendable SAWs. Our algorithm produces a kinetic distribution of the extendable SAWs, for which the critical exponent ν is such that for , for and for . Keywords: self-avoiding walk, connective constant, critical exponent ν, random generation.
... subm. to DMTCS c by the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France Fig. 1: A self-avoiding walk on the square lattice, and a (quasi-)random SAW of length 1,000,000, constructed by Kennedy using a pivot algorithm [17]. ...
... The pivot algorithm was invented by Lal [27] but first studied in detail by Madras and Sokal [30]. In particular, we use the recent implementation of one of us [4,5], which improved on earlier important work by Kennedy [24]. ...
We study terminally attached self-avoiding walks and bridges on the
simple-cubic lattice, both by series analysis and Monte Carlo methods. We
provide strong numerical evidence supporting a scaling relation between
self-avoiding walks, bridges, and terminally attached self-avoiding walks, and
posit that a corresponding amplitude ratio is a universal quantity.
... We generated θ-point walks in the full plane with increasing lengths (n = 5000, 10 000, 20 000, 40 000) and obtained 10 000 samples for each n. The method used is the extended reptation algorithm [30,31]; remark that very efficient algorithms are available for the non-interacting SAW [32,33], but they are expected to perform much less efficiently for the ISAW. In order to reduce the severity of finite-chain effects, we restricted the analysis of the driving function to the first n/10 steps (we obtain similar results with n/20 steps; an independent check that corrections in this regime are small is given in Fig. 3 for the radial distribution functions, see Sec. ...
We study the connection between polymers at the θ temperature on the lattice and Schramm-Loewner chains with constant step length in the continuum. The second of these realize a useful algorithm for the exact sampling of tricritical polymers, where finite-chain effects are excluded. The driving function computed from the lattice model via a radial implementation of the zipper method is shown to converge to Brownian motion of diffusivity κ=6 for large times. The distribution function of an internal portion of walk is well approximated by that obtained from Schramm-Loewner chains. The exponent of the correlation length ν and the leading correction-to-scaling exponent Δ_{1} measured in the continuum are compatible with ν=4/7 (predicted for the θ point) and Δ_{1}=72/91 (predicted for percolation). Finally, we compute the shape factor and the asphericity of the chains, finding surprising accord with the θ-point end-to-end values.
... Even if v f is properly chosen, the corresponding pivot move could be unfeasible because R would map some residues onto positions that are either already occupied by the opposite branch (i.e. a clash occurs), or do not actually correspond to lattice vertices (in this case the elements of R are not all integers). Anyway, the application of a pivot move in any case requires a significant computational effort [42], hampering its extensive employment in Monte Carlo methods. Local moves are an effective means to explore the neighborhood of a given conformation, but in optimization problems like protein folding it is very difficult to edge away from local minima by their application in a random way. ...
models of proteins have been widely used as a practical means to computationally investigate general properties of the system. In lattice models any sterically feasible conformation is represented as a self-avoiding walk on a lattice, and residue types are limited in number. So far, only two- or three-dimensional lattices have been used. The inspection of the neighborhood of alpha carbons in the core of real proteins reveals that also lattices with higher coordination numbers, possibly in higher dimensional spaces, can be adopted. In this paper, a new general parametric lattice model for simplified protein conformations is proposed and investigated. It is shown how the supporting software can be consistently designed to let algorithms that operate on protein structures be implemented in a lattice-agnostic way. The necessary theoretical foundations are developed and organically presented, pinpointing the role of the concept of main directions in lattice-agnostic model handling. Subsequently, the model features across dimensions and lattice types are explored in tests performed on benchmark protein sequences, using a Python implementation. Simulations give insights on the use of square and triangular lattices in a range of dimensions. The trend of potential minimum for sequences of different lengths, varying the lattice dimension, is uncovered. Moreover, an extensive quantitative characterization of the usage of the so-called "move types" is reported for the first time. The proposed general framework for the development of lattice models is simple yet complete, and an object-oriented architecture can be proficiently employed for the supporting software, by designing ad-hoc classes. The proposed framework represents a new general viewpoint that potentially subsumes a number of solutions previously studied. The adoption of the described model pushes to look at protein structure issues from a more general and essential perspective, making computational investigations over simplified models more straightforward as well.
... It was invented by Lal [12], but the true power of the method was only appreciated after the ground-breaking work of Madras and Sokal [13]. Recently, the implementation of the pivot algorithm has been improved to make it even more powerful [14][15][16]. The recent improvements make it an extremely attractive prospect to utilize the pivot algorithm whenever possible. ...
We calculate the connective constant for self-avoiding walks on the simple
cubic lattice to unprecedented accuracy, using a novel application of the pivot
algorithm. We estimate that \mu = 4.684 039 931(27). Our method also provides
accurate estimates of the number of self-avoiding walks, even for walks with
millions of steps.
