Phase Transitions and Critical Phenomena: An Essay in Natural Philosophy (Thales to Onsager) Prof. David A. Edwards Department of Mathematics University of Georgia Athens, Georgia 30602 http://www.math.uga.edu/~davide/ http://davidaedwards.tumblr.com/ [email protected] §1. Introduction. In this essay we will present a detailed analysis of the concepts and mathematics that underlie statistical mechanics. For a similar discussion of classical and quantum mechanics the reader is referred to [E-1] and [E-4]. For a similar discussion of quantum field theory the reader is referred to [E-2]. For a discussion of quantum geometrodynamics the reader is referred to [E- 3] Scientific theories go through many stages in their development, some eventually reaching the stage at which one might say that "the only work left to be done is the computing of the next decimal." We shall call such theories climax theories in analogy with the notion of a climax forest (other analogies are also appropriate). A climax theory has two parts: one theoretical, which has become part of pure mathematics; and the other empirical, which somehow relates the mathematics to experience. The earliest example of a climax theory is Euclidean geometry. That such a development of geometry is even possible is not obvious. One can easily imagine an Egyptian geometer explaining to a Babylonian number theorist why geometry could never become a precise science like number theory because pyramids and other such bodies are intrinsically irregular and fuzzy (similar discussions occur often today between biologists and physicists). Archimedes' fundamental work showing that the four fundamental constants related to circles and spheres (C =αr, A = βr2 , S = ϒr 2, V= δr3 ) are all simply related (1/2α=β = 1/4ϒ= 3/4δ= π), together with his estimate that 3+10/71< π < 3+1/7 will serve for us as the paradigm of what science should be all about. Namely, rigorous derivation from a well defined theory yielding results that are empirically satisfactory. Gauss took geometry a step further by systematically developing non-Euclidean geometries and by attempting to determine which of these was empirically most adequate by measuring the sum of the angles of a triangle formed by three mountain tops. A further step was taken by Einstein who developed a four- dimensional Pseudo-Riemannian geometry of space-time and showed that it provided a better organizing principle of experience than the previous geometries. Thus, the only legitimate philosophical problems that we recognize in a mature scientific theory are the same ones that occur in geometry. Of course, raising a theory to the level at which it attains a Euclidean form requires a great effort. For example, while the Greeks (e.g., Aristotle) were very conscious of the importance of motion, they were only able to mathematically handle celestial mechanics (Ptolemy), and a full mathematical theory of dynamics was only successfully attained with Newton. Newton describes his program in the preface to his opus the Mathematical Principles of Natural Philosophy: ...therefore I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this - from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena; and to this end the general propositions in the first and second books are directed. In the third book I give an example of this in the explication of the System of the World; for by the propositions mathematically demonstrated in the former books in the third I derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces by other propositions which are also mathematical, I deduce the motions of the planets, the comets, the moon, and the sea. I wish we could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another. These forces being unknown, philosophers have hitherto attempted the search of Nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. [S] Newton clearly had a mighty vision. We will be surveying some of the progress made since Newton in deriving “the rest of the phenomena of Nature”. §2. Statistical Mechanics. §2.1 The Reductionist Program. At least from the time of the Greeks, it has been a dream of natural philosophers to establish a single model from which all of the phenomena of nature could be rigorously derived. One of the most enticing of such models was atomism. Under this hypothesis, a ground zero description of the world consists of atoms in motion in a void. All other phenomena are to be under stood as collective properties of the atoms. As originally understood, the atomic theory was a realistic theory - the atoms were real things independent of man. We will take a more idealistic approach. Namely, for us an atomic theory will entail the assumption of certain modes of perceiving. The rest of perception should then follow in a manner analogous to the way a long distance view of a Seurat painting follows from a close-up view. The point of atomic theories is to be able to derive the complex phenomena of appearances from simple initial assumptions. For example, one goal of atomic theory would be to derive the temperatures at which water freezes and boils. There were a number of atomic theories in antiquity, for example, those due to Leucippus, Democritus, Plato, Epicurus,-and Lucretius. These theories postulated a space which was either a passive void or an active plenum together with various fundamental particles. Plato's doctrine of geometrical atomism was certainly the most sophisticated. Consider the following passage from the Timaeus [S]. In the first place, then, it is of course obvious to anyone that fire, earth, water, and air are bodies; and all body has depth. Depth, moreover, must be bounded by surface; and every surface that is rectilinear is composed of triangles. Now all triangles are derived from two, each having one right angle and the other angles acute. Of these triangles, one has on either side the half of a right angle, the division of which is determined by equal sides (the right-angled isosceles); the other has unequal parts of a right angle allotted to unequal sides (the right- angled scalene). This we assume as the first beginning of fire and of the other bodies, following the account which combines likelihood with necessity; the principles yet more remote than these are known to Heaven and to such men as Heaven favours. Now,the question to be determined is this: What are the most perfect bodies that can be constructed, four in number, unlike one another, but such that some can be generated out of one another by resolution? If we can hit upon the answer to this, we have the truth concerning the generation of earth and fire and of the bodies which stand as proportionals between them. For we shall concede to no one that there are visible bodies more perfect than these, each corresponding to a single type. We must do our best, then to construct the four types of body that are most perfect and declare that we have grasped the constitution of these things sufficiently for our purpose. Now, of the two triangles, the isosceles is of one type only; the scalene, of an endless number. Of this unlimited multitude we must choose the best, if we are to make a beginning on our own principles. Accordingly, if anyone can tell us of a better kind that he has chosen for the construction of these bodies, his will be the victory not of an enemy, but of a friend. For ourselves, however, we postulate as the best of these many triangles one kind, passing over all the rest; that, namely, a pair of which compose the equilateral triangle. The reason is too long a story; but if anyone should put the matter to the test and discover that it is not so, the prize is his with all good will. So much, then, for the choice of the two triangles, of which the bodies of fire and of the rest have been wrought: the one isosceles (the half-square),the other having the greater side triple in square of the lesser (the half-equilateral). We must now be more precise upon a point that was not clearly enough stated earlier. It appeared as though all the four kinds could pass through one another into one another; but this appearance is delusive: for the triangles we selected give rise to four types, and whereas three are constructed out of the triangle with unequal sides, the fourth alone is consttucted out of the isosceles. Hence it is not possible for all of them to pass into one another by resolution, many of the small forming a few of the greater and vice versa. But three of them can do this; for these are all composed of one triangle and when the larger bodies are broken up several small ones will be formed of the same triangles, taking on their proper figures; and again when several of the smaller bodies are dispersed into their triangles, the total number made up by them will produce a single new figure of larger size, belonging to a single body.