Two classical subjects in statistical physics are kinetic theory and phase transitions. The latter has been traditionally studied in the equilibrium framework, where the goal is to characterize the ordered phases that arise below the critical temperature in systems with short-range interactions. A more recent development has been to investigate how this order is formed dynamically. In the present and following chapter, we focus on this dynamics.

Phenomenology of coarsening

The theory of phase transitions was originally motivated by the goal of understanding ferromagnetism. The Ising model played a central role in this effort and provided a useful framework for investigating dynamics. The basic entity in the Ising model is a spin variable that can take two possible values, s = ±1, at each site of a lattice. A local ferromagnetic interaction between spins promotes their alignment, while thermal noise tends to randomize their orientations. The outcome of this competition is a disordered state for sufficiently high temperature, while below a critical temperature the tendency for alignment prevails and an ordered state arises in which the order parameter – the average magnetization – is non-zero.

Suppose that we start with an Ising model in an equilibrium disordered phase and lower the temperature. To understand how the system evolves toward the final state, we must endow this model with a dynamics and we also need to specify the quenching procedure. Quenching is usually implemented as follows:

1. • Start at a high initial temperature Ti > Tc, where spins are disordered; here Tc is the critical temperature.

2. • Instantaneously cool the system to a lower temperature Tf.

Suppose that gas molecules impinge upon and adsorb on a surface, or substrate. If the incident molecules are monomers that permanently attach to single adsorption sites on the surface and there are no interactions between adsorbed monomers, then the density ρ of occupied sites increases with time at a rate proportional to the density of vacancies, namely, dρ/dt = 1 − ρ. Thus ρ(t) = 1 − e−t, and vacancies disappear exponentially in time. However, if each arriving molecule covers more than one site on the substrate, then a vacant region that is smaller than the molecular size can never be filled. The substrate reaches an incompletely filled jammed state that cannot accommodate additional adsorption. What is the filling fraction of this jammed state? What is the rate at which the final fraction is reached? These are basic questions of adsorption kinetics.

Random sequential adsorption in one dimension

A simple example with non-trivial collective behavior is the random sequential adsorption of dimers – molecules that occupy two adjacent sites of a one-dimensional lattice (Fig. 7.1). We model the steady influx of molecules by adsorption attempts that occur one at a time at random locations on the substrate. An adsorption attempt is successful only if a dimer lands on two adjacent empty sites. If a dimer lands on either two occupied sites or on one occupied and one empty site, the attempt fails. After each successful attempt, the coverage increases.

The goal of statistical physics is to study collective behaviors of interacting many-particle systems. In equilibrium statistical physics, the simplest interaction is exclusion – for example, hard spheres that cannot overlap. This model depends on a single dimensionless parameter, the volume fraction; the temperature is irrelevant since the interaction energy is zero when the spheres are non-overlapping and infinite otherwise. Despite its apparent simplicity, the hard-sphere gas is incompletely understood except in one dimension. A similar state of affairs holds for the lattice version of hard spheres; there is little analytical understanding of its unusual liquid–gas transition when the spatial dimension d ≥ 2.

In this chapter we explore the role of exclusion on the simplest non-equilibrium models that are known as exclusion processes. Here particles occupy single lattice sites, and each particle can hop to a neighboring site only if it is vacant (see Fig. 4.1). There are many basic questions we can ask: What is the displacement of a single particle? How does the density affect transport properties? How do density gradients evolve with time? In greater than one dimension, exclusion does not qualitatively affect transport properties compared to a system of independent particles. Interestingly, exclusion leads to fundamentally new transport phenomena in one dimension.

Statistical physics is an unusual branch of science. It is not defined by a specific subject per se, but rather by ideas and tools that work for an incredibly wide range of problems. Statistical physics is concerned with interacting systems that consist of a huge number of building blocks – particles, spins, agents, etc. The local interactions between these elements lead to emergent behaviors that can often be simple and clean, while the corresponding few-particle systems can exhibit bewildering properties that defy classification. From a statistical perspective, the large size of a system often plays an advantageous, not deleterious, role in leading to simple collective properties.

While the tools of equilibrium statistical physics are well-developed, the statistical description of systems that are out of equilibrium is less mature. In spite of more than a century of effort to develop a formalism for non-equilibrium phenomena, there still do not exist analogs of the canonical Boltzmann factor or the partition function of equilibrium statistical physics. Moreover, non-equilibrium statistical physics has traditionally dealt with small deviations from equilibrium. Our focus is on systems far from equilibrium, where conceptually simple and explicit results can be derived for their dynamical evolution.

Non-equilibrium statistical physics is perhaps best appreciated by presenting wide-ranging and appealing examples, and by developing an array of techniques to solve these systems. We have attempted to make our treatment self-contained, so that an interested reader can follow the text with a minimum of unresolved methodological mysteries or hidden calculational pitfalls.

Non-equilibrium statistical physics describes the time evolution of many-particle systems. The individual particles are elemental interacting entities which, in some situations, can change in the process of interaction. In the most interesting cases, interactions between particles are strong and hence a deterministic description of even a few-particle system is beyond the reach of any exact theoretical approach. On the other hand, many-particle systems often admit an analytical statistical description when their number becomes large. In that sense they are simpler than few-particle systems. This feature has several different names – the law of large numbers, ergodicity, etc. – and it is one of the reasons for the spectacular successes of statistical physics and probability theory.

Non-equilibrium statistical physics is also quite different from other branches of physics, such as the “fundamental” fields of electrodynamics, gravity, and high-energy physics that involve a reductionist description of few-particle systems, as well as applied fields, such as hydrodynamics and elasticity, that are primarily concerned with the consequences of fundamental governing equations. Some of the key and distinguishing features of non-equilibrium statistical physics include the following:

1. • there are no basic equations (like Maxwell equations in electrodynamics or Navier–Stokes equations in hydrodynamics) from which the rest follows;

2. • it is intermediate between fundamental and applied physics;

3. • common underlying techniques and concepts exist in spite of the wide diversity of the field;

4. • it naturally leads to the creation of methods that are useful in applications outside of physics (for example the Monte Carlo method and simulated annealing).

The previous chapter focused on population dynamics models, where the reactants can be viewed as perfectly mixed and the kinetics is characterized only by global densities. In this chapter, we study diffusion-controlled reactions, in which molecular diffusion limits the rate at which reactants encounter each other. In this situation, spatial gradients and spatial fluctuations play an essential role in governing the kinetics. As we shall see in this chapter, the spatial dimension plays a crucial role in determining the importance of these heterogeneities.

Role of the spatial dimension

When the spatial dimension d exceeds a critical dimension dc, diffusing molecules tend to remain well mixed. This efficient mixing stems from the transience of diffusion in high spatial dimension, which means that a molecule is almost as likely to react with a distant neighbor as with a near neighbor. Because of this efficient mixing, spatial fluctuations play a negligible role, ultimately leading to mean-field kinetics. Conversely, when d < dc, nearby particles react with high probability. This locality causes large-scale heterogeneities to develop, even when the initial state is homogeneous, that invalidate a mean-field description of the kinetics.

To illustrate the role of the spatial dimension in a simple setting, consider the evolution of a gas of identical diffusing particles that undergo either irreversible annihilation or coalescence (see also Section 1.2). Suppose that each particle has radius R and diffusivity D.

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.