We consider a coarse-grained (CG) model with pairwise interactions, suitable to describe low-density solutions of star-branched polymers of functionality f. Each macromolecule is represented by a CG molecule with (f+1) interaction sites, which captures the star topology. Potentials are obtained by requiring the CG model to reproduce a set of distribution functions computed in the microscopic model in the zero-density limit. Explicit results are given for f=6,12 and 40. We use the CG model to compute the osmotic equation of state of the solution for concentrations c such that , where is the overlap concentration. We also investigate in detail the phase diagram for f=40, identifying the boundaries of the solid intermediate phase. Finally, we investigate how the polymer size changes with c. For polymers become harder as f increases at fixed reduced concentration . On the other hand, for , polymers show the opposite behavior: At fixed , the larger the value of f, the larger their size reduction is.
We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps N ranging up to . We show that the mean square winding angle of random walks converges to the theoretical form when . For self-avoiding walks on the square lattice, we show that the ratio converges slowly to the Gaussian value 3. For self avoiding walks on the cubic lattice we find that the ratio exhibits non-monotonic dependence on N and reaches a maximum of 3.73(1) for . We show that to a good approximation, the square winding angle of a self-avoiding walk on the cubic lattice can be obtained from the summation of the square change in the winding angles of independent segments of the walk, where the i-th segment contains steps. We find that the square winding angle of the i-th segment increases approximately as , which leads to an increase of the total square winding angle proportional to .
The thermal properties of coarse-grained knotted copolymer rings fluctuating in a highly screening solution are investigated on a simple cubic lattice using the Wang-Landau Monte Carlo algorithm. The rings contain two kinds of monomers A and B with opposite charges that are subjected to short-range interactions. In view of possible applications in medicine and the construction of intelligent materials, it is shown that the behavior of copolymer rings can be tuned by changing both their monomer configuration and topology. We find several phase transitions depending on the monomer distribution. They include the expansion and collapse of the knotted polymer as well as rearrangements leading to metastable states. The temperatures at which these phase transitions are occurring and other features can be tuned by changing the topology of the system. The processes underlying the observed transitions are identified. In knots formed by diblock copolymers, two different classes of behaviors are detected depending on whether there is an excess of monomers of one kind or not. Moreover, we find that the most stable compact states are formed by copolymers in which units of two A monomers are alternated by units of two B monomers. Remarkably, these compact states are in a lamellar phase. The transition from the lamellar to the expanded state produces in the specific heat capacity a narrow and high peak that is centered at temperatures that are much higher than those of the peaks observed in all other monomer distributions.
The Self-Avoiding Trail (SAT) is a random walk model represented by an N-step lattice path, with forbidden step overlaps. The SAT is a variant of the Self-Avoiding Walk (SAW) model, where a site is visited only once. There is a consensus that SAT is in the same universality class of SAW, implying that the scaling behavior of conformational quantities, represented by the exponents ν0 and Δ1, is the same in both models. However, in comparison to SAW, SAT results exhibit most pronounced finite-size effects and inconsistent estimates of ν0, depending on the lattice coordination number and conformational quantity. The scaling behavior of the persistence length, λN, is in disagreement with recent estimates for the SAW model. Also, the influence of coordination number on the lattice-dependent constants requires a better understanding. In this study, we revisit the SAT model with data generated by the pivot algorithm up to N=8000 steps, for the hexagonal, square and triangular lattices. Accordingly, we find compelling evidences that ν0 and Δ1 values are the same for the SAT and SAW models, regardless the lattice and conformational quantity used, while the most pronounced finite-size effect comes from the higher amplitude of the correction to scaling terms. Similarly to SAW, we find the λN convergence to lattice dependent constant λ∞(SAW) for the SAT model. Based on the ratio λ∞(SAW)/λ∞(SAT)∝z/2, where z is the lattice coordination number, we find that the factor z/2 quantifies the relation between SAW and SAT lattice dependent constants.
The pivot algorithm is the most efficient known method for sampling polymer configurations for self-avoiding walks and related models. Here we introduce two recent improvements to an efficient binary tree implementation of the pivot algorithm: an extension to an off-lattice model, and a parallel implementation.
We consider a coarse-grained (CG) model with pairwise interactions, suitable to describe low-density solutions of star-branched polymers of functionality f. Each macromolecule is represented by a CG molecule with (f+1) interaction sites, which captures the star topology. Potentials are obtained by requiring the CG model to reproduce a set of distribution functions computed in the microscopic model in the zero-density limit. Explicit results are given for f=6,12 and 40. We use the CG model to compute the osmotic equation of state of the solution for concentrations c such that , where is the overlap concentration. We also investigate in detail the phase diagram for f=40, identifying the boundaries of the solid intermediate phase. Finally, we investigate how the polymer size changes with c. For polymers become harder as f increases at fixed reduced concentration . On the other hand, for , polymers show the opposite behavior: At fixed , the larger the value of f, the larger their size reduction is.
We study the Domb-Joyce model of weakly self-avoiding walks on the simple cubic lattice via Monte Carlo simulations. We determine to excellent accuracy the value for the interaction parameter which results in an improved model for which the leading correction-to-scaling term has zero amplitude.
The universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self-avoiding walks is estimated by high-resolution Monte Carlo simulations. By studying chains of length of up to N=225≈34×106 monomers, we find that the ratio takes the value RG/RH=1.5803940(45), which is several orders of magnitude more accurate than the previous state of the art. This is facilitated by a sampling scheme which is quite general and which allows for the efficient estimation of averages of a large class of observables. The competing corrections to scaling for the hydrodynamic radius are clearly discernible. We also find improved estimates for other universal properties that measure the chain dimension. In particular, a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the Flory exponent of ν=0.58759700(40).
We investigate the occurrence and size of knots in a continuum polymer model with finite memory via Monte Carlo simulations. Excluded volume interactions are local and extend only to a fixed number of successive beads along the chain, ensuring that at short length scales the excluded volume effect dominates, while at longer length scales the polymer behaves like a random walk. As such, this model may be useful for understanding the behavior of polymers in a melt or semi-dilute solution, where exactly the same crossover is believed to occur. In particular, finite memory walks allow us to investigate the role of local interactions in the transition from highly knotted ideal polymers to almost unknotted self-avoiding polymers. Even though knotting decreases substantially when a few next-nearest neighbor interactions are considered, we find that the knotting probability of a polymer chain of modest length of 500 steps only decays slowly as a function of the range of the excluded volume interaction. In this context, we also find evidence that for length scales up to the interaction length the knotting behavior of the finite memory walk resembles that of a self-avoiding walk (effectively suppressing small knots), while for larger length scales it resembles that of a random walk.
We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps N ranging up to . We show that the mean square winding angle of random walks converges to the theoretical form when . For self-avoiding walks on the square lattice, we show that the ratio converges slowly to the Gaussian value 3. For self avoiding walks on the cubic lattice we find that the ratio exhibits non-monotonic dependence on N and reaches a maximum of 3.73(1) for . We show that to a good approximation, the square winding angle of a self-avoiding walk on the cubic lattice can be obtained from the summation of the square change in the winding angles of independent segments of the walk, where the i-th segment contains steps. We find that the square winding angle of the i-th segment increases approximately as , which leads to an increase of the total square winding angle proportional to .
This chapter introduces two major classes of numerical methods used to study self-avoiding walks (SAWs) on deterministic and random fractals, namely, Monte Carlo (MC) methods and the exact enumeration (EE) technique. The advantages and disadvantages of both approaches are highlighted in connection to the different substrates and the statistical quantities of interest. Self-avoiding walks (SAWs) constitute the simplest, yet non-trivial model for studying the static behavior of a linear polymer embedded in a good solvent. Many MC methods have been developed for studying SAWs on regular lattices. These diverse algorithms can be classified according to static, quasistatic, and dynamic MC methods. The exact enumeration (EE) technique allows enumeration and evaluation of all SAW configurations on a given substrate. The chapter also discusses SAWs on three different substrates and compares them with the results obtained for SAWs on regular lattices. The first two substrates, Sierpinski triangular and square lattices, are representative members of the class of deterministic fractals, whereas percolation is the standard model for random fractals. Deterministic fractals consist of two distinct subgroups: the finitely (Sierpinski triangular) and infinitely (Sierpinski square) ramified fractals. The behavior of SAWs is drastically affected by the underlying substrate but certain similarities emerge: SAWs on Sierpinski triangular and square lattices display a kind of intermediate behavior between SAWs on the corresponding regular lattices and on percolation. The spatial characteristics of SAWs on Sierpinski square lattices are closer to those observed for regular lattices, while SAWs on Sierpinski triangular lattices are more similar to their counterparts on percolation. Scaling forms known for SAWs on regular lattices seem to remain valid also on deterministic and random fractals, while the corresponding relations between the scaling exponents need to be modified in some cases.
We perform a Monte Carlo study of N-step self-avoiding walks, attached to
the corner of an impenetrable wedge in two dimensions (d=2), or the tip of an
impenetrable cone in d=3, of sizes ranging up to steps. We find that
the critical exponent which determines the dependence of the
number of available conformations on N for a cone/wedge with opening angle
, is in good agreement with the theory for d=2. We study the
end-point distribution of the walks in the allowed space and find similarities
to the known behavior of random walks (ideal polymers) in the same geometry.
For example the ratio between the mean square end-to-end distances of a polymer
near the wedge and a polymer in free space depends linearly on
as is known for ideal polymers. We show that the end-point
distribution of polymers attached to a wedge does not separate into a product
of angular and radial functions, as it does for ideal polymers in the same
geometry. The angular dependence of the end-position of polymers near the wedge
differs from theoretical predictions.
We critically discuss and review the general ideas behind single- and
multi-site coarse-grained (CG) models as applied to macromolecular solutions in
the dilute and semi-dilute regime. We first consider single-site models with
zero-density and density-dependent pair potentials. We highlight advantages and
limitations of each option in reproducing the thermodynamic behavior and the
large-scale structure of the underlying reference model. As a case study we
consider solutions of linear homopolymers in a solvent of variable quality.
