# Mathematical Statistical Mechanics

COLIN J. THOMPSON
Pages: 290
https://www.jstor.org/stable/j.ctt13x1022

1. Front Matter
(pp. i-iv)
2. Preface
(pp. v-vi)
C. J. T.
(pp. vii-ix)
(pp. x-x)
5. CHAPTER 1 Kinetic Theory
(pp. 1-31)

Among the earliest recorded speculations on the nature of matter and the material world were those of Thales of Miletus (about 640–547 B.C.), many of whose ideas may have had their origin in ancient Egyptian times. He suggested that everything was composed of water and substances derived from water by physical transformation. About 500 B.C. Heraclitus suggested that the fundamental elements were earth, air, fire, and water and that none could be obtained from another by physical means. A little later, Democritus (460–370 B.C.) claimed that matter was composed of “atoms,” or minute hard particles, moving as separate...

6. CHAPTER 2 Thermodynamics
(pp. 32-53)

The main aim of statistical mechanics is to derivemacroscopicproperties of matter from the laws governing themicroscopicinteractions of the individual particles.

Thermodynamics consists of laws governing the behavior of macroscopic variables and as such is a necessary prerequisite for statistical mechanics. It is a subject in its own right and has the advantage, from the mathematical point of view, that it can be completely axiomatized. Since our main concern here, however, is statistical mechanics, we shall present only the “primitive concepts” and “laws” and, for later reference, some particular thermo-dynamic relations. A detailed account of the axiomatization...

7. CHAPTER 3 The Gibbs Ensembles and the Thermodynamic Limit
(pp. 54-77)

The task of statistical mechanics is to derive macroscopic properties of matter from the laws governing the behavior of the individual particles. The macro scopic properties are expressed in terms of thermodynamic variables discussed in Chapter 2. In statistical mechanics one must first define the thermodynamic variables and then check that the laws of thermodynamics are valid. Unlike thermodynamics, however, which consists only of laws and relations between thermodynamic quantities (such as energy, specific heat, etc.), statistical mechanics aims to derive analytic expressions for these quantities.

Before we can proceed, we must specify the microscopic description. For simplicity we consider...

8. CHAPTER 4 Phase Transitions and Critical Points
(pp. 78-115)

One of the most interesting and challenging problems in equilibrium statistical mechanics is the problem of phase transitions. We are all familiar with boiling water and the transition of a gas to a liquid under compression, which are simple examples of phase transitions. Mathematically, the problem is to explain, or derive, the existence of phase transitions and the behavior of thermodynamic quantities in the neighborhood of the transition point from the statistical-mechanical ensembles.

Thermodynamically we describe, for example, the liquid-gas transition as follows.

Consider a gas at a fixed temperatureT.If the gas is compressed at sufficiently smallT....

9. CHAPTER 5 The Ising Model: Algebraic Approach
(pp. 116-144)

The Ising model is perhaps the simplest system that undergoes a nontrivial phase transition, and for reasons that will become clear, now occupies an almost unique place in theoretical physics. The literature on the Ising model is enormous; since the original contribution of Ising (1925) almost 500 research papers have been published on the subject.

The 1925 paper of Ising was essentially his Ph. D. thesis. He succeeded in solving the one-dimensional model exactly and found to nobody’s surprise these days that there was no phase transition. In this paper Ising gives credit to his supervisor Wilhelm Lenz for inventing...

10. CHAPTER 6 The Ising Model: Combinatorial Approach
(pp. 145-176)

The basic idea behind the combinational approach is extremely simple and was noted many years ago by Van der Waerden (1941). The problem is to evaluate the partition function ZN(Equation 1.7 of Chapter 5), which can be written in the form

${Z_N} = \mathop \sum \limits_{(\mu )} {\mathop \prod \limits_{P,Q} ^*}\exp \left( {v{\mu _P}{\mu _Q}} \right),$(1.1)

where the starred product is over nearest-neighbor lattice points P and Q, v=J\kT, and the sum is over all configurations of the lattice$\left( {{\mu _P} = \pm 1} \right)$. Expanding the exponential in Equation 1.1 and noting, since$\mu _P^2 = + 1,$, that

${\left( {{\mu _P}{\mu _O}} \right)^n} = \{ _{{\mu _P}}^1{\mu _O}$

ifnis even

ifnis odd, (1.2)

we have

exp$\left( {v{\mu _P}{\mu _Q}} \right)$= cosh ν +${\mu _P}{\mu _Q}$sinh ν

= cosh$\nu \left( {1 + \omega {\mu _P}{\mu _Q}} \right),$...

11. CHAPTER 7 Some Applications of the Ising Model to Biology
(pp. 177-208)

The Ising and related models have been applied with some success to a number of biological systems. We shall discuss three examples here: hemoglobin, allosteric enzymes, and deoxyribonucleic acid (DNA). These examples are by no means exhaustive, but they illustrate how general lattice combinatoric problems and their methods of solution may be applied to biological problems.

The common feature in the examples we shall consider here is “cooperativity,” which is playing a role of increasing importance in biology these days.

Hemoglobin, for example, which is the oxygen carrier in red blood cells, has four distinct binding sites for oxygen with...

12. APPENDIX A Measure-theoretic Statement of Liouville’s Theorem
(pp. 211-213)
13. APPENDIX B Ergodic Theory and the Microcanonical Ensemble
(pp. 214-217)
14. APPENDIX C Lebowitz–Penrose Theorem for Magnetic Systems
(pp. 218-228)
15. APPENDIX D Algebraic Derivation of the Partition Function for a Two-dimensional Ising Model
(pp. 229-243)
16. APPENDIX E Combinatorial Solution of the Dimer and Ising Problems in Two Dimensions
(pp. 244-262)
17. Bibliography
(pp. 263-274)
18. Index
(pp. 275-278)