A function $f({x}_{1}, {x}_{2},\dots{}, {x}_{n})$ is a generalized homogeneous function (GHF) if we can find numbers ${a}_{1}, {a}_{2}, \dots{}, {a}_{n}$ such that for all values of the positive number $\ensuremath{\lambda}$, $f({\ensuremath{\lambda}}^{{a}_{1}}{x}_{1}, {\ensuremath{\lambda}}^{{a}_{2}}{x}_{2}, \dots{}, {\ensuremath{\lambda}}^{{a}_{n}}{x}_{n})={\ensuremath{\lambda}}^{{a}_{f}}f({x}_{1}, {x}_{2}, \dots{}, {x}_{n})$. We organize the properties of GHFs in four theorems. These are used to systematically examine the consequences of various scaling hypotheses. An advantage of this approach is that the same formalism may be used to treat thermodynamic functions, static correlation functions, dynamic correlation functions, and "universality." The simple case of thermodynamic scaling (two independent variables) is first generalized to static and dynamic correlation functions (three and four variables), and then to scaling with a parameter (for which the critical subspace becomes higher dimensional). In this last case, where a second GHF hypothesis is made, the necessity of crossover lines is demonstrated. The assumption of homogeneity is clearly separated from any extra assumptions that may also be called scaling (or "strong scaling"), but are independent of and different from that of homogeneity. One practical insight gained from the present approach is that all experimentally measured exponents are expressible as the ratio of two scaling powers, ${a}_{f}$ (which refers to the function) and ${a}_{j}$ (which refers to the path of approach to the critical point). A second practical advantage is that, since a GHF can be scaled with respect to any of its arguments, one can immediately write a variety of scaling functions for each type of scaling hypothesis. The GHF approach thereby permits data to be plotted in a variety of convenient fashions, and is found to facilitate computation of the relevant scaling functions (in particular, the GHF approach led directly to the recent calculation of the Heisenberg model scaling function by Milos\ifmmode \breve{}\else \u{}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley).
A function f(x1,x2,…,xn) is a generalized homogeneous function (GHF) if we can find numbers a1,a2,…,an such that for all values of the positive number \ensuremathλ, f(\ensuremathλa1x1,\ensuremathλa2x2,…,\ensuremathλanxn)=\ensuremathλaff(x1,x2,…,xn). We organize the properties of GHFs in four theorems. These are used to systematically examine the consequences of various scaling hypotheses. An advantage of this approach is that the same formalism may be used to treat thermodynamic functions, static correlation functions, dynamic correlation functions, and "universality." The simple case of thermodynamic scaling (two independent variables) is first generalized to static and dynamic correlation functions (three and four variables), and then to scaling with a parameter (for which the critical subspace becomes higher dimensional). In this last case, where a second GHF hypothesis is made, the necessity of crossover lines is demonstrated. The assumption of homogeneity is clearly separated from any extra assumptions that may also be called scaling (or "strong scaling"), but are independent of and different from that of homogeneity. One practical insight gained from the present approach is that all experimentally measured exponents are expressible as the ratio of two scaling powers, af (which refers to the function) and aj (which refers to the path of approach to the critical point). A second practical advantage is that, since a GHF can be scaled with respect to any of its arguments, one can immediately write a variety of scaling functions for each type of scaling hypothesis. The GHF approach thereby permits data to be plotted in a variety of convenient fashions, and is found to facilitate computation of the relevant scaling functions (in particular, the GHF approach led directly to the recent calculation of the Heisenberg model scaling function by Milos\ifmmode \breve{}\else \u{}\fi{}evi\ifmmode \acute{c}\else '{c}\fi{} and Stanley).
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.
The goal of this work is to provide an analysis of spaces of critical points for multicomponent systems. First, we propose the geometric concept of order O for critical points; we distinguish it from a previous definition of a "multicritical" point. Specifically, we may define the intersection of spaces of critical points of order O to be a space of critical points of order (O+1). Ordinary critical points are defined to be of order O=2, so that the tricritical points introduced by Griffiths are of order O=3. We discuss more general examples of critical spaces of order O=3 which are known for a wide variety of systems; we also propose several examples of models of magnetic systems showing critical points of order O=4---i.e., systems having intersecting lines of tricritical points. The analysis of critical and coexistence spaces also provides a new form of the Gibbs phase rule suitable for complex magnetic models. Next we define---for the critical points of order O of which examples have been given---special directions in terms of which to make a scaling hypothesis. We give the hypothesis for simple systems and then for tricritical points, and then, in a subsequent paper, part II, the special directions are used to make a scaling hypothesis at spaces of critical points of any order. Certain predictions (e.g., scaling laws and "single-power" scaling functions) follow in a simple and straightforward fashion. We consider the scaling hypothesis at a critical space of order O in terms of a group of transformations. We can define a set of invariants of the group. It is possible, for $\mathcal{O}\ensuremath{\ge}3$, to make a second scaling hypothesis for the space of order $\mathcal{O}\ensuremath{-}1$ using certain of these invariants as independent variables. This is advantageous because certain "double-power" scaling functions then follow directly; these predict that for O=3, experimental data collapse from a volume onto a line. This prediction is to be contrasted with ordinary scaling functions, which predict that data collapse by only a single dimension (e.g., from a volume onto a surface or from a surface onto a line).
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium . The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis , generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.
We make the scaling hypothesis at a space of critical points of arbitrary order, using the geometric classification of critical points by order proposed in a preceding paper, part I. It follows from the definition of order that there are regions in which several scaling hypotheses are simultaneously valid, one at each space of critical points of order $\mathcal{O}\ensuremath{\ge}2$ which is connected to the point of highest order. The successive hypotheses are conveniently framed in terms of invariants of the groups of transformations about the points of higher order. This procedure facilitates the formation of multiple-power scaling functions. We suggest that the regions of validity of later hypotheses are bounded by hypersurfaces which scale and which are therefore most easily visualized in the spaces of invariants. A general critical point of order 4 is treated as a preliminary example. We then apply the hypothesis in detail to the critical point of order 4 in a specific example: the Ising model with variable interplanar interaction. This is done in two ways: the first shows specifically the relationship to the case of a metamagnet and to scaling with a parameter for change of lattice dimension; the second explicitly demonstrates that the connectivity between the various surfaces of ordinary (O=2) critical points is the same as for the edges of a tetrahedron. As a consequence of this scaling hypothesis, we can make predictions about the shapes of the coexistence and critical surfaces meeting at the critical point of order 4.