Secondly, we extend the discussion to multi-component systems presenting, as a
test case, results for mixtures of colloids and polymers. Specifically, we
found the CG model with zero-density potentials to be unable to predict
fluid-fluid demixing in a reasonable range of densities for mixtures of
colloids and polymers of equal size. For larger colloids, the polymer volume
fractions at which phase separation occurs are largely overestimated. CG models
with density-dependent potentials are somewhat less accurate than models with
zero-density potentials in reproducing the thermodynamics of the system and,
although they presents a phase separation, they significantly underestimate the
polymer volume fractions along the binodal. Finally, we discuss a general
multi-site strategy, which is thermodynamically consistent and fully
transferable with the number of sites, and that allows us to overcome most of
the limitations discussed for single-site models.
Toxic fibrillar aggregates of Islet Amyloid PolyPeptide (IAPP) appear as the
physical outcome of a peptidic phase-transition signaling the onset of type-2
diabetes mellitus in different mammalian species. In particular, experimentally
verified mutations on the amyloidogenic segment 20-29 in humans, cats and rats
are highly correlated with the molecular aggregation propensities. Through a
microcanonical analysis of the aggregation of IAPP_{20-29} isoforms, we show
that a minimalist one-bead hydrophobic-polar continuum model for protein
interactions properly quantifies those propensities from free-energy barriers.
Our results highlight the central role of sequence-dependent hydrophobic
mutations on hot spots for stabilization, and so for the engineering, of such
biological peptides.
In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable Xk
. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n.A few years ago, Bassetti and Diaconis showed that, for a sampler à la Knuth that generates walks crossing a k × k square and consisting of North and East steps, the relative variance is only \O(\sqrt k)\. In this note we take one step further and show that, for walks consisting of North, South and East steps, the relative variance jumps to \2^{k(k+1)}/(k+1)^{2k}\. This is exponential in the average length of the walks, which is of order k2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k2 (which is again the average length of these walks).Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following the paper by Bassetti and Diaconis, is one of very few examples where the variance of the estimator can be found.
We model protein folding as a physical stochastic process as follows. The unfolded protein chain is treated as a random coil described by SAW (self-avoiding walk). Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. The resulting model is termed CSAW (conditioned self-avoiding walk). Conceptually, the mathematical basis is a generalized Langevin equation. In practice, the model is implemented on a computer by combining SAW and Monte Carlo. To illustrate the flexibility and capabilities of the model, we consider a number of examples, including folding pathways, elastic properties, helix formation, and collective modes.
Conditioned self-avoiding walk (CSAW) is a model of protein folding that combines the features of self-avoiding walk (SAW) and the MonteCarlo method. It simulates the Brownian motion of a chain-molecule in the presence of interactions. The model is designed so that one can add desired features step by step. It can be used as guide to structure prediction, as well as theoretical laboratory.
The topological effects on the thermal properties of several knot configurations are investigated using Monte Carlo simulations. In order to check whether the topology of the knots is preserved during the thermal fluctuations we propose a method that allows very fast calculations and can be easily applied to arbitrarily complex knots. As an application, the specific energy and heat capacity of the trefoil, the figure-eight and the 81 knots are calculated at different temperatures and for different lengths. Short-range repulsive interactions between the monomers are assumed. The knot configurations are generated on a three-dimensional cubic lattice and sampled by means of the Wang–Landau algorithm and of the pivot method. The results obtained show that the topological effects play a key role for short-length polymers. Three temperature regimes of the growth rate of the internal energy of the system are distinguished.
We extend our previously developed coarse-graining strategy for linear polymers with a tunable number n of effective atoms (blobs) per chain [G. D'Adamo et al., J. Chem. Phys. 137, 024901 (2012)] to polymer systems in thermal crossover between the good-solvent and the θ regimes. We consider the thermal crossover in the region in which tricritical effects can be neglected, i.e., not too close to the θ point, for a wide range of chain volume fractions Φ = c∕c∗ (c∗ is the overlap concentration), up to Φ ≈ 30. Scaling crossover functions for global properties of the solution are obtained by Monte Carlo simulations of the Domb-Joyce model with suitably rescaled on-site repulsion. They provide the input data to develop a minimal coarse-grained model with four blobs per chain (tetramer model). As in the good-solvent case, the coarse-grained model potentials are derived at zero density, thus avoiding the inconsistencies related to the use of state-dependent potentials. We find that the coarse-grained model reproduces the properties of the underlying, full-monomer system up to some reduced density Φ which increases when lowering the temperature towards the θ state. Close to the lower-temperature crossover boundary, the tetramer model is accurate at least up to Φ ≃ 10, while near the good-solvent regime reasonably accurate results are obtained up to Φ ≃ 2. The density region in which the coarse-grained model is predictive can be enlarged by developing coarse-grained models with more blobs per chain. We extend the strategy used in the good-solvent case to the crossover regime. This requires a proper treatment of the length rescalings as before, but also a proper temperature redefinition as the number of blobs is increased. The case n = 10 is investigated in detail. We obtain the potentials for such finer-grained model starting from the tetramer ones. Comparison with full-monomer results shows that the density region in which accurate predictions can be obtained is significantly wider than that corresponding to the tetramer case.
