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The Statistical Mechanics of Lattice Gases Volume I
By Barry Simon PRINCETON UNIVERSITY PRESS
Copyright © 1993 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08779-5
CHAPTER 1
Preliminaries
1.1 Models to be Discussed
Lattice models are caricatures invented to illuminate various aspects of elementary statistical mechanics, especially the phenomena of phase transitions and spontaneously broken symmetry. The simplest of all models is the Ising (or Lenz-Ising) model. This model was suggested to Ising by his thesis adviser, Lenz. Ising [1925] solved the one-dimensional model, an easy task (we will solve it three times: once in this section, once using transfer matrices in Section II.5, and once using high-temperature expansions in Section V.6), and on the basis of the fact that the one-dimensional model had no phase transition, he asserted there was no phase transition in any dimension. As we shall see, this is false. It is ironic that on the basis of an elementary calculation and erroneous conclusion, Ising's name has become among the most commonly mentioned in the theoretical physics literature. But history has had its revenge. Ising's name, which is correctly pronounced "E-zing," is almost universally mispronounced as "I-zing"!
To describe the Ising model, we pick an integer v and let Zv be the family of v-tuples of integers α = (α1 ···, αv) called sites. Fix l = 1,2,· ··. Let Λl = {[alpha | |αi| ≤ < l} so Λl has (2l + 1)v [equivalent to] |Λl| sites. At each site, we place a "spin," σα, which can take the values ±1 corresponding to "spin up" or "spin down." We imagine the spin represents a little magnet which can point in one of two directions. Thus, in finite volume, a configuration of one system corresponds to giving |Λl| numbers, each ±1; that is, there are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] distinct configurations (values of "σα"). Given a configuration [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we define its energy to be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)
In (1.1.1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so the sum is over nearest neighbors in Zv. The symbol <αγ> is a reminder that normally we have a convention that, in sums like (I.1.1), each pair is counted only once. Thus, for example, if Jαγ is a symmetric matrix vanishing on the diagonal
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In (I.1.1), J is a parameter which we most often take positive. J > 0 means that the more pairs of spins which point parallel (i.e., σασμ = 1), the lower the energy. Since we will see that low energy states have a higher weighting in statistical mechanics, this means that spins have a preferred possibility of pointing parallel, so in our magnetic picture, the system is ferromagnetic. Indeed, the Hamiltonian ([equivalent to] energy function or operator) (I.1.1) is often called the nearest-neighbor, spin-1/2 Ising ferromagnet; the meaning of the phrase "spin-1/2" will be made clear below.
According to the fundamental laws of statistical mechanics as laid down by our forefathers (Maxwell, Boltzmann, and Gibbs), if our system is placed in equilibrium with a heat bath at temperature, T, then configurations occur with probabilities, the probability of a configuration being proportional to exp(-βH(σα)) where β = (kT)-1 and k is Boltzmann's constant. Thus, the expected value, < f(σ) >, of some function, f, of the configuration is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.2)
the sum being over the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] possible configurations. Z is the normalization constant
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.3)
called the partition function. The set of weights e-βH/Z is called the Gibbs distribution; see the appendix for a discussion of a "justification" of the Gibbs choice.
The partition function is of importance because of its connection with some basic thermodynamic objects. It is quite natural to associate the expected value of H
< H(σα) > = - d(ln Z)/dβ (1.1.4)
with the internal energy of the system, in which case basic thermodynamics (see, e.g., Sommerfeld [1956], Pippard [1964]) says that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is the "free energy," a fundamental thermodynamic "potential."
To further discuss the physics of the Ising model, it is useful to introduce an additional parameter, h, representing an external magnetic field. Choosing units so that the magnetic moment of a spin is 1, we see that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Now Z,< >, F depend on Λl, β and h (we imagine fixing J) . The magnetization of the magnet is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1.5)
In general, F is a kind of generating function for truncated correlation functions.
An observed aspect of ferromagnets in nature is that (at least at low enough temperatures) they have memories; that is, if a magnetic field is turned on and then off, the system remains magnetized in the direction of the field. Thus, one would like to think that, for β sufficiently large,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.6)
but this is false. There are two related ways of seeing this. First, we can note that the finite-volume expectation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is continuous in β, h so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is left invariant under the map σα -> -σα (all α), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Alternatively, in finite-volume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is analytic in β, h and in particular it is C1 in h. Moreover, by an obvious symmetry, it is an even function of h; hence, its derivative at h = 0 is zero.
The reconciliation of the obvious analyticity of the Gibbs formalism and the observed discontinuities in nature (discontinuities which, we might add, come at such well-defined values of parameters, you can set your thermometers — indeed, we do set our thermometers—by them) is an interesting problem. The accepted solution is that true discontinuities in derivatives only occur in the limit of infinite Λ. The picture that results is that, in nature, where material is finite in size, a function like the density of water at fixed pressure and varied temperature is really a smooth function of T (or would be, if one waited for thermal equilibrium at each value of T). However, it is close to a discontinuous function; that is, the density jumps over such a short temperature interval (an interval whose size shrinks with increasing volume of our sample), that for all practical purposes, the density is discontinuous. Under such circumstances, it is obviously convenient to study objects in the idealization of infinite volume.
