Complexity Explorer Santa Few Institute



Statistical Mechanics: Entropy, Order Parameters and Complexity

Preface: The purview of science grows rapidly with time. It is the responsibility of each generation to join new insights to old wisdom, and to distill the key ideas for the next generation. This is my distillation of the last fifty years of statistical mechanics—a period of grand synthesis and great expansion. This text is careful to address the interests and background not only of physicists, but of sophisticated students and researchers in mathematics, biology, engineering, computer science, and the social sciences. It therefore does not presume an extensive background in physics, and (except for Chapter 7) explicitly does not assume that the reader knows or cares about quantum mechanics. The text treats the intersection of the interests of all of these groups, while the exercises encompass the union of interests. Statistical mechanics will be taught in all of these fields of science in the next generation, whether wholesale or piecemeal by field. By making statistical mechanics useful and comprehensible to a variety of fields, we enrich the subject for those with backgrounds in physics. Indeed, many physicists in their later careers are now taking excursions into these other disciplines. To make room for these new concepts and applications, much has been pruned. Thermodynamics no longer holds its traditional key role in physics. Like fluid mechanics in the last generation, it remains incredibly useful in certain areas, but researchers in those areas quickly learn it for themselves. Thermodynamics also has not had significant impact in subjects far removed from physics and chemistry: nobody finds Maxwell relations for the stock market, or Clausius–Clapeyron equations applicable to compression algorithms. These and other important topics in thermodynamics have been incorporated into a few key exercises. Similarly, most statistical mechanics texts rest upon examples drawn from condensed matter physics and physical chemistry—examples which are then treated more completely in other courses. Even I, a condensedmatter physicist, find the collapse of white dwarfs more fun than the low-temperature specific heat of metals, and the entropy of card shuf- fling still more entertaining. The first half of the text includes standard topics, treated with an interdisciplinary slant. Extensive exercises develop new applications of statistical mechanics: random matrix theory, stock-market volatility, the KAM theorem, Shannon entropy in communications theory, and Dyson’s speculations about life at the end of the Universe. The second half of the text incorporates Monte Carlo methods, order parameters, linear response and correlations (including a classical derivation of the fluctuation-dissipation theorem), and the theory of abrupt and continuous phase transitions (critical droplet theory and the renormalization group). This text is aimed for use by upper-level undergraduates and graduate students. A scientifically sophisticated reader with a familiarity with partial derivatives and introductory classical mechanics should find this text accessible, except for Chapter 4 (which demands Hamiltonian mechanics), Chapter 7 (quantum mechanics), Section 8.2 (linear algebra), and Chapter 10 (Fourier methods, introduced in the Appendix). An undergraduate one-semester course might cover Chapters 1–3, 5–7, and 9. Cornell’s hard-working first-year graduate students covered the entire text and worked through perhaps half of the exercises in a semester. I have tried to satisfy all of these audiences through the extensive use of footnotes: think of them as optional hyperlinks to material that is more basic, more advanced, or a sidelight to the main presentation. The exercises are rated by difficulty, from 1 (doable by inspection) to 5 (advanced); exercises rated 4 many of them computational laboratories) should be assigned sparingly. Much of Chapters 1–3, 5, and 6 was developed in an sophomore honors ‘waves and thermodynamics’ course; these chapters and the exercises marked 1 and 2 should be accessible to ambitious students early in their college education. A course designed to appeal to an interdisciplinary audience might focus on entropy, order parameters, and critical behavior by covering Chapters 1–3, 5, 6, 8, 9, and 12. The computational exercises in the text grew out of three different semester-long computational laboratory courses. We hope that the computer exercise hints and instructions on the text web site [129] will facilitate their incorporation into similar courses elsewhere. The current plan is to make individual chapters available as PDF files on the Internet. I also plan to make the figures in this text accessible in a convenient form to those wishing to use them in course or lecture presentations

J. P. Sethna
Physics, Emergence, Mathematics, Thermodynamics, Statistical Physics