This book aims to elucidate the current status of research in phase transition dynamics. Because the topics treated are very wide, a unified phenomenological time-dependent Ginzburg–Landau approach is used, and applied to dynamics near the critical point. Into the simple Ginzburg–Landau theory for a certain order parameter, we introduce a new property or situation such as elasticity in solids, viscoelasticity in polymers, shear flow in fluids, or heat flow in He near the superfluid transition. By doing so, we encounter a rich class of problems on mesoscopic spatial scales. A merit of this approach is that we can understand such diverse problems in depth using universal concepts.

The first four chapters (Part one) deal with static situations, mainly of critical phenomena, and introduce some new results that would stand by themselves. However, the main purpose of Part one is to present the definitions of many fundamental quantities and introduce various phase transitions. So it should be read before Parts two and three which deal with dynamic situations. Chapter 5 is also introductory, reviewing fundamental dynamic theories, the scheme of Langevin equations and the linear response theory. Chapter 6 treats critical dynamics in (i) classical fluids near the gas–liquid and consolute critical points and (ii) He near the superfluid transition. Chapter 7 focuses on rather special problems in complex fluids: (i) effects of viscoelasticity on composition fluctuations in polymer systems; and (ii) volume phase transitions and heterogeneity effects in gels.

General aspects of static critical behavior [1]–[5] will be summarized using fractal concepts in Section 2.1. The mapping relations between the critical behavior of one- and two-component fluids and that of Ising systems will be discussed in Sections 2.2 and 2.3. They are useful in understanding a variety of thermodynamic experiments in fluids and will be the basis of the dynamical theories developed in Chapter 6. As another kind of critical behavior of xy symmetry, He near the superfluid transition will be treated in our scheme in Section 2.4. Gravity effects on the critical behavior in one-component fluids and He will also be discussed.

General aspects

First we provide the reader with snapshots of critical fluctuations whose characteristic features are strikingly similar in both Ising spin systems and fluids. Figure 2.1 shows a 128 × 128 spin configuration generated by a Monte Carlo simulation of a 2D Ising spin system in a disordered phase very close to the critical point. Figure 2.2 displays particle positions realized in a molecular dynamics simulation of a 2D one-component fluid in a one-phase state close to the gas–liquid critical point. In the latter simulation, the pair potential u(r) is of the Lenard-Jones form (1.2.1) cut off at r/σ = 2.5 and characterized by ∈ and σ. The temperature and average number density are T = 0.48 ∈ and n = 0.325σ–-2, respectively.

In the dynamics of one- and two-component fluids near the critical point and 4He and 3He–4He near the superfluid transition, the dynamic equations of the gross variables are nonlinear Langevin equations with reversible nonlinear mode coupling terms. These terms represent nonlinear dynamic interactions between the fluctuations, which cause critical divergence of the kinetic coefficients. We will give intuitive pictures of the physical processes leading to such enhancement of transport and review the mode coupling and dynamic renormalization group theories. New results are presented on various adiabatic processes including the piston effect and supercritical fluid hydrodynamics near the gas–liquid critical point and on nonequilibrium effects of heat flow near the superfluid transition.

Hydrodynamic interaction in near-critical fluids

In the dynamics of nearly incompressible binary fluid mixtures it is usual to take the concentration deviation δχ as the order parameter ψ. In one-component fluids it is convenient to take the entropy deviation δs (per unit mass) as ψ, because δs is decoupled from the sound mode in the hydrodynamic description. In these fluids, the dynamics of the order parameter is slowed down but the kinetic coefficients are enhanced near the critical point. These features originate from random convection of the critical fluctuations by the transverse velocity field fluctuations [1]–[7].

Intuitive picture of random convection

The order parameter undergoes diffusive relaxation resulting from convective motion due to the velocity field fluctuations.

Microstate and macrostate

In the formulation of statistical mechanics we are concerned with the theoretical prediction of thermodynamic properties of a physical system which contains a large number of particles such as electrons, atoms, and molecules by some statistical average processes performed over an appropriately prepared statistical sample. There are many different ways to prepare the sample. J. Willard Gibbs (1901) coined the name ensemble for the statistical sample, and three different ensembles are introduced, i.e., the microcanonical, canonical, and grand canonical ensembles.

We can choose for the physical system any thermodynamical system such as a single-component gas, liquid, or solid, as well as a mixture of many components, as long as the system is in the condition of thermodynamical equilibrium. In order to establish a fundamental principle of statistical mechanics, however, we naturally choose as simple a system as possible, such as the one-component dilute gas made up of structureless monatomic molecules. We then extend the fomulation step by step to more complicated systems. In this chapter, formulation of the three Gibbs ensembles will be developed.

The microcanonical ensemble is a collection of identical replicas of a given physical system which is a gas made up of noninteracting structureless particles. Firstly, the system is assumed to be contained in a box of volume V, the number of particles is equal to N, and the total energy is given in a narrow range between E and E + dE.

The thermodynamic system and processes

A physical system containing a large number of atoms or molecules is called the thermodynamic system if macroscopic properties, such as the temperature, pressure, mass density, heat capacity, etc., are the properties of main interest. The number of atoms or molecules contained, and hence the volume of the system, must be sufficiently large so that the conditions on the surfaces of the system do not affect the macroscopic properties significantly. From the theoretical point of view, the size of the system must be infinitely large, and the mathematical limit in which the volume, and proportionately the number of atoms or molecules, of the system are taken to infinity is often called the thermodynamic limit.

The thermodynamic process is a process in which some of the macroscopic properties of the system change in the course of time, such as the flow of matter or heat and/or the change in the volume of the system. It is stated that the system is in thermal equilibrium if there is no thermodynamic process going on in the system, even though there would always be microscopic molecular motions taking place. The system in thermal equilibrium must be uniform in density, temperature, and other macroscopic properties.

The zeroth law of thermodynamics

If two thermodynamic systems, A and B, each of which is in thermal equilibrium independently, are brought into thermal contact, one of two things will take place: either (1) a flow of heat from one system to the other or (2) no thermodynamic process will result.