It should be mentioned that this notion of infinite-volume reconciling phase transitions and statistical mechanics took time to be accepted by the theoretical physics community. Indeed, it was only with Onsager's solution of the Ising model in 1944 in two dimensions and zero field that the notion was more or less universally accepted (Peierls's work in 1936 should have done the trick, but it wasn't widely appreciated at the time). In fact, at the van der Waals Centenary Conference in 1937, there was a spirited debate on whether phase transitions are consistent with the formalism of statistical mechanics. After the debate, a vote was taken on whether the infinite-volume limit could provide the answer. While the infinite-volume limit did win, it was a close vote! (See the discussion on pp. 432–33 in Pais's beautiful biography of Einstein [1982].)
Thus, a basic object for which one wants to prove existence and then study it is the free energy per unit volume,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1.7)
Chapter II proves the existence of this limit and some general features of the functional dependence of f. We note that independently of its interest for phase transitions, the existence of the limit (I.1.7) is of great physical significance. After all, if one needs to know the specific heat of iron, one looks it up in a table. One doesn't bother to find out the precise geometric configuration used by the experimenter in measuring this specific heat; rather, one takes three points on faith: That the experimenter was competent enough to take a big enough piece of material; that surfaces effects aren't important; and that for large pieces of material, the specific heat is proportional to the volume, that is, that limits like (I.1.6) exist! In finite volume, there are, in principle, fluctuations about the predictions of thermodynamics, so that one only expects thermodynamic arguments to be exact in this infinite-volume limit. For this reason, the infinite-volume limit is often called the thermodynamic limit.
It was a basic discovery of Gibbs (when, prior to his work on statistical mechanics, he worked on the thermodynamics of mixtures; see Wightman [1979] for a delightful discussion of the history) that thermodynamic functions are concave or convex in many basic parameters (convex functions are discussed in Section 1.3); in particular, f is a concave function of h for each fixed β. Such functions automatically have one-sided derivatives (see Prop. 1.3.1(f)), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists. The symmetry h ->-h now says f(β, -h) = f(β, h) so (D±hf) (β, -h) = -(D±h f) (β, -h). Given the formula in (1.1.5), it is natural to replace (I.1.6) by the conjecture
(D+h f)(β, A = 0) ≠ (D-h f)(β, h = 0). (1.1.8)
Such a discontinuity in the derivative of the fundamental thermodynamic function is called a first-order phase transition. If some higher-order derivative fails to exist or is discontinuous, we say there is a higher-order phase transition. Of course, it could happen that f fails to be C1 in one parameter and only fails to be C2, or even is analytic, in another. It is thus often useful to speak of a first-order phase transition in h (or some other parameter).
(I.1.5) suggests it should be useful to study infinite-volume limits of finite-volume Gibbs expectations. Here, a new phenomenon enters: The limit (1.1.7) always exists and the limit is the same if HΛ is modified by changing "boundary conditions." The limit of states might not exist (or more properly, we only know that it exists at special values of parameters, like high temperatures [small β], or under special restrictions on the interactions like ferromagnetism) and, in general, the limit may depend on boundary conditions. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is simpler than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it should be particularly useful to consider infinite-volume states invariant under translations (so < σα > is independent of a), so that (1.1.5) suggests that -β-1 [partial derivative]f/[partial derivative] = < σα > for such states. What the detailed analysis (see chapter III) shows is that the values of β < σα >, as one runs through all "suitable" infinite-volume translation invariant states, is precisely the set of numbers between -(D+h f)(h) and -(D-h f)(h). A similar relation holds for other functions than σα: for example, if we want to know about expectation values of σασα+δ for some fixed δ, we must look at the free energy per unit volume, f(β, η), associated to the Hamiltonian
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and look at derivatives with respect to η. Thus, one sees a basic fact: First-order phase transitions in some parameter are equivalent to the existence of more than one distinct translation invariant infinite-volume state.
If there are states at h = 0 with < σα > ≠ 0, we have states which do not have the symmetry σα -> -σα of the basic interaction (rather, σα -> -σα takes the state to another suitable infinite-volume limit). This illustrates the fundamental notion of spontaneous broken symmetry, a notion which (in a related context; see Section 1.2) is fundamental to all modern theories of elementary particle physics.
There is an alternative interpretation of the Ising model which associates a different physics to it: namely, consider a "gas" of particles. At each site, α, we either place a particle, in which case we set ρα = 1 or we don't, in which case we set ρα = 0. The particles attract one another in the sense that we gain energy, K > 0, if two particles are nearest neighbors, that is,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We do not wish to study this model in a picture where the particle number
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is fixed (or in which the density NΛ/Λ is fixed), but rather want to allow variable particle numbers. This means that one is supposed to work in the so-called grand canonical ensemble. Rather than the density, a direct physical quantity, one has a parameter, μ, the chemical potential. One forms the grand canonical partition function
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(Continues...)
Excerpted from The Statistical Mechanics of Lattice Gases Volume I by Barry Simon. Copyright © 1993 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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