Chapter 1: Introductionscaling, power laws, and why a smooth sum can turn singular

1.1Scaling and dimensional analysis

We say that two measurable quantities obey scaling when they depend on each other in a power-law fashion, \(y \propto x^{a}\). A familiar example is Kepler's third law,

relating the period \(T\) of a planet to the radius \(R\) of its circular orbit.

A second example: the phase speed \(c\) of waves on shallow water of depth \(h\). Neglecting surface tension and viscous effects, \(c^2 = gh\), so that \(c \propto \sqrt{h}\). This is the scaling law for this example, and it is only valid when \(h \ll \lambda\), with \(\lambda\) the wavelength. The more general dispersion relation is

and for \(h \ll \lambda\) the Taylor expansion \(\tanh(2\pi h/\lambda) \approx 2\pi h/\lambda\) gives back \(c^2 \approx gh\).

Scaling from dimensional analysis

Scaling laws can often be deduced from dimensional analysis alone. Writing \([\,\cdot\,]\) for "the physical dimension of",

The fluid density \(\rho\) is the only quantity carrying mass, so it cannot appear. The most general consistent form is

where \(f(x)\) is a dimensionless function that dimensional analysis cannot determine. In the limit \(h/\lambda \to 0\), \(c \sim (gh)^{1/2} f(0) \propto \sqrt{h}\): we recover the scaling law without any detailed knowledge of fluid dynamics.

This step quietly assumes the limit is regular, i.e., that \(f(0)\) exists and is finite and nonzero. The cases where this regularity assumption breaks down, where the limit is singular, are the central topic of this course.

The choice of prefactor is an arbitrary convention. We could just as well have written \(c = (g\lambda)^{1/2}\tilde f(h/\lambda)\). Then in the limit \(h/\lambda \to 0\) the only way \(\lambda\) can cancel out is if \(\tilde f\) has a square-root behavior for small arguments:

with \(f\) analytic as \(x \to 0\), so that \(f(0)\) is well defined.

1.2Power laws in statistical physics

In statistical physics we meet power laws \(y = Cx^{\alpha}\) whose exponent \(\alpha\) need not be a simple fraction; it can even be irrational. A broad class of phenomena with power-law behavior is phase transitions where there is a critical point.

The liquid–gas critical point

Above the critical temperature \(T_c\) one can pass continuously from a gas to a liquid at constant temperature; there is no real way to distinguish the two. Below \(T_c\), as the density is increased at fixed temperature, it is not possible to go from the gaseous phase to the liquid phase without passing through a two-phase (coexistence) regime. Near the top of the coexistence curve, its shape is a power law. For sulfur hexafluoride (SF\(_6\)),

The exponent \(\beta\) is a critical exponent. Remarkably, it does not depend on the particular fluid: this is our first glimpse of universality.

The magnetic critical point

A ferromagnet can be regarded as a set of magnetic dipoles, or spins, residing on the vertices of a crystal lattice. The spins exchange energy through interactions among themselves, as well as with other degrees of freedom of the crystal lattice, e.g., via spin–orbit coupling. For systems in equilibrium one can define a temperature \(T\): if one waits a sufficiently long time, equilibrium is established between the lattice and the spins, and both sets of degrees of freedom are described by the same \(T\).

• At high temperatures and zero external field the system is in the paramagnetic phase: spins point in all directions with equal frequency, and the average net magnetization is zero.
• Below \(T_c\) and in zero external field, the spins tend to align along a particular direction in space even in the absence of an external field. There is a net magnetization \(M(T) \neq 0\), and the system is in the ferromagnetic phase. Which direction is chosen is a nontrivial question we return to later.

The magnetization is the order parameter: it rises continuously from zero as \(T\) is lowered through \(T_c\), so this is a continuous phase transition.

