Microstate and macrostate

In the formulation of statistical mechanics we are concerned with the theoretical prediction of thermodynamic properties of a physical system which contains a large number of particles such as electrons, atoms, and molecules by some statistical average processes performed over an appropriately prepared statistical sample. There are many different ways to prepare the sample. J. Willard Gibbs (1901) coined the name ensemble for the statistical sample, and three different ensembles are introduced, i.e., the microcanonical, canonical, and grand canonical ensembles.

We can choose for the physical system any thermodynamical system such as a single-component gas, liquid, or solid, as well as a mixture of many components, as long as the system is in the condition of thermodynamical equilibrium. In order to establish a fundamental principle of statistical mechanics, however, we naturally choose as simple a system as possible, such as the one-component dilute gas made up of structureless monatomic molecules. We then extend the fomulation step by step to more complicated systems. In this chapter, formulation of the three Gibbs ensembles will be developed.

The microcanonical ensemble is a collection of identical replicas of a given physical system which is a gas made up of noninteracting structureless particles. Firstly, the system is assumed to be contained in a box of volume V, the number of particles is equal to N, and the total energy is given in a narrow range between E and E + dE.

The thermodynamic system and processes

A physical system containing a large number of atoms or molecules is called the thermodynamic system if macroscopic properties, such as the temperature, pressure, mass density, heat capacity, etc., are the properties of main interest. The number of atoms or molecules contained, and hence the volume of the system, must be sufficiently large so that the conditions on the surfaces of the system do not affect the macroscopic properties significantly. From the theoretical point of view, the size of the system must be infinitely large, and the mathematical limit in which the volume, and proportionately the number of atoms or molecules, of the system are taken to infinity is often called the thermodynamic limit.

The thermodynamic process is a process in which some of the macroscopic properties of the system change in the course of time, such as the flow of matter or heat and/or the change in the volume of the system. It is stated that the system is in thermal equilibrium if there is no thermodynamic process going on in the system, even though there would always be microscopic molecular motions taking place. The system in thermal equilibrium must be uniform in density, temperature, and other macroscopic properties.

The zeroth law of thermodynamics

If two thermodynamic systems, A and B, each of which is in thermal equilibrium independently, are brought into thermal contact, one of two things will take place: either (1) a flow of heat from one system to the other or (2) no thermodynamic process will result.

This book may be used as a textbook for the first or second year graduate student who is studying concurrently such topics as theory of complex analysis, classical mechanics, classical electrodynamics, and quantum mechanics.

In a textbook on statistical mechanics, it is common practice to deal with two important areas of the subject: mathematical formulation of the distribution laws of statistical mechanics, and demonstrations of the applicability of statistical mechanics.

The first area is more mathematical, and even philosophical, especially if we attempt to lay out the theoretical foundation of the approach to a thermodynamic equilibrium through a succession of irreversible processes. In this book, however, this area is treated rather routinely, just enough to make the book self-contained.

The second area covers the applications of statistical mechanics to many thermodynamic systems of interest in physics. Historically, statistical mechanics was regarded as the only method of theoretical physics which is capable of analyzing the thermodynamic behaviors of dilute gases; this system has a disordered structure and statistical analysis was regarded almost as a necessity.

Emphasis had been gradually shifted to the imperfect gases, to the gas–liquid condensation phenomenon, and then to the liquid state, the motivation being to be able to deal with correlation effects. Theories concerning rubber elasticity and high polymer physics were natural extensions of the trend. Along a somewhat separate track, starting with the free electron theory of metals, energy band theories of both metals and semiconductors, the Heisenberg–Ising theories of ferromagnetism, the Bloch–Bethe–Dyson theories of ferromagnetic spin waves, and eventually the Bardeen–Cooper–Schrieffer theory of super-conductivity, the so-called solid state physics, has made remarkable progress.