Here proposed is a revised version of the scaling theory of equilibrium critical phenomena. This revised theory is based on hypotheses which differ only slightly from those set forth in original formulations. The conclusions reached are essentially equivalent to those of Stell's weak scaling theory. One obtains relations which require prior knowledge of three exponents for the derivation of the rest. An additional exponent, denoted w, enters certain relations containing exponents which pertain to the correlation function. This exponent must be greater than or equal to one in order to satisfy certain generally accepted inequalities. An argument for why w should be greater than or equal to one is formulable within the framework of the original hypotheses of scaling theory. While the revised theory has the drawback of containing an extra unknown, it is nevertheless capable of reconciling scaling theory with other theories of critical exponents.
Here proposed is a revised version of the scaling theory of equilibrium critical phenomena. This revised theory is based on hypotheses which differ only slightly from those set forth in original formulations. The conclusions reached are essentially equivalent to those of Stell's weak scaling theory. One obtains relations which require prior knowledge of three exponents for the derivation of the rest. An additional exponent, denoted w, enters certain relations containing exponents which pertain to the correlation function. This exponent must be greater than or equal to one in order to satisfy certain generally accepted inequalities. An argument for why w should be greater than or equal to one is formulable within the framework of the original hypotheses of scaling theory. While the revised theory has the drawback of containing an extra unknown, it is nevertheless capable of reconciling scaling theory with other theories of critical exponents.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.
A single assumption is made about the dynamic correlation function, that close to the critical point and for large r and t, it is a generalized homogeneous function. After proving and using certain simple theorems on this class of functions, contact is made with previous formulations of dynamic and static scaling. It is found that 1. this single assumption is sufficient to obtain all previous results of both the dynamic and static scaling hypotheses, 2. simple expressions, which can be written down ‘by inspection’ for all critical point exponents, enable all the exponent relations to be derived immediately, 3. we have additional ways of plotting data in the form of ‘scaling functions’, 4. it becomes possible to easily and accurately calculate the scaling function for models (such as the Heisenberg model) for which no low‐temperature series expansions exist, and 5. the two‐exponent relations (e. g., dν=2−α) are not consequences of our approach unless we make an additional assumption, an assumption made implicitlyby many other workers. If we don't make this additional assumption, the two‐exponent relations are found to have correction terms which transform them into ordinary thermodynamic scaling laws.
By the early 1960's advances in statistical physics had established the existence of universality classes for systems with second-order phase transitions and characterized these by critical exponents which are different to the classical ones. There followed the discovery of (now famous) scaling relations between the power-law critical exponents describing second-order criticality. These scaling relations are of fundamental importance and now form a cornerstone of statistical mechanics. In certain circumstances, such scaling behaviour is modified by multiplicative logarithmic corrections. These are also characterized by critical exponents, analogous to the standard ones. Recently scaling relations between these logarithmic exponents have been established. Here, the theories associated with these advances are presented and expanded and the status of investigations into logarithmic corrections in a variety of models is reviewed.
By the early 1960's advances in statistical physics had established the existence of universality classes for systems with second-order phase transitions and characterized these by critical exponents which are different to the classical ones. There followed the discovery of (now famous) scaling relations between the power-law critical exponents describing second-order criticality. These scaling relations are of fundamental importance and now form a cornerstone of statistical mechanics. In certain circumstances, such scaling behaviour is modified by multiplicative logarithmic corrections. These are also characterized by critical exponents, analogous to the standard ones. Recently scaling relations between these logarithmic exponents have been established. Here, the theories associated with these advances are presented and expanded and the status of investigations into logarithmic corrections in a variety of models is reviewed.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
Arguments are presented which allow one to characterize weak-scaling correlations in terms of thermodynamic homogeneity. Using as a guide the present indications on the spin-spin correlation function of the three-dimensional Ising model, it is argued that the thermodynamic effects of the smooth deviations from strong scaling in dimension d>2 can be accounted for most easily by looking at the correlations at a second effective length $\ensuremath{\zeta}=\ensuremath{\zeta}(\ensuremath{\omega})$, much smaller than the correlation length. This depends on a constant $\ensuremath{\omega}$, a nonzero value of which indicates deviation from strong scaling and $\ensuremath{\zeta}(\ensuremath{\omega})$ is determined as that length over which an effective strong-scaling correlation in dimension ($d\ensuremath{-}\ensuremath{\omega}$) yields the expected "scaling" relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$. It is found that $\ensuremath{\zeta}(\ensuremath{\omega})$ is much larger than the second length which appears in the weak-scaling theory of Stell. The constant $\ensuremath{\omega}$ enters all relations between critical exponents which depend explicitly on dimensionality and it is shown that a value $\ensuremath{\omega}\ensuremath{\simeq}\frac{1}{12}$ is consistent with the exponents for the three-dimensional Ising model. It is argued that weak scaling should be expected for general pair correlations and it is shown how a further inequality between critical exponents involving energy-density correlations can be derived. The limitation of our scheme to values of d smaller than that at which the mean-field behavior sets in is explicitly pointed out.