General discussion of stochastic models

In the hydrodynamic theories, one writes equations which describe the system at hand on long time and length scales, using local conservation laws and the presence of broken symmetries to determine the identity of these slowly varying quantities. However, one would like a way to take account of the effects of the more rapidly varying degrees of freedom in such theories. Implicitly, these more rapid degrees of freedom are present in the integrals which determine the transport coefficients in the hydrodynamic equations by way of Kubo relations, but no method is presented to calculate the required integrands. One way to take account of the other degrees of freedom is to take on the entire many body problem all the way down to the atomic or electronic level, as one does in molecular dynamics simulations of various sorts. However, it is useful to have some approximate analytical ways to attack this problem as well. Here we review the basis for the most common such approach, the Langevin equation. For most of this discussion, we will assume that the identity of the slow variable or variables is known and that the separation of time scales is extreme so that the faster variables are essentially instantaneous in a sense we will discuss. These assumptions are not often particularly well justified. However, the resulting formulation has yielded very useful insights and is an important part of the subject.

Here we briefly review some aspects of the statistical mechanics of liquids. The distinction between a liquid and a gas is not sharp except in the neighborhood of the transition between them which we will discuss in Chapter 10. As a working definition we will consider a system to be a liquid if it lacks the geometrical structure associated, for example, with crystals and for which the density expansions discussed in the last chapter do not converge. This distinction can be made somewhat sharper when we have discussed the relevant correlation functions and phase diagrams. It is to be noted that for atomic and molecular systems which can be treated classically and for which the two body interactions contain a hard core at short distances, there will always be a region of the thermodynamic phase space (for example in the PT diagram) for which the system will behave as a liquid according to this definition.

We begin the discussion by defining correlation functions which are very useful for characterizing the structure of liquids and also for making measurements and formulating theories to describe them. The considerations here apply equally well to the imperfect gases discussed in the last chapter, but they are particularly useful and necessary for the discussion of liquids. We next describe experimental techniques which directly measure some of these correlation functions. Finally we briefly describe two distinct theoretical approaches to the description of liquids: analytical formulations based on approximate summations of series like those described in the last chapter, and numerical simulation.

Historically, the first and most successful case in which statistical mechanics has made the connection between microscopic and macroscopic description is that in which the system can be said to be in equilibrium. We define this carefully later but, to proceed, may think of the equilibrium state as the one in which the values of the macroscopic variables do not drift in time. The macroscopic variables may have an obvious relation to the underlying microscopic description (as for example in the case of the volume of the system) or a more subtle relationship (as for temperature and entropy). The macroscopic variables of a system in equilibrium are found experimentally (and in simulations) to obey historically empirical laws of thermodynamics and equations of state which relate them to one another. For systems at or near equilibrium, statistical mechanics provides the means of relating these relationships to the underlying microscopic physical description.

We begin by discussing the details of this relation between the microscopic and macroscopic physical description in the case in which the system may be described classically. Later we run over the same ground in the quantum mechanical case. Finally we discuss how thermodynamics emerges from the description and how the classical description emerges from the quantum mechanical one in the appropriate limit.

The problems of statistical mechanics are those which involve systems with a larger number of degrees of freedom than we can conveniently follow explicitly in experiment, theory or simulation. The number of degrees of freedom which can be followed explicitly in simulations has been changing very rapidly as computers and algorithms improve. However, it is important to note that, even if computers continue to improve at their present rate, characterized by Moore's “law,” scientists will not be able to use them for a very long time to predict many properties of nature by direct simulation of the fundamental microscopic laws of physics. This point is important enough to emphasize.

Suppose that, T years from the present, a calculation requiring computation time t0 at present will require computation time t(T) = t02−T/2 (Moore's “law,” see Figure 1). Currently, state of the art numerical solutions of the Schrödinger equation for a few hundred atoms can be carried out fast enough so that the motion of these atoms can be followed long enough to obtain thermodynamic properties. This is adequate if one wishes to predict properties of simple homogeneous gases, liquids or solids from first principles (as we will be discussing later). However, for many problems of current interest, one is interested in entities in which many more atoms need to be studied in order to obtain predictions of properties at the macroscopic level of a centimeter or more.

Here we introduce interactions between particles, beginning with the classical case. In practice we will call a system an imperfect gas when it is sufficiently dilute so that an expansion of the pressure in a power series in the density converges reasonably quickly. This series is called the virial series and we will introduce it in this chapter. This definition of an imperfect gas thus can depend on the temperature. If the power series in the density does not converge we may refer loosely to the system as a liquid, as long as it does not exhibit long range order characteristic of various solids and liquid crystals. The experimental distinction between a gas and a liquid will be discussed more precisely in Chapter 10.

We will develop the virial series for a classical gas in two different, but equivalent, ways here. In the first method we develop a series for the partition function Z using the grand canonical distribution. By making a partial summation of this series we get a series in the fugacity. In the second method we study a series for the free energy F = −kBT ln Z and use the canonical ensemble. Though the two methods are equivalent, we discuss them both in order to provide an opportunity to introduce several concepts common in the statistical mechanical literature.