We simulate self-avoiding walks on a cubic lattice and determine the second virial coefficient for walks of different lengths. This allows us to determine the critical value of the renormalized four-point coupling constant in the three-dimensional N-vector universality class for N = 0. We obtain \bar{g}^* = 1.4005(5) , where \bar{g} is normalized so that the three-dimensional field-theoretical beta function behaves as \beta(\bar{g}) = - \bar{g} + \bar{g}^2 for small \bar{g} . As a byproduct, we also obtain precise estimates of the interpenetration ratio Psi*, Psi* = 0.246 85(11) and of the exponent nu, nu = 0.5876(2).
Understanding thermodynamic properties, including the phase behavior of polymer solutions, polymer melts, and blends, has been a long-standing challenge. This chapter presents an overview of polymer models, and describes the most important aspects of Monte Carlo (MC) simulations of these models. It discusses the basic aspects of molecular dynamics simulation of polymer melts and blends with both coarse-grained and chemically detailed models. The chapter presents recent work on simple short alkanes and solutions of alkanes in supercritical carbon dioxide, to clarify to what extent a comparison of Monte Carlo results on phase behavior and experimental data is sensible. Monte Carlo simulations are, as indicated by the name, based on the idea of evolving a system by drawing random numbers. The chapter focuses on a set of techniques commonly used to determine the phase behavior of oligomer melts and blends to give an example of how MC techniques are applied in practice.
The principle of polymer separation in size exclusion chromatography (SEC) is studied based on a classical equilibrium partitioning theory. The task is to examine the correlation between the mean span dimension of polymer chains and their equilibrium partition coefficients with confining pores. Using an extended formulation of the recently developed confinement analysis from bulk structures (CABS) method, we calculate the partition coefficients for both linear and branched polymer chains with cylindrical pores—a model pore geometry that is considered to be more realistic for voids in SEC columns than the commonly considered slit model. The partition coefficients plotted as a function of the mean span dimension relative to the pore diameter are truly universal for wide pores and nearly so for flexible polymer chains with different architectures (linear, star, two-branch-point, and comb) in the range of the partition coefficient relevant to SEC separation. We also examine the correlation between the mean span dimension and the SEC retention volume using the experimental data by Sun et al. [Macromolecules 2004, 37, 4304−4312]. It is found that when the mean span dimension is plotted as a function of the retention volume, results for both linear and branched polyethylene molecules lie nearly on the master curve determined by linear polystyrene standards. Our findings support the equilibrium thermodynamic separation principle in SEC. Since the mean span dimension is a purely geometric size parameter applicable to any chain architecture, its use in the interpretation of SEC data is appealing.
We study the end-to-end distribution function for dilute polymers. We present a computation to order O(epsilon(2)), epsilon=4-d, and discuss in detail its asymptotic behavior for small and large distances. The theoretical predictions are compared with Monte Carlo results, finding good agreement. We show that the McKenzie-Moore-des Cloizeaux phenomelogical ansatz provides a very precise approximation to the exact end-to-end distribution function. (C) 2000 American Institute of Physics. [S0021-9606(00)50517-0].
We introduce a model of self-repelling random walks where the short-range interaction between two elements of the chain decreases as a power of the difference in proper time. The model interpolates between the lattice Edwards model and the ordinary random walk. We show by means of Monte Carlo simulations in two dimensions that the exponentv
MF obtained through a mean-field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self-avoiding walks. We also compute the exponent γ and present a numerical study of the scaling functions.
We compute the exponent gamma for self-avoiding walks in three dimensions. We get gamma = 1.1575 +- 0.0006 in agreement with renormalization-group predictions. Earlier Monte Carlo and exact-enumeration determinations are now seen to be biased by corrections to scaling. Comment: 8 pages
We present simulation results for long () self-avoiding walks in four dimensions. We find definite indications of logarithmic corrections, but the data are poorly described by the asymptotically leading terms. Detailed comparisons are presented with renormalization group flow equations derived in direct renormalization and with results of a field theoretic calculation. Comment: NLATEX, 20 pages
The result of a collaboration between a theoretician and an experimentalist, this book is devoted to the static properties of flexible polymers in solution. It presents the vast progress made by both theory and experiment in recent years. Despite the variety in the chemical composition and physical properties of long polymer chains, when in solution they show a universality in their behaviour. On the experimental side, the use of photon and neutron scattering has led to a better understanding, while the use of computer simulation has also produced interesting results. This work is the result of a collaboration between a theoretician and an experimentalist, who have both worked for many years on polymer solutions.
Imagine that you are standing at an intersection in the centre of a large city whose streets are laid out in a square grid. You choose a street at random and begin walking away from your starting point, and at each intersection you reach you choose to continue straight ahead or to turn left or right.
