This volume is based on courses in Statistical Mechanics and Thermodynamics which I taught for several years at the Department of Physics, Worcester Polytechnic Institute. My objective is to treat classical statistical mechanics and its modern applications, especially to systems of interacting particles and to time-dependent phenomena. The development is based on Gibbs' ensemble formulation, though other approaches are also treated.
I emphasize a cyclic treatment of the main themes rather than an axiomatic- deductive approach. I could have begun with a fully-detailed formal treatment of ensemble mechanics, as found in Gibbs' volume, and then give material realizations. I instead interleave my formal discussion with examples of concrete analytic systems. The material examples are then used to illustrate the formal definitions. I prefer to discuss core ideas and simple applications before leaping into more complicated problems. This approach gives students a chance to identify the central features of a method before they are buried in ancillary detail. The cyclic approach also provides the instructor with material examples as a skeleton on which to hang a more formal treatment.
There are several points in which the development of topics presented here differs markedly from the pattern followed in other modern works:
First, I have adhered to Gibbs [1] rather than Boltzmann [2] or Schrodinger [3] in asserting the primacy of the canonical over the microcanonical ensemble, not the other way around. I believe that this choice maintains pedagogical simplicity and keeps a direct connection between theory and reality.
Had I begun with the microcanonical ensemble, I could have provided the necessarily elaborate demonstration that the microcanonical statistical weight for the whole of a large isolated physical system
implies the canonical statistical weight
for a small part of the large isolated system. However, real systems of fixed temperature are generally not small parts of equilibrium systems which have fixed energy, so this derivation is unphysical. Also, this derivation sacrifices the major advantage which Gibbs assigned to his canonical ensemble treatment, namely that the canonical ensemble is equally valid for small and large systems, while the transition from eq. 0.1 to eq. 0.2 is only useful for large systems.
From a logical-theoretical standpoint, equations 0.1 and 0.2 are equivalently desireable, in that either gives a single new postulate beyond Newtonian and quantum mechanics. One may cultivate a preference for one or the other of these equations on such grounds as ``simplicity'', but this is theology, not science. Operationally, eq. 0.2 is to be preferred to eq. 0.1, in that the world contains many examples of thermostatted systems (for which eq. 0.2 is apparently exact), but no examples of isolated systems (for which eq. 0.1 is believed to be correct.[4] Gibbs emphasizes that the canonical ensemble is as useful for systems containing few particles as it is for systems containing many particles, in contrast to the microcanonical ensemble, which is only applicable to many particle systems. Gibbs' arguments on this topic are treated below.
Second, in developing most of the material presented here, quantum mechanics has been reduced to its historically subordinate role. Most research in statistical mechanics of physical systems does not use quantum theory directly. Admittedly, if one wishes to compute the forces within a pair or cluster of atoms, quantum mechanics is indispensable. Similarly, if one needs to calculate the allowed energies of a system rather than treating the list of energies of a system as a postulated given, one may need quantum mechanics. However, interference effects are seldom obvious except at low temperatures. The correct counting of states of indistinguishable particles at normal densities and room temperature was obtained by Gibbs in the last century, using strictly classical arguments. In cold dense systems, quantum corrections may become large; these effects are treated separately in Part II, Chapter 16.
Third, in some ways I am a firm believer in dotting i's and crossing t's. A good example is the treatment of the so-called Gibbs' Paradox, which is not due to Gibbs and which Gibbs did not view as involving a paradox. It is not difficult to find authors who mix results from the canonical and microcanonical ensembles without bothering to obtain logical consistency. I have tried, probably without complete success, to avoid inconsistencies here.
Colleagues in physics may find this text acceptable for a Senior-year special topics course, though my emphasis on condensed fluid phases may appear excessively broadening for students of physics. The difference between chemistry and physics -- that a graduate text in one field becomes an undergraduate text in the other -- is not meant as an aspersion on the abilities of chemistry students, only a criticism of their preparation. A Senior in Physics will have had at least some formal exposure to the theories -- classical mechanics, quantum mechanics, electromagnetism -- which form the core of all further work in physics. In contrast, an entering graduate student in physical chemistry who has completed the customary curriculum for his bachelor's degree can readily be a tabula rasa insofar as a quantitative knowledge of post-1890 advances in physical chemistry is concerned.
Physicists may find my treatment of algebraic manipulations to be more explicit than is typical of graduate texts in physics. Recall that I am writing for students in chemistry as well as physics, the former receiving far less undergraduate practice in algebraic manipulation than the latter. A chemistry undergraduate may well complete his year-and-a-half or two years of calculus, and then use absolutely no mathematics beyond long division (perhaps some simple derivatives in analytic chemistry) until he reaches senior year thermodynamics. Thermodynamics demands a clear understanding of partial derivatives, but few thermodynamic texts call on the student to perform significant algebraic manipulation. An entering graduate student in chemistry may well (expecially if he had an inferior physical chemistry text) never have performed a volume integral while doing chemistry. Also, my experience suggests that many physics students are prepared to accept a sketch of a calculation as a definitive proof, without feeling any obligation to check that the missing steps are correct.
I note two extremes in methods for causing students to learn theoretical topics. In one approach, held by a sufficiently large majority that its supporters oft refer to it as the ``only approach'', students are expected to learn primarily form the act of solving problems. Those who respond to the other approach, while interested in seeing worked examples, tend to find homework problems to be an obstacle which must be cleared before one can spend time learning the material, i. e. thinking about what one has read. Graduate students are warned that the number of persons who think that they need to ponder problems rather than working them is larger than the number of people who do need to ponder problems rather than working them.
I believe that these philosophies represent underlying differences in the processes of thought of their supporters. I am myself an adherent of the latter philosophy who hopes that enough problems have been presented to satisfy those who want to work problems. I have also used the homework problems to introduce significant results that do not elsewise appear in the text, so perusal of unworked problems may prove worthwhile. The concept of naming problems is due to Mr. Mark Swanson, who initially advocated the procedure as a method for tagging rules in large complex games.
On a parallel line, some students will say ``tell us what is true, not what is not true'', while others find the mathematician's emphasis on elaborated counter-examples to be critical in sharpening their thinking about what definitions mean. The mathematicians appear to have the better of this argument.
George D. J. Phillies
Worcester, Massachusetts
