Cluster expansion for a polymer chain

Domb, C.; Joyce, G S

doi:10.1088/0022-3719/5/9/009

Cited by 183 publications

(168 citation statements)

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“…In one dimension (C ϭ2), the partition function Z(⑀,T) is exactly evaluated by the standard transfer matrix method: 36 Thermodynamic properties of the pure solvent are then obtained by differentiating the Gibbs free energy G ϭϪkT ln Z or the chemical potential w ϭϪkT ln(e Ϫ⑀/kT 1 ) of solvent. The volume per molecule is given by ‫ץ(‬ w /‫ץ‬p) T , which is in a dimensionless form:…”

Section: ͑23͒mentioning

confidence: 99%

Hydrophobic effect in the pressure-temperature plane

Koga

2004

The Journal of Chemical Physics

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The free energy of the hydrophobic hydration and the strength of the solvent-mediated attraction between hydrophobic solute molecules are calculated in the pressure-temperature plane. This is done in the framework of an exactly soluble model that is an extension of the lattice model proposed by Kolomeisky and Widom ͓A. B. Kolomeisky and B. Widom, Faraday Discuss. 112, 81 ͑1999͔͒. The model takes into account both the mechanism of the hydrophobic effect dominant at low temperatures and the opposite mechanism of solvation appearing at high temperatures and has the pressure as a second thermodynamic variable. With this model, two boundaries are identified in the pressure-temperature plane: the first one within which the solubility, or the Ostwald absorption coefficient, decreases with increasing temperature at fixed pressure and the second one within which the strength of solvent-mediated attraction increases with increasing temperature. The two are nearly linear and parallel to each other, and the second boundary lies in the low-temperature and low-pressure side of the first boundary. It is found that a single, near-linear relation between the hydration free energy and the strength of the hydrophobic attraction holds over the entire area within the second boundary in the pressure-temperature plane.

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Section: ͑23͒mentioning

confidence: 99%

Hydrophobic effect in the pressure-temperature plane

Koga

2004

The Journal of Chemical Physics

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“…We implement the Bethe lattice condition either by explicitly considering sites far from the boundary or, alternatively, by using a formulation appropriate to periodic lattices and then introducing approximations which become exact when the lattice does not support any loops. 19 By initially treating a periodic lattice we eliminate anomalous surface effects. We are thus assured that our results are characteristic of the interior of the tree and should be associated with what is now commonly called a Bethe lattice.…”

Section: Introductionmentioning

confidence: 99%

Dimer statistics on a Bethe lattice

Harris

Cohen

2006

The Journal of Chemical Physics

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We discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice.We reproduce the well-known exact result for noninteracting hard-core dimers by both a very simple geometrical argument and a general algebraic formulation for lattice statistical problems. The algebraic formulation enables us to discuss loop corrections for finite dimensional lattices. For the Bethe lattice we also obtain the exact solution when either (a) the dimers interact via a short-range interaction or (b) the underlying lattice is anisotropic. We give the exact solution for a special limit of dimers on a Bethe lattice in a quenched random potential in which we consider the maximal covering of dimers on random clusters at site occupation probability p. Surprisingly the partition function for "maximal coverage" on the Bethe lattice is identical to that for the statistics of branched polymers when the activity for a monomer unit is set equal to −p. Finally we give an exact solution for the number of residual vacancies when hard-core dimers are randomly deposited on a one dimensional lattice. We discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice. We reproduce the well-known exact result for noninteracting hard-core dimers by both a very simple geometrical argument and a general algebraic formulation for lattice statistical problems. The algebraic formulation enables us to discuss loop corrections for finite dimensional lattices. For the Bethe lattice we also obtain the exact solution when either ͑a͒ the dimers interact via a short-range interaction or ͑b͒ the underlying lattice is anisotropic. We give the exact solution for a special limit of dimers on a Bethe lattice in a quenched random potential in which we consider the maximal covering of dimers on random clusters at site occupation probability p. Surprisingly the partition function for "maximal coverage" on the Bethe lattice is identical to that for the statistics of branched polymers when the activity for a monomer unit is set equal to −p. Finally we give an exact solution for the number of residual vacancies when hard-core dimers are randomly deposited on a one dimensional lattice.

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“…For λ = 1, we have c (1) n (x) = c n (x), and we will usually omit the superscript (1) when λ = 1. For 0 < λ < 1, (2.2) defines a much-studied model of weakly self-avoiding walks (sometimes called the Domb-Joyce model after [64]) in which walks with self intersections receive less weight than walks that are self-avoiding.…”

Section: Asymptotic Behaviourmentioning

confidence: 99%

The Lace Expansion and its Applications

Saint-Flour

Slade

Picard³

2006

Lecture Notes in Mathematics

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ii The Lace Expansion and its ApplicationsThe Lace Expansion and its Applications iii Preface Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. Although these models have been studied by mathematicians for about half a century, exciting new developments continue to occur and the subject is flourishing. Much progress has been made, but it remains a major challenge for mathematical physics and probability theory to obtain a complete and mathematically rigorous understanding of the scaling theory of these models at criticality.These lecture notes concern the lace expansion, which is a powerful tool for the analysis of the critical scaling of several models above their upper critical dimensions, namely:• the self-avoiding walk on Z d for d > 4, • lattice trees and lattice animals onResults include proofs of existence of critical exponents, with mean-field values, and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion. There are two distinct goals for these notes. The first goal is to provide a written accompaniment to my lectures at the Saint-Flour summer school in 2004, and at the Pacific Institute for the Mathematical Sciences -University of British Columbia summer school in 2005. The notes contain an introduction to the lace expansion and several of its applications, with sufficient background and depth to prepare a newcomer to do research using the lace expansion. Basic graduate level probability theory will be used, but no previous knowledge of the lace expansion or super-Brownian motion is assumed. The second goal is to provide a survey of the field, so that an interested reader can follow up by consulting the original literature. In pursuit of the second goal, these notes include more material than can be covered during a summer school course.Following a brief initial section concerning random walk, the notes can be divided into four parts, whose contents are summarized as follows.Part I, which concerns the self-avoiding walk, consists of Sections 2-6. A complete and self-contained proof is given of the convergence of the lace expansion for the nearest-neighbour model in dimensions d 4, and for the spread-out model of self-avoiding walks which take steps of length at most L, with L 1, in dimensions d > 4. The convergence proof presented here seems simpler than all previous lace expansion convergence proofs. As a consequence of convergence, it is shown that the critical exponent γ for the generating function of the number of n-step self-avoiding walks exists and is equal to 1. A survey is then given of the many extensions of this result that have been obtained using the lace expansion. Part II, which concerns lattice trees and lattice animals, consists of Sections 7-8. It is shown how a minor modification of the expansion for the self-avoiding walk can be applied to give expansions for lattice trees and lattice animals, and an indicatio...

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Cluster expansion for a polymer chain

Cited by 183 publications

References 23 publications

Hydrophobic effect in the pressure-temperature plane

Hydrophobic effect in the pressure-temperature plane

Dimer statistics on a Bethe lattice

The Lace Expansion and its Applications

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