Accurate Monte Carlo data are used to determine the critical exponent nu (=0.590+or-0.001) or inverse fractal dimension of the three-dimensional self-avoiding walk. The leading correction to scaling term is non-analytic and is near 0.5.
A pivot algorithm is used to numerically investigate the shape properties of linear polymers, with and without excluded volume, in two and three dimensions. The high computational efficiency of the pivot algorithm has allowed us to obtain data which is four to eight times more accurate than our previous Brownian dynamics simulations, while employing considerably less computational resources. The accurate data for long polymer chains highlight the inadequacy of the d expansion of polymer shapes to first order only.
Monte Carlo simulations employing the pivot algorithm are used to generate excluded-volume chains on two- and three-dimensional lattices. The radius of gyration distribution function is calculated from the resulting configurations. The distributions are in excellent agreement with the scaled form postulated by Lhuillier.
A computationally efficient method to evaluate the interaction energy of a polymer configuration is presented. New polymer conformations are generated via the pivot algorithm. The method involves a hierarchical search scheme for determining intersections and nonbonded nearest‐neighbor contacts. The search uses a hierarchy of spheres which enclose segments of the polymer chain. At each level of the hierarchy the spheres enclose successively smaller polymer segments. This technique is found to be very efficient in terms of CPU time. It scales as O(N), where N is the polymer length, and is independent of ϕ, the interaction energy parameter. The method is readily applicable to off‐lattice polymer models and results for both lattice and off‐lattice simulations are presented.
We present a field-theoretic analysis of high-precision Monte Carlo data for the Domb - Joyce model on the sc lattice. We vary the repulsion between two segments at the same point from zero (random walk) to infinity (self-avoiding walk). Eventually, we even include a repulsion between segments at neighbour points to increase the excluded volume beyond that of self-avoiding walks. The data for the end-to-end distance, the radius of gyration and the partition function clearly show the existence of two branches of universal behaviour. These two branches can be identified with the weak- and strong-coupling branch of the renormalization group, respectively. A quantitative analysis shows the ability of the standard field theoretic approach to describe the data, including the data for strong coupling, i.e. renormalized coupling u greater than its fixed point value 0305-4470/30/20/010/img6. We conclude, in contrast with some claims in the literature, that the standard formalism of the renormalized field theory can be used even for 0305-4470/30/20/010/img7 (strong-coupling branch). In addition, exploiting the fast approach to asymptotic behaviour at the transition between weak and strong coupling, we obtain very precise estimates for the critical exponents of self-avoiding walks.
Simple random walk is well understood. However, if we condition a random walk not to intersect itself, so that it is a self-avoiding walk, then it is much more di-- cult to analyse and many of the important mathematical problems remain unsolved. This paper provides an overview of some of what is known about the self-avoiding walk, includ- ing some old and some more recent results, using methods that touch on combinatorics, probability, and statistical mechanics.
Monte Carlo simulations employing the pivot algorithm are used to generate ideal and excluded-volume chains on two- and three-dimensional lattices. The second, fourth, sixth, and eighth moments of the average monomer–monomer separation are calculated from the resulting configurations. The coefficients in the expansion of the structure factor are computed from universal ratios of these. The values found for excluded-volume chains are smaller than the ideal chain values and the differences are greater in two dimensions than in three dimensions.
Chains embedded in the tetrahedral lattice have been produced by means of Monte Carlo simulation for athermal and for theta conditions. Nonreversal random walks (random walks without backfolding bonds) have been generated as a reference. Probability distributions of an asphericity factor δ∗, of a prolatness factor S∗, and of shape factors sfi∗ have been evaluated, the quantities being based on the orthogonal components of the squared radius of gyration taken along the principal axes of inertia. In addition, the correlation between δ∗ and other shape descriptors as well as between δ∗ and quantities characteristic of the size of configurations have been evaluated. In accordance with existing literature, the distributions H(δ∗) and H(S∗) are found to be very broad. The distributions H(sf1∗) of the small and H(sf3∗) of the large shape factor (sf1∗ ⩽ sf2∗ ⩽ sf3∗, sf1∗+sf2∗+sf3∗ = 1) are clearly distinct from each other, while the distribution of sf2∗ overlaps with that of sf1∗ and (slightly) with H(sf3∗). Distributions of theta chains (“unperturbed” polymer) coincide fairly well with respective distributions of nonreversal random walks (which in turn are nearly identical to those of (off-lattice) random walks. As a matter of course, other shape descriptors are directly correlated with the asphericity factor δ∗. Actually, for all systems evaluated, the global size of configurations is strongly correlated with δ∗ as well: The larger the asymmetry the larger are the dimensions of the configuration under consideration. © 1998 American Institute of Physics.
