Principles and applications of a direct position space renormalization group for lattice models of polymers—called the Cluster Renormalization (CR)—are reviewed in detail. Particular emphasis is placed on the study of crossover phenomena and determination of universality classes in polymer models. The first part is largely a pedagogical description of the scaling concept and the idea of fractal dimensionality for polymers, and discusses the relation between polymer statistics and critical phenomena. The generalized lattice animal model for polymers is then introduced and it is shown that within a grand canonical ensemble polymers can be described as critical objects. This description enables us to apply CR to polymer models. Next, essentials of the CR are presented. Discussion of the applications begins with a single‐parameter CR study of the scaling properties of models of linear polymers, branched polymers and polymer networks. Finally various applications of two‐parameters CR to crossover phenomena are discussed. These include a study of the crossover from a random walk to a renormalization group approach for random walks, and a discussion of the effects of branches and loops on the universality classes of polymers. It also deals with the effects of screening in solutions of branched polymers and the crossover from lattice animals to percolation.
Principles and applications of a direct position space renormalization group for lattice models of polymers—called the Cluster Renormalization (CR)—are reviewed in detail. Particular emphasis is placed on the study of crossover phenomena and determination of universality classes in polymer models. The first part is largely a pedagogical description of the scaling concept and the idea of fractal dimensionality for polymers, and discusses the relation between polymer statistics and critical phenomena. The generalized lattice animal model for polymers is then introduced and it is shown that within a grand canonical ensemble polymers can be described as critical objects. This description enables us to apply CR to polymer models. Next, essentials of the CR are presented. Discussion of the applications begins with a single‐parameter CR study of the scaling properties of models of linear polymers, branched polymers and polymer networks. Finally various applications of two‐parameters CR to crossover phenomena are discussed. These include a study of the crossover from a random walk to a renormalization group approach for random walks, and a discussion of the effects of branches and loops on the universality classes of polymers. It also deals with the effects of screening in solutions of branched polymers and the crossover from lattice animals to percolation.
Main concepts in critical phenomena such as universality, critical dimension, analogies with the statistics of polymers and scaling theory, together with renormalization group ideas and rudimentary methods in field theory are presented through the simple and intuitive properties of random walks. This work, which requires little previous knowledge of mathematics besides elementary calculus, has served as the basis of a program to introduce undergraduate students to research in theoretical physics.
Abstract Chapter 16 explores the statistical mechanics of discrete lattice models and their use in describing critical phenomena. The chapter begins with a review of phase diagrams, coexistence curves, crititical points, and the behavior of certain thermodynamic quantities in the neighborhood of a critical point. The notion of universality classes is introduced. To set up the class of lattice models, magnetic systems are described; simplifications are made to these models to yield the so-called Ising model. Solution of this model is demonstrated within the mean-field theory approximation, which is shown to yield a qualitative description of critical behavior. Exact solutions are derived in one and two dimensions, and predictions of critical exponents from them are obtained. Following this, the renormalization group is introduced, and approaches for understanding critical behavior and universality from renormalization group transformations are discussed. The chapter concludes with a discussion of the use of discrete models in areas ranging from molecular crystals to tissue morphogenesis
This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.
Condensed media show a large variety of critical phenomena, ranging from critical opalescence at the end point of the liquid–gas coexistence curve, the Curie transition of ferromagnetic materials, to the superfluid transition of helium, the behaviour of solutions of polymers, the conductivity of random media, … To this list should be added systems with local symmetries, such as those suggested by particle physics in order to understand quark confinement. Surprisingly all these phenomena can be classified into a few universality classes, characterized by specific large distance behaviour with the same critical exponents. In this chapter, we sketch the methods and ideas of the renormalization procedure. We illustrate the concepts with simple approximations in the language of classical spin models in the first section, treating in more detail the XY -model as an example in the second section. Scaling laws. Real space renormalization Homogeneity and scale invariance The discussion given in previous chapters suggests that, close to a continuous transition, critical systems exhibit universal properties. Correlations at large distance are not sensitive to the details of microscopic interactions. Their behaviour is described by a specific dimensional analysis governed by some essential characteristics of the system, such as the dimension of space, the nature of the order parameter and the underlying symmetries. The mean field approximation gave a first idea of a simple critical behaviour. Fluctuations have only a quantitative effect in dimension greater than the upper critical one.
The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.
We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous nonequilibrium active to absorbing state phase transitions whose asymptotic features are governed respectively by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions dc = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder. Key words: Percolation, epidemic processes, directed percolation, dynamic isotropic percolation, active to absorbing phase transitions, renormalization group theory, dynamic critical phenomena, crossover