In an Ising ferromagnet each spin can only point up or down along one axis (the \(z\) axis): simpler, but not that simple. Parallel spins (\(\uparrow\uparrow\)) have lower energy and antiparallel spins (\(\uparrow\downarrow\)) have higher energy. In an Ising antiferromagnet the antiparallel arrangement has lower energy; on certain crystal lattices and in zero applied field its thermodynamics is identical to that of the ferromagnet. For the three-dimensional Ising antiferromagnet DyAlO\(_3\),

The superfluid (λ) transition in \(^4\)He

For pressures from near zero to about 25 atmospheres, liquid \(^4\)He undergoes a continuous transition to a superfluid at a temperature of about 2 K. The superfluid shows dissipationless flow through fine capillaries, among other unusual properties. The transition is called the λ-transition after the shape of the heat-capacity curve, and its temperature \(T_\lambda\) depends on pressure. Near \(T_\lambda\) the heat capacity has a cusp,

a very small critical exponent. Because \(x^{n} = e^{n\ln x} \approx 1 + n\ln x\) for \(|n| \ll 1\) (and \(x\) not too small), such a power law is nearly indistinguishable from a logarithm,

The self-avoiding random walk

For an ordinary random walk, the root-mean-square distance \(R\) traveled after \(N\) steps obeys \(R \propto \sqrt N\). This is diffusion, \(\partial\rho/\partial t \propto \nabla^2\rho\), in disguise: space squared is proportional to time.

A self-avoiding walk (SAW) is not permitted to intersect itself. It is a minimal model of a polymer chain in solution, and here

Since \(\nu > 1/2\), the SAW naturally goes farther than an ordinary walk. The exponent is completely independent of the chemistry and depends only on the "spaghetti" nature of the polymer. Renormalization-group calculations give \(\nu = 0.5880 \pm 0.0015\).

Dynamic critical phenomena

Nontrivial power laws can also appear in transport coefficients of a system near a critical point. A transport coefficient is a phenomenological parameter relating a current to a driving force. Ohm's law \(\vec J = \overleftrightarrow{\sigma}\,\vec E\) relates the current density to the electric field through the conductivity tensor; other examples are diffusion coefficients and the viscosity. For shear flow between plates of area \(A\),

and the shear viscosity \(\eta_s\) is found to diverge weakly near the critical point of a binary fluid (nitroethane–3-methylpentane):

1.3Some important questions

This course is primarily concerned with phase transitions where no latent heat is generated. For the most part we omit first-order transitions in the Ehrenfest classification, in which a transition is called \(n\)th order if some \(n\)th derivative of the free energy with respect to one of its arguments is discontinuous at the transition.

The phenomena we describe, critical phenomena, occur at what Ehrenfest would call second-order transitions. We prefer the term continuous phase transition. Ehrenfest's scheme fails because, when it was formulated, it was not known that quantities such as the specific heat actually diverge at continuous transitions instead of simply jumping. We will see that this failure is tied to the failure of mean-field theory, exemplified by the Weiss theory of ferromagnetism and the van der Waals equation for fluids.

Many different observable quantities exhibit scaling behavior and have associated critical exponents. They fall into two categories: (1) thermodynamic variables, such as the specific heat, and (2) spatial ordering, e.g., how the local magnetic moment varies from point to point.

In equilibrium statistical mechanics, for a system in contact with a large reservoir at temperature \(T\), thermodynamics follows from the Helmholtz free energy \(F\) and its derivatives, given by Gibbs' formula

where \(\mathcal H\) is the Hamiltonian, the trace is the sum over all degrees of freedom appearing in \(\mathcal H\), and \(Z\) is the partition function. Since \(\mathcal H\) is usually a non-singular function of the degrees of freedom, \(Z\) is a sum of terms \(e^{-\mathcal H/k_BT}\), each an analytic function of the parameters in the Hamiltonian. How, then, can such a sum give rise to the non-analytic behavior described above? Is it even clear a priori that a single partition function can describe several phases?

The answer, developed in Chapter 2, is the thermodynamic limit: a finite sum of analytic terms is analytic, but the limit \(N \to \infty\) need not be. The demo below makes this concrete with a model whose partition function can be summed exactly for any \(N\).

The external parameters enter \(F\) through \(\mathcal H\), as well as through \(T\), via \(e^{-F/k_BT} = \operatorname{Tr}\, e^{-\mathcal H/k_BT}\).