Dynamic Monte Carlo simulations, using the pivot algorithm, have been performed on a three‐dimensional simple cubic lattice in order to study statistical properties of self‐avoiding walks of lengths up to about 7200 steps. The scaling properties of the end‐to‐end distribution function and its second moment have been investigated and compared with previous works. The distribution function is found to be in agreement with the enhanced Gaussian scaling form of des Cloizeaux. Some properties of the pivot algorithm, such as the acceptance fraction and the integrated autocorrelation times in three dimensions, are also discussed.
Monte Carlo simulations employing the pivot algorithm are used to generate random and self‐avoiding walks on two‐ and three‐dimensional lattices. The moments of the end‐to‐end distance distribution function are calculated from the resulting configurations. It is found that the moments and the shape of the vector distribution function are in excellent agreement with the scaling form derived by des Cloizeaux.
By use of a dynamic Monte Carlo method, known as wiggle or pivot algorithm, chains of chain length N up to 10 000 segments are generated on a tetrahedral lattice. A new chain is obtained by rotating one part of a chain around a randomly selected bond by ±120°. If segments of the transformed subchain overlap with segments of the second (unmoved) part, the new chain is rejected and the old one retained. The starting configurations usually were prepared from an all-trans configuration by equilibrating except for short chains where they were generated using a simple step-by-step procedure. The most important results are as follows: The exponent α in the power law describing the decrease of the acceptance fraction f with increasing number of bonds n = (N - 1), f ∼ n-α, equals 0.1146. The position-dependent acceptance fraction, f(x), roughly may be approximated by f(x) = [x(1 - x)n]-α, x being the length of the shorter subchain divided by n. By use of a simple double-logarithmic plot (in the range 100 ≤N ≤ 10 000), the critical exponent in the scaling relation of the mean-square end-to-end distance 〈h2〉 ∼ nν′ was estimated as ν′ = 1.1802, and for the mean-square radius of gyration 〈s2〉 ∼ nν″ a value ν″ = 1.1832 was found. These values are not equal and slightly exceed those predicted by renormalization group theory (ν′ = ν″ = ν = 1.176). This is in full accordance with other recent Monte Carlo results. However, taking into account a correction (C0 + C1n-Δ) to nν and using ν = 1.176 and Δ = 0.47 as proposed by renormalization group theory, 〈h2〉 and 〈s2〉 may be very well described by 〈h2〉 = nν(4.367 - 0.982n-Δ) and 〈s2〉 = nν(0.698 - 0.242n-Δ).
We propose a consistent description for the mean-square length of self-avoiding walks on lattices and in the continuum, based on careful numerical studies in two and three dimensions. Existing exact enumerations for various lattices are combined with new precise Monte Carlo results and are analyzed according to the best available theoretical models. Small but persistent differences from two-parameter dimensions for walks on different lattices are discussed. The Domb-Barrett equation for the mean-square end-to-end length of a walk is adjusted to reflect the most accurate estimates of the coefficients and exponents, and a similar equation is proposed for two-dimensional walks. Finally, new results for self-avoiding walks in the continuum are interpreted within the framework of the Domb-Joyce model as being consistent with lattice results.
The authors study the effect of repulsion for self-avoiding walks and random walks from excluded sets. They show, in particular, that the mean displacement away from an excluded infinite needle of self-avoiding random walks in three dimensions has to diverge along the privileged axis as Nsigma , where N is the number of steps and sigma is a sub-leading critical exponent for the two-point function. This exponent has been determined by using a high-precision Monte Carlo simulation ( sigma =0.370+or-0.011). Its knowledge is used to improve the measure of universal quantities, like the exponent nu ( nu =0.5867+or-0.0025, in agreement with the in -expansion estimate and with experimental data) and amplitude ratios. They verify also that for simple random walks the excluded needle introduces instead logarithmic violations to scaling.
The authors present new results which indicate that the leading correction-to-scaling exponent in the mean squared end-to-end distance in two dimensions is the analytic term. A potential source of the various correction-to-scaling terms reported in the literature is pointed out.
A new Monte Carlo method is proposed which allows for the efficient generation of equilibrium conformations of polymer chains in two and three dimensions. The method treats each site (monomer) as a potential pivot around which a new conformation may be generated by rotating a portion of the chain. The method does not suffer from the severe attrition associated with the simple sampling of self-avoiding walks and may be extended to treat the interacting polymer chain. The authors find in two dimensions that nu =0.748+or-0.005 (exact=0.750) and in three dimensions nu =0.595+or-0.005 (series expansion and renormalisation group predict nu approximately 0.588). The end-end distances calculated for shorter chains are in good agreement with the exact values from enumeration techniques.
The authors correct a combinatorial error in the Cardy-Saleur conformal-invariance prediction of a universal amplitude ratio for two-dimensional self-avoiding walks. They present high-precision Monte Carlo data that confirm the corrected prediction.
