Morikazu Toda's research while affiliated with Yokohama National University and other places

A gas whose equation of state is described by the Boyle-Charles law and a paramagnetic substance which obeys Curie’s law are examples of so-called ideal systems. These systems are composed of elements with negligible interactions and the treatment of these systems can be reduced essentially to that of a single element. The harmonically vibrating lattices have strong interactions among particles, but they are ideal systems in view of the existence of normal modes or phonons. In contrast to these ideal systems, systems which are by no means reducible to ideal systems exist and thus have strong interactions among constituent elements which can never be ignored. They are sometimes called cooperative systems and exhibit, among other things, a cooperative phenomenon called a phase transition. For example, a gas condenses to the liquid state by compression or by cooling, and a paramagnetic substance becomes ferromagnetic by cooling below the Curie temperature. The ideal Bose gas undergoes a Bose condensation, because it can be regarded effectively as a cooperative system having attractive interactions among atoms by virtue of the symmetry of the wave function. Through a phase transition, a substance acquires a new structure or a new property which is absent before the phase transition. Sometimes the biological function of a biomaterial can be regarded as a result of a phase transition.

In this chapter, we start with certain principles and describe the general methods of statistical mechanics [2.1–17]. If we assume that every quantum-mechanical state (microscopic state) has the same weight (the principle of equal probability), then we can establish a standpoint where mechanical laws are combined with probability theory. By considering a system in contact with a larger system, we can describe a system with constant temperature or constant pressure. Thus, we develop the statistical mechanics for an equilibrium state (statistical mechanics in a narrow sense) and we can also find a microscopic interpretation of the laws in thermodynamics.

In the preceding chapter, we described the general principles of statistical mechanics which can be applied to many-particle systems. In quantum mechanics identical particles are indistinguishable and particles are classified into two groups, Bose particles and Fermi particles, according to the symmetry character of their wave functions. Quantum states must fulfill the demand of symmetricity, which means that the number of quantum states depends on the symmetry. This circumstance is taken into account in the so-called quantum statistics. In this chapter, the method of quantum statistics and its application to quantum ideal gases are discussed. Classical statistics is discussed as a limit of quantum statistics and the condition of its validity is classified. Application of classical statistics to nonideal gases is also given.

In the preceding chapters, we have described the general methods of statistical mechanics mostly on the basis of quantum mechanics. But in this chapter, we shall describe the ergodic problems based on classical and quantum mechanics, the reason for which will first be made clear.

In this chapter, we introduce the subjects to be treated in terms of statistical mechanics. The principles of statistical mechanics, the topics of Chap. 2, are well established for the equilibrium state, while the kinetic theory of gases was developed to interpret the thermal properties of matter in the bulk. Though we will not be involved in kinetic theory in this chapter, we will clarify some important relations which can be derived by the use of averages with respect to configuration and motion of molecules.

We have seen how linear irreversible processes can generally be described by the response function, the relaxation function, and especially the wave- vector-dependent complex admittance. The formulas by which these functions can be evaluated are derived from statistical mechanics, based on assumptions about the material structure. Making use of these formulas, we can connect these functions with new quantities, i.e., the correlation functions. The fluctuation-dissipation theorem developed from the Einstein relation and the Nyquist formula, as described in Sect. 1.6, will thus be generally proved. Statistical mechanics of equilibrium states, as discussed in [4.1], has offered us a means to understand static properties of materials by providing methods of calculating, e.g., the specific heat, the static dielectric constant and the static magnetic susceptibility. Were dynamical quantities such as the response function calculable, we would have at hand an even more powerful theory. This is the problem we are going to treat in this and the subsequent chapter.

In 1827 the botanist Brown discovered under his microscope vigorous irregular motion of small particles originating from pollen floating on water [1.1]. He also observed that very fine particles of minerals undergo similar incessant motion as if they were living objects. This discovery must have been a great wonder at that time. The idea of combining such a motion – Brownian motion – with molecular motion became fairly widespread in the latter half of the nineteenth century when atomism had not yet been fully recognized as reality. It was the celebrated work of Einstein, which appeared in 1905, that gave the first clear theoretical explanation of such a phenomenon which could be directly verified quantitatively by experiments and thus established the very basic foundation of the atomic theory of matter [1.2]. Einstein did not known that Brownian motion had actually been observed many years before when he first came upon this idea to verify the reality of the atomic concept. At any rate, Einstein’s theory had a great impact at that time, finally convincing people of the theory of heat as molecular motion, and so paved the way to modern physics of the twentieth century. It also greatly influenced pure mathematics, that is, the theory of stochastic processes.

We have seen that macroscopic properties in a linear irreversible process are determined by the response function, the relaxation function, the complex admittance or the double-time correlation functions. This chapter briefly describes techniques for calculating these functions. Of course, there are many methods of calculation, each of which has its own merits and demerits and has particular key points to be considered. A simple example for the determination of the response function by using the kinetic theoretical method was given in .