This chapter defines more general statistical systems in an arbitrary number of space dimensions. For convenience, it uses the ferromagnetic language — even though, as a consequence of universality, the results that are derived in this work apply to much more general statistical systems. In addition to complete and connected correlation functions (whose decay properties at large distance — called cluster properties — are recalled), it introduces vertex functions, which are related to the thermodynamic potential. The free energy and thermodynamic potential, like connected correlation functions and vertex functions, are related by a Legendre transformation of which a few properties are discussed. Exercises are provided at the end of the chapter.
This chapter defines more general statistical systems in an arbitrary number of space dimensions. For convenience, it uses the ferromagnetic language — even though, as a consequence of universality, the results that are derived in this work apply to much more general statistical systems. In addition to complete and connected correlation functions (whose decay properties at large distance — called cluster properties — are recalled), it introduces vertex functions, which are related to the thermodynamic potential. The free energy and thermodynamic potential, like connected correlation functions and vertex functions, are related by a Legendre transformation of which a few properties are discussed. Exercises are provided at the end of the chapter.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.
Introduction We noted in the previous chapter that the singular behavior of thermodynamic functions at a critical point (the termination of a coexistence line) can be characterized by a set of critical exponents {α, β, γ, …}. Experimental observations indicate that these exponents are quite universal , i.e. independent of the material under investigation, and to some extent, of the nature of the phase transition. For example, the vanishing of the coexistence boundary in the condensation of CO 2 has the same singular behavior as that of the phase separation of protein solutions into dilute and dense components. This universality of behavior needs to be explained. We also noted that the divergence of the response functions, as well as direct observations of fluctuations via scattering studies, indicate that fluctuations have long wavelengths in the vicinity of the critical point, and are correlated over distances ξ ≫ a , where a is a typical interparticle spacing. Such correlated fluctuations involve many particles and a coarse-graining approach, in the spirit of the theory of elasticity, may be appropriate to their description. Here we shall construct such a statistical field theory . We shall frame the discussion in the language of a magnetic system whose symmetries are more transparent, although the results are of more general applicability. Consider a material such as iron, which is experimentally observed to be ferromagnetic below a Curie temperature T c , as in Fig. 1.4.
The structural properties of all diagrammatic expansions developed so far can be analyzed systematically with the help of functional equations. 5.1 Generating Functionals In Chapter 3 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (2.14), and the generating functional itself, contain many disconnected parts. Ultimately, however, we shall be interested only in the connected parts of Z[j]. Remember that a meaningful description of a very large thermodynamic system can only be given in terms of the free energy which is directly proportional to the total volume. In the limit of an infinite volume, also called thermodynamic limit, one has then a well-defined free energy density. The partition function, on the other hand, has no proper infinite-volume limit. We can observe this property directly in the diagrammatic expansion of Z[j]. Each component of a disconnected diagram is integrated over the entire space, thus contributing a volume factor. The expansion of Z[j] therefore diverges at an infinite volume. In thermodynamics, we form the free energy from the logarithm of the partition function, which carries only a single overall volume factor and contains only connected diagrams. Therefore we expect the logarithm of Z[j] to provide us with the desired generating functional W[j]: W[j] = log Z[j]. (5.1) In this chapter we shall see that the functional derivatives of W[j] produce, indeed, precisely the connected parts of the Feynman diagrams in each correlation function. Consider the connected correlation functions G (n) c (x1,...,xn) defined by the functional derivatives G (n) c (x1,...,xn) = δ δ
A general perturbation theory of the statistics of spin interactions is developed in the form of a linked cluster expansion with particular emphasis on the Ising model. The theory applies to the evaluation of the expectation value of arbitrary spin functions as well as of the free energy. The thermodynamical consistency of the perturbation expansion is shown to arise from (1) variational principles satisfied by the free energy after a "renormalization procedure" has been carried out and (2) a generalized "Ward Identity" between renormalized quantities. These results are used to discuss inconsistencies in recent high-density theories of ferromagnetism and an improved theory obtained by the summation of all the convolution diagrams (nodal expansion) is briefly presented. The applicability of the method to general quantum mechanical many-body problems, including the theory of the Heisenberg model of ferromagnetism, is shown.
This chapter examines the universal properties of phase transitions in the quasi-Gaussian or mean-field approximations. It studies the singularities of thermodynamic functions at the transition point as well as the large-distance behaviour of the two-point correlation function. It summarizes the universal properties in the form of Landau′s theory. It stresses the peculiarities of models with continuous symmetries at low temperature due to the appearance of Goldstone modes. Finally, it evaluates corrections to the quasi-Gaussian approximation and shows that the approximation is only consistent in space dimension larger than four. Exercises are provided at the end of the chapter.