Linear and star-branched chains with F = 4 − 12 arms and N = 125 − 7685 segments covering points of a tetrahedral lattice were generated by use of a pivot algorithm. For large N, the acceptance fractions f̄ of attempted moves may be described by a power law f̄ = A · (N − 1)−α. Clearly, the factor A decreases with increasing functionality F, but the exponent α is independent of the number of arms and equal to the value obtained for linear chains, α ≈ 0,1. Due to the influence of the hard-core (centre) of the star, the acceptance fraction f̄ for small N is lower than predicted by the scaling law, yielding a stronger dependence on F than for large chains. Meansquare dimensions, i. e. mean-square radius of gyration, mean-square end-to-end distance and mean-square centre-to-end distance obey a power law dependence on chain-length (N − 1), the exponent being ≈ 1,184 for 245 ≤ N ≤ 7685 in all cases; alternatively, the (quadratic) dimensions may be described by a corrected scaling law (N − 1)2v · (C0 + C1 · (N − 1)−Δ) with v = 0,588 and Δ ≈ 0,5 as proposed by renormalization group theory for linear chains. The shape asymmetry of star-branched polymers (with the same total number of segments each) decreases with increasing number of arms, but is still appreciable for F = 12, the highest number of arms examined.
The asymptotic behaviour of isolated long chain polymers in good solvents is modelled well by the self-avoiding walk on a regular crystal lattice. In this paper we investigate the typical scaling length and the probability distribution function for the position of the end point and an interior point for self-avoiding walks on a simple cubic lattice. We find that for N-step walks the typical internal scaling length, RN(n), associated with the position of the nth step, has a similar functional form to the external scaling length, RN. The associated exponent ν is found to be the same for both scaling lengths, within the confidence limits of our simulations. The internal probability distribution function, pN(n,r), is shown to be skew-Gaussian and furthermore has a similar form to the external distribution function: pN(n,r) ≈ [r/RN(n)]aexp{−[r/RN(n)δ}, where a = 2.42 ± 0.08 and δ = 2.50 ± 0.09.
The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one effectively independent sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents are=0.74960.0007 ind=2 and= 0.5920.003 ind=3 (95% confidence limits), based on SAWs of lengths 200N10000 and 200N 3000, respectively.
We propose a new class of dynamic Monte Carlo algorithms for generating self-avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension, and prove that these algorithms are ergodic in all cases. We also prove the ergodicity of a variant of the pivot algorithm.
We consider a simplified model of vorticity configurations in the inertial range of turbulent flow, in which vortex filaments are viewed as random walks in thermal equilibrium subjected to the constraints of helicity and energy conservation. The model is simple enough so that its properties can be investigated by a relatively straightforward Monte-Carlo method: a pivot algorithm with Metropolis weighting. Reasonable values are obtained for the intermittency dimensionD, a Kolmogorov-like exponent γ, and higher moments of the velocity derivatives. Qualitative conclusions are drawn regarding the origin of non-gaussian velocity statistics and regarding analogies with polymers and with systems near a critical point.
We analyze Monte Carlo data of self-avoiding walks with up to about 8000 steps on a simple cubic lattice with emphasis on the question of the discrepancy between the scaling exponents obtained by renormalization group calculations and Monte Carlo simulations. This discrepancy has been recently investigated by Dayantis and Palierne [Phys. Rev. B 49, 3217 (1994)] for self-avoiding walks of up to 3000 steps, and has been shown to originate from the finite size of chains generated in Monte Carlo simulations. Our analysis demonstrates this conjecture and shows that the exponents decrease for longer chains toward the renormalization group value.
We discuss possible sources of systematic errors in the computation of critical exponents by renormalization-group methods, extrapolations from exact enumerations and Monte Carlo simulations. A careful Monte Carlo determination of the susceptibility exponent gamma for three-dimensional self-avoiding walks has been used to test the claimed accuracy of the various methods. Comment: Talk presented at Lattice '97 (Higgs, Yukawa, SUSY and Spin Models), Edinburgh (UK), July 1997. LaTeX2e, 3 pages, uses espcrc2
We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents and as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation . In two dimensions, we confirm the predicted exponent and the hyperscaling relation; we estimate the universal ratios \ / \ = 0.14026 \pm 0.00007 , \ / \ = 0.43961 \pm 0.00034 and (68\% confidence limits). In three dimensions, we estimate with a correction-to-scaling exponent (subjective 68\% confidence limits). This value for agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for . Earlier Monte Carlo estimates of , which were , are now seen to be biased by corrections to scaling. We estimate the universal ratios \ / \ = 0.1599 \pm 0.0002 and ; since , hyperscaling holds. The approach to is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. Comment: 87 pages including 12 figures, 1029558 bytes Postscript (NYU-TH-94/09/01)
Monte Carlo Methods for the Self-Avoiding Walk, in Monte Carlo and Molecu-lar Dynamics Simulations in Polymer Science
- A Sokal
A. Sokal, Monte Carlo Methods for the Self-Avoiding Walk, in Monte Carlo and Molecu-lar Dynamics Simulations in Polymer Science, Kurt Binder, ed. (Oxford University Press, 1995), hep-lat/9405016.
Monte Carlo" computer simulations of chain molecules I
M. Lal, "Monte Carlo" computer simulations of chain molecules I, Molec. Phys. 17: 57-64,
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