Can we have a theoretical framework to systematize empirical facts of macrosystems without referring to microscopic details?

What is Phenomenology?

A phenomenological description of macrosystems is a description solely in terms of quantities that may be observed, controlled, and described at the macroscopic scale. If we could make a closed (complete) system of theory in terms of such quantities, the result is called phenomenology.

The phenomenology of macroscopic systems in equilibrium (see 12.2) we now have is called thermodynamics. The reader must clearly recognize that, in contrast to (the supposedly more fundamental) statistical mechanics, thermodynamics survived the quantum revolution unscathed; and it actually helped launch the revolution. Quantum mechanics has no problem with thermodynamics (for now), but even if quantum mechanics is replaced by something else in the future, thermodynamics will remain intact.

In physics, “phenomenology” has not necessarily been respected; it could almost be a pejorative (e.g., in high-energy physics). However, notice that when underlying microscopic descriptions are impossible or only approximate, phenomenology may be the only realistic rational means for human beings to grasp the world. As we will see soon, thermodynamics correctly captures a certain universal feature of the macroscopic world which is quite insensitive to the microscopic details. A phenomenology is not an approximate way to understand the world nor a crude version of something more accurate; thermodynamics is not an approximation to a certain more advanced and accurate theory. Perhaps, intelligent beings could emerge only in such worlds that allow phenomenological descriptions.

However, one may well claim that as an empirical science, quantum mechanics is much more precisely validated than thermodynamics.

The best way to study is not to study when you do not wish to.

This book grew from course lecture notes for an advanced undergraduate level thermal physics course called “Statistical Physics, a jogging course,” aimed at the top third of students. However, I believe every student graduating from physics should have some understanding of the topics in this book. Therefore, I tried to give detailed derivations of all the formulas. Since I learned mathematics and physics without attending any course (beyond 300 in the US level), I certainly wished to have books filled with details and with all the problems solved.

However, I learned some willpower was needed to use such books effectively, because learning is always “active learning”; If one wishes to build one's muscle, some load is required. Therefore, always try to guess the next line or step in the derivation/transformation of formulas before reading the lines. Think what you would do if you were to encounter the problem as the first person in the world.

Each chapter has a Summary, Key words, and The reader should be able to at the beginning. “Summary” tells the reader what the section will discuss. Since it is placed at the beginning of each chapter, the reader will not be able to understand fully what it summarizes, but still some flavor of the section can be sensed. The summary should be checked again after finishing the section. “Key words” is a list of concepts/technical terms the reader must be able to explain to her intelligent lay friends. “The reader should be able to” is a list of minimal (mostly practical) items the reader should be able to do with comfort.

Each chapter has a modular structure, consisting of about ten numbered short units with titles such as “21.14 Sanov's Theorem and Entropy Maximization.” This should make the main line of each unit explicit and easy to grasp. A few units with dagger marks (†) are slightly advanced. There are other units which are details or side issues that the reader can skip or browse through, these are indicated by an asterisk (∗).

This book requires a rudimentary knowledge of quantum mechanics. The purpose of this Appendix is to make the reader somewhat familiar with the basic structure/principles of quantum mechanics. How we proceed is quite similar to our approach to statistical thermodynamics; mathematics and empirical facts are synthesized. For statistical thermodynamics the reader has only to understand the time-independent Schrödinger equation and energy eigenvalue problem A.24 (the harmonic oscillator is discussed in A.29). Those who are not familiar with the bra-ket notation should read Appendix 17A first. The following exposition includes an outline of Fourier analysis in the Dirac notation.

How We Proceed

Everyone reading Dirac's The Principles of Quantum Mechanics should realize that respecting both the salient empirical facts and the mathematical aesthetics leads to the most natural exposition of an empirical theory. Therefore, the following exposition of mechanics is constructed around the empirical facts that suggest/justify the mathematical principles.

Remarks for Those Who are Familiar with Introductory Mechanics

For the reader who knows an outline of (analytical and quantum) mechanics, an outline of this Appendix is given here.

The first part A.3–A.8 is a preparation of our language – Fourier analysis. This part explains the Fourier transformation in Dirac's notation.

Based on experimental facts and the thought experiments they inspire, Schrödinger's equation of motion is derived (A.14). At the same time the fundamental postulates of quantum mechanics are summarized (Postulate 1–6). The perturbation method (A.31) and the minimax principle for eigenvalues (A.32) as well as the uncertainty relation (A.28) are explained. The harmonic oscillator eigenenergies are obtained (A.29). For convenience 1/2 spins are also outlined (A.30).

We then proceed to the classical approximation. The reader should be familiar with the rudiments of analytical mechanics. First, the Feynman path integral expression of the system evolution is derived (A.33), from which classical mechanics is deduced as a macroscopic approximation (A.34). The correspondence between Heisenberg's and Hamilton's equations of motion introduces the Poisson bracket and the canonical quantization condition (A.37).

Remark on the notation. In the first half of this Appendix, although the notation x is used for coordinates, this can be a vector of any dimension; xy should be interpreted as the scalar product, dx is a volume element.