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2015 | Book

Classical Mechanics Read chapter Read first chapter

Statistical Mechanics for Engineers

Author: Isamu Kusaka

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides a gentle introduction to equilibrium statistical mechanics. The particular aim is to fill the needs of readers who wish to learn the subject without a solid background in classical and quantum mechanics. The approach is unique in that classical mechanical formulation takes center stage. The book will be of particular interest to advanced undergraduate and graduate students in engineering departments.

Table of Contents

Frontmatter
1. Classical Mechanics
Abstract
A macroscopic object we encounter in our daily life consists of an enormously large number of atoms. While the behavior of these atoms is governed by the laws of quantum mechanics, it is often acceptable to describe them by means of classical mechanics. In this chapter, we familiarize ourselves with the basic concepts of classical mechanics. From the outset, we assume that concepts such as mass, time, displacement, and force are understood on the basis of our everyday experiences without further elaboration. The role of classical mechanics, then is to explore the precise relationship among these objects in mathematical terms.
Isamu Kusaka
2. Thermodynamics
Abstract
We know from experience that a macroscopic system behaves in a relatively simple manner. For example, when liquid water is heated under atmospheric pressure, it will boil at 100 °C. If the vapor so produced is cooled at the same pressure, it will condense at 100 °C. These statements hold true regardless of the initial conditions from which the body of water under consideration has evolved. This situation is in stark contrast to that in classical mechanics, in which initial conditions play a far more prominent role. In fact, our experience tells us that results of measurements we make of a macroscopic body are quite insensitive to the detailed microscopic state of the body. Thermodynamics is built on this empirical observation and systematically elucidates interconnections among these insensitive observations. In this chapter, we will review the framework of thermodynamics before attempting to interpret it from the classical mechanical point of view in Chaps. 3 and 4.
Isamu Kusaka
3. Classical Statistical Mechanics
Abstract
According to classical mechanics, equations of motion supplemented by initial conditions uniquely determine the subsequent evolution of a given system. For typical systems of our interest, however, the number of mechanical degrees of freedom is of the order of 1024. One cannot possibly write down 1024 equations of motion, much less solve them. It is also impossible to specify the initial conditions for such a system with a required accuracy. Moreover, even if we could somehow accomplish all of this, it would be entirely impossible to comprehend the resulting list of coordinates and momenta at any instant. Despite a hopeless scenario this observation might suggest, behavior of a macroscopic system is surprisingly regular as we have seen in thermodynamics. It is as if laws governing behavior of a macroscopic system are quite different from those governing its behavior at a microscopic level. In this chapter, we examine the connection between these two distinct ways of looking at a macroscopic system.
Isamu Kusaka
4. Various Statistical Ensembles
Abstract
A canonical ensemble describes a system held at a constant temperature. From the canonical partition function follows the Helmholtz free energy. As we saw in thermodynamics, however, it is highly desirable to be able to describe a system held under a different set of constraints, such as constant temperature and pressure or constant temperature and chemical potentials of some species. These situations call for different free energies in thermodynamics, to which correspond different statistical ensembles in statistical mechanics. In this chapter, we construct such ensembles and illustrate their applications with several simple examples. Our first task is to establish the notion of microcanonical ensemble, which is suitable for describing an isolated system. The other ensembles of more practical importance, including the canonical ensemble, can be derived straightforwardly from the microcanonical ensemble.
Isamu Kusaka
5. Simple Models of Adsorption
Abstract
As an illustration of canonical and grand canonical ensembles, we discuss a few variants of a simple model of adsorption. Despite their simplicity, these models provide important insights into diverse phenomena ranging from oxygen binding to hemoglobin to vapor–liquid phase coexistence.
Isamu Kusaka
6. Thermodynamics of Interfaces
Abstract
Fundamental equations of thermodynamics are commonly written for homogeneous systems without explicitly accounting for effects of the container wall. Similarly, we treat a system consisting of multiple coexisting phases as if it is made of homogeneous parts separated by sharp interfaces. This is an acceptable practice provided that the number of molecules in the vicinity of the container wall or the interface is negligibly small compared to the number of the molecules in the bulk. The approach becomes inappropriate for microscopic systems in which the majority of molecules are near a wall or an interface. For example, inhomogeneity extends throughout the entire system of interest when a fluid is confined to a narrow pore of several atomic diameters. In this case, the phase behavior changes dramatically compared to that in the bulk. It is also possible that the phenomena of interest are dictated by the properties of the interface. Examples include formation of microemulsion, wetting of a solid surface, condensation of a vapor phase, and crystallization from a melt or a solution. In this Chapter, we examine how thermodynamics can be extended to explicitly account for effects of