Mean-field theories always lead to exponents given by rational fractions, but the observed values do not agree with these predictions:

Resolving this discrepancy produced a new way of looking at physics: K. G. Wilson's renormalization group (RG), recognized by the 1982 Nobel Prize in Physics.

The critical exponents are often independent of the specific system. The exponent \(\beta\) of the liquid–gas critical point and the exponent \(\beta\) of the ferromagnetic critical point are the same within experimental accuracy, as the log–log plot in the coexistence-curve demo shows. That two apparently different physical systems can share precisely the same set of critical exponents is called universality, and phenomena with the same set of exponents form a universality class. In general, members of a universality class have only three things in common:

1. the symmetry group of the Hamiltonian (not of the lattice, if one is present);
2. the dimensionality of space;
3. whether or not the forces are short ranged.

1.4Historical development

We will follow the historical development of the subject approximately, but start from a precise statistical-mechanical definition of a phase transition. That definition emphasizes the thermodynamic limit, in which the system size is taken to be infinitely large; strictly speaking, phase transitions arise only in this limit.

Two physical systems anchor the classical theory: the liquid–gas system, described by van der Waals, and the ferromagnet, described by Weiss. Their mathematical descriptions become identical near a critical point, and this similarity forms the basis of Landau's theory of phase transitions, the most succinct statement of the classical approach. Landau theory is a mean-field theory: a variable such as the magnetization is replaced by its average value, and fluctuations about that value are ignored. Near a critical point fluctuations are not negligible, so the theory is not self-consistent there.

Widom scaling

The notion of scaling laws was first reached on a phenomenological basis by B. Widom. For a magnet, the equation of state relates the magnetization \(M\), the temperature, and the external field \(H\): \(H = f(M,t)\) with \(t = (T-T_c)/T_c\), a function of two variables. Widom found that near a critical point it can be written as

a function of a single variable. This implies relations among the exponents of different thermodynamic quantities (specific heat, susceptibility, magnetization). His striking discovery was that \(\delta\), \(\beta\), and \(T_c\) can be chosen so that experimental data from different materials all satisfy the equation with the same function \(\Phi\).

Kadanoff, Wilson, and beyond

Where do scaling laws come from? L. P. Kadanoff proposed that Widom's form follows if, near a critical point, the system looks the same on all length scales. The argument is not quite correct, but it supplies the physical insight on which the renormalization group is built. Starting in 1971, Wilson developed the RG in both condensed matter and high-energy physics; many applications used perturbation expansions in small parameters such as \(\epsilon \equiv 4-d\), with \(d\) the dimension of space, or \(1/n\), with \(n\) the number of components of the order parameter.

1. 1873van der Waals: equation of state for fluids, with a liquid–gas critical point.
2. 1907Weiss: molecular-field theory of ferromagnetism.
3. 1937Landau: a unified mean-field theory of continuous transitions built on the order parameter.
4. 1965Widom: scaling form of the equation of state.
5. 1966Kadanoff: block-spin picture of self-similarity near \(T_c\).
6. 1971Wilson: the renormalization group; \(\epsilon\)-expansion follows in 1972 with Fisher.
7. 1982Nobel Prize in Physics to Wilson.

The RG reaches beyond equilibrium. Goldenfeld's preface describes realizing, in the winter of 1983–84, that certain non-equilibrium phenomena can be analyzed with the RG even though there is no partition function to compute. Examples include phase separation in a binary alloy below its critical point, where the X-ray scattering intensity obeys dynamical scaling,

and partial differential equations with similarity solutions \(u(x,t) \sim t^{\alpha} f(xt^{\beta})\) as \(t\to\infty\), whose exponents sometimes cannot be found by dimensional analysis: precisely the singular case flagged in §1.1. These return in Chapter 10.

Further reading

• N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992), Chapter 1.
• G. I. Barenblatt, Scaling (Cambridge University Press, 2003): dimensional analysis, self-similarity, and when it fails.
• D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 2000): transport coefficients and viscosity at an undergraduate level.