In the previous chapter, we discussed the phase-separation kinetics or domain coarsening in the kinetic Ising model after a temperature quench from a homogeneous high-temperature phase to a two-phase low-temperature regime. Because of the complexity of the ensuing coarsening process, considerable effort has been devoted to developing continuum, and analytically more tractable, theories of coarsening. While a direct connection to individual spins is lost in such a continuum formulation, the continuum approach provides many new insights that are hard to obtain by a description at the level of individual spins.

Models

We tacitly assume that the order parameter is a scalar unless stated otherwise. We generally have in mind magnetic systems and will use the terminology of such systems; this usage reflects tradition rather than the dominant application of coarsening. There is a crucial distinction between non-conservative and conservative dynamics, and we begin by describing generic models for these two dynamics.

Non- conservative dynamics

The basic ingredients that underlie non-conservative dynamics are the following.

1. • The primary variable is a continuous coarse-grained order parameter m(x, t) ≡ l−dΣσ, the average magnetization in a block of linear dimension l that is centered at x, rather than a binary Ising variable σ = ±1. Here l should be much greater than the lattice spacing a and much smaller than the system size to give a smoothly varying coarse-grained magnetization on a scale greater than l. This coarse graining applies over a time range where the typical domain size is large compared to the lattice spacing.

2. […]

Non-equilibrium statistical physics courses usually begin with the Boltzmann equation and some of its most prominent consequences, particularly, the derivation of the Navier–Stokes equation of fluid mechanics and the determination of transport coefficients. Such derivations are daunting, often rely on uncontrolled approximations, and are treated in numerous standard texts. A basic understanding can already be gained by focusing on idealized collision processes whose underlying Boltzmann equations are sufficiently simple that they can be solved explicitly. These include the Lorentz gas, where a test particle interacts with a fixed scattering background, and Maxwell molecules, where the collision rate is independent of relative velocity. We also present applications of the Boltzmann equation approach to granular and traffic flows.

Kinetic theory

Non-equilibrium statistical physics originated in kinetic theory, which elucidated the dynamics of dilute gases and provided the starting point for treating more complex systems. Kinetic theory itself started with the Maxwell–Boltzmann velocity distribution, which was found before the Boltzmann equation – whose equilibrium solution is the Maxwell–Boltzmann distribution – had even been formulated.

The Maxwell–Boltzmann distribution

Let's start by deriving the Maxwell–Boltzmann velocity distribution for a classical gas of identical molecules that is in equilibrium at temperature T. Molecules scatter elastically when they are sufficiently close due to a short-range repulsive intermolecular potential. Let P(v)dv be the probability to find a molecule within a range dv about v.

Can statistical mechanics be used to describe phase transitions?

A phenomenological description of a phase transition does not raise any special difficulty a priori. For instance, to describe the solidification of a gas under pressure, one can make a simple theory for the gaseous phase, e.g., an ideal gas corrected by a few terms of the virial expansion. Then, for the solid, one can use the extraction energies of the atoms, and the vibration energies around equilibrium positions. These calculations will provide a thermodynamic potential for each phase. The line of coexistence between the two phases in the pressure—temperature plane will be determined by imposing the equality of the two chemical potentials μI (T, P) = μII (T, P).

If this method may turn out to be useful in practice, it does not answer any of the questions that one can raise concerning the transition between the two states. Indeed the interactions between the molecules are not statistical in nature: they are independent of the temperature, or of the pressure; the Hamiltonian is a combination of kinetic energy and well-defined interaction potentials between pairs of molecules. How can one see in such a description, following the principles established by Boltzmann, Gibbs and their successors, that at equilibrium the same molecules can form a solid or a fluid, a superconductor, a ferromagnet, etc., without any modification of the interactions?

These lecture notes do not attempt to cover the subject in its full extent. There are several excellent books that go much deeper into renormalization theory, or into the physical applications to critical phenomena and related topics. In writing these notes I did not mean either to cover the more recent and exciting aspects of the subject, such as quantum criticality, two-dimensional conformal invariance, disordered systems, condensed matter applications of the AdS/CFT duality borrowed from string theory, and so on.

A knowledge of the renormalization group and of field theory remains a necessary part of today's physics education. These notes are simply an introduction to the subject. They are based on actual lectures, which I gave at Sun Yat-sen University in Guangzhou in the fall of 2008. In order not to scare the students, I felt that a short text was a better introduction. There are even several parts that can be dropped by a hasty reader, such as GKS inequalities or high-temperature series. However, high-T series lead to an easy way of connecting geometrical criticality, such as self-avoiding walks and polymers or percolation to physics. I have chosen not to use Feynman diagrams; not that I think that they are unnecessary, I have used them for ever. But since I did not want to require a prior exposition to quantum field theory, I would have had to deal with a long detour, going through connected diagrams, one-particle irreducibility, and so on.

The initial formulation of the renormalization group appeared in the study of quantum electrodynamics (QED) in the high-energy limit where the momenta of the particles (electrons, positrons, photons) were much larger than the rest mass of the electron (Gell-Mann and Low, 1954). It is then natural to neglect the electron mass from the beginning (rather than calculating Feynman diagrams and taking their high-energy limit). However, the theory with massless electrons has to be handled with some care: the usual definition of the renormalized charge, normally defined as the electron—photon vertex in the limit of vanishing momenta, leads to an infrared divergence. Then the definition of the renormalized charge has to be modified as being the value of the same vertex but at some arbitrary non-zero momenta. This introduces a new length scale to the theory (the inverse of an arbitrary momentum in units ħ = c = 1). Physical quantities should not depend on this arbitrariness, and different charges of the electron, related to different scales of definition, lead to the same physics.

It was only in the beginning of the 1970s that the study of dilatation invariance in field theory, extended by K. Wilson to the vicinity of the critical point and similar physical problems, threw light on the true meaning of the renormalization group.