Self avoiding walks

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

The basic features of folding can be understood in tenns of two fundamental equilibrium temperatures that detennine tire phases of tire system [7]. At sufficiently high temperatures (JcT greater tlian all tire attractive interactions) tire shape of tire polypeptide chain can be described as a random coil and hence its behaviour is tire same as a self-avoiding walk. As tire temperature is lowered one expects a transition at7 = Tq to a compact phase. This transition is very much in tire spirit of tire collapse transition familiar in tire theory of homopolymers [10]. The number of compact... 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

H. Meirovitch, S. Livne. II. Critical behavior of single self-avoiding walks. J Chem Phys 4507 515, 1988. 

N. Madras, A. D. Sokal. The pivot algorithm A highly efficient Monte Carlo method for the self-avoiding walk. J Stat Phys 50 109, 1988. 

A. B. Harris. Self-avoiding walks on random lattices. Z Phys B 49 347-349, 1983. 

For a self-avoiding walk (SAW) the coil end-to-end distance, R, scales with,... 

Sokal, A.D. Monte Carlo methods for the self-avoiding walk. In Monte Carlo and Molecular Dynamics Simulations in Polymer Science (ed. K. Binder), Oxford University Press, New York, 1995, pp. 47-124. 

Nidras, P.P., Brak, R. New Monte Carlo algorithms for interacting self-avoiding walks. J. Phys. A Math. Gen. 1997, 30,1457-69. 

II. Random Walks, Restricted Walks, and Self-Avoiding Walks.230... 

G. Passage from a Restricted to a Self-Avoiding Walk.247... 

A self-avoiding walk on a lattice is a random walk subject to the condition that no lattice site may be visited more than once in the walk. Self-avoiding walks were first introduced as models of polymer chains which took into account in a realistic manner the excluded volume effect1 (i.e., the fact that no element of space can be occupied more than once by the polymer chain). Although the mathematical problem of... 

Despite the shortage of rigorous analytical results, information has been accumulating regarding various properties of self-avoiding walks, and it now seems possible to put forward a coherent pattern of general behavior. This is the aim of the present paper. No attempt will be made to provide complete references to the extensive literature, but a sufficient number of key papers will be cited to enable anyone interested to do so for himself,... 

H. RANDOM WALKS, RESTRICTED WALKS, AND SELF-AVOIDING WALKS... 

A self-avoiding walk corresponds to the limit of a restricted walk as r- oo. Such a walk is no longer Markovian, and no general analytical methods have been found for determining its properties. However, evidence has been accumulating to indicate that the basic characteristics differ in an essential manner from those of restricted walks. We now proceed to discuss this evidence in further detail. 

Four independent approaches have been used to investigate the properties of self-avoiding walks. 

In the previous section we considered the approach to a self-avoiding walk by means of a restricted walk of order r, governed by a Markovian transition matrix. Although the size of this matrix increases rapidly, it is possible to calculate the distribution of its eigenvalues for small r in the hope of detecting a general pattern of behavior as r increases and tends to infinity. This approach has been used by Domb and Hioe14 and more recently by Mazur.15 It is difficult to draw direct conclusions from the method but confirmatory evidence can be provided for a type of behavior... 

The methods described above should be regarded as complementary rather than alternative. In the absence of adequate exact knowledge it is useful to derive evidence from as many sources as possible, and the pattern of behavior of self-avoiding walks proposed in the next section is the result of a synthesis of the information provided by the different methods. 

Fig. 1. Ratio plot for triangular lattice, (a) Self-avoiding walks / x and C /C i...

The majority of attention in earlier discussions of self-avoiding walks was focused on the behavior of mean square length with increasing n, and on the question whether converges to a limit, as for restricted... 

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