The classical virial series will clarify more precisely than we were able to do in the last two chapters the conditions under which a gas can be treated as perfect or ideal.

We begin with some thermodynamic considerations and then proceed to a discussion of critical phenomena. In discussing critical phenomena we first describe the phenomenology, then some general considerations concerning Landau–Ginzburg free energy functionals and mean field theory and finally an introduction to the renormalization group.

Thermodynamic considerations

Consider a system at fixed pressure P and temperature T. (We will not be concerned with magnetic properties yet.) We will suppose that the system contains some integer number s of molecular species. We will also suppose that we have a means of distinguishing two or more phases of this system. Though this is an assumption it requires some examples and discussion. We distinguish between phases by consideration of their macroscopic properties so a spatial as well as a temporal average is involved. For example, we distinguish gas from liquid by the difference in average density and magnet from paramagnet by the existence of a finite average magnetization in the former. (We will discuss some more subtle cases later.) But if we wish to consider (as we do here) the possible coexistence of more than one phase then a problem arises concerning the length scale over which we ought to average spatially. If, in a system containing two coexisting phases, we find the average properties by averaging over the entire system, then we will always get just one number and not two and will have no means of distinguishing the phases.

This book is based on a course which I have taught over many years to graduate students in several physics departments. Students have been mainly candidates for physics degrees but have included a scattering of people from other departments including chemical engineering, materials science and chemistry. I take a “reductionist” view, that implicitly assumes that the basic program of physics of complex systems is to connect observed phenomena to fundamental physical laws as represented at the molecular level by Newtonian mechanics or quantum mechanics. While this program has historically motivated workers in statistical physics for more than a century, it is no longer universally regarded as central by all distinguished users of statistical mechanics some of whom emphasize the phenomenological role of statistical methods in organizing data at macroscopic length and time scales with only qualitative, and often only passing, reference to the underlying microscopic physics. While some very useful methods and insights have resulted from such approaches, they generally tend to have little quantitative predictive power. Further, the recent advances in first principles quantum mechanical methods have put the program of predictive quantitative methods based on first principles within reach for a broader range of systems. Thus a text which emphasizes connections to these first principles can be useful.

The level here is similar to that of popular books such as those by Landau and Lifshitz, Huang and Reichl. The aim is to provide a basic understanding of the fundamentals and some pivotal applications in the brief space of a year.

General discussion

In general, by a hydrodynamic description of a many body fluid we mean a description valid at long wavelengths and low frequencies and which is based on closure of the local conservation laws of the fluid by use of a linear relation between fluxes and the gradients of densities. The coefficients of the linear relation are transport coefficients and they are phenomenological parameters of the hydrodynamic theory, calculable in principle from a theory describing the system at shorter length and time scales. The resulting hydrodynamic theory is generally a set of nonlinear partial differential equations of which the Navier–Stokes equations for the hydrodynamics of a simple fluid are a familiar example.

The reason that hydrodynamic theories accurately describe slow motions on large length scales is that global conservation laws link long distances to long times. Physically, for example, conservation of mass results in a diffusion equation in which the distance which particles diffuse increases with the square root of the time (see Problem 10.1). Although this link guarantees that some of the slow variables of the system are described by the hydrodynamic equations, it does not ensure that all of the slow variables can be so described. Near critical points associated with second order phase transitions, there are very slow changes in the fluid which are not described by the conservation laws of hydrodynamics, but which arise because of the very slow development and decay of large, almost stable regions looking like one of the (two or more) phases between which the system is slowly fluctuating.

As discussed in Chapter 6, when the temperature is lowered in a classical liquid until the thermal wavelength becomes comparable to the interparticle spacing, then the semiclassical approximation is no longer adequate and quantum effects must be considered. In practice, most classical liquids freeze at all positive pressures before this temperature is reached. The exceptions are the helium liquids (3He and 4He) for which the quantum effects are large enough to prevent freezing as the temperature is lowered while the liquid is kept in equilibrium with its vapor. Conveniently, 3He is a Fermi system and 4He is a Bose system. In these systems as well, phase transitions occur at low enough temperatures. But quantum effects are significant even before these transitions occur. Another system which may for some purposes be regarded as an isotropic liquid with large quantum effects is the collection of electrons in (at least some) metals. Here too, a phase transition to the superconducting state intervenes in many cases at low enough temperatures. Finally neutron stars may contain regions in which neutrons are in a liquid state with large quantum effects and white dwarf stars contain a degenerate electron gas which can be regarded as a quantum liquid. In general, the reason that quantum liquids are so hard to observe is that interactions tend to result in symmetry breaking phase transitions in high density systems at temperatures low enough to permit quantum effects to be observed.

For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates {qi}. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement pi → (ħ/i)(∂/∂qi). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ(∂Ψ/∂t) = HΨ which requires as initial condition the knowledge of the wave function Ψ({qi}, t) at some initial time t0. (Boundary conditions on Ψ({qi}, t) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψν(q), is a basis for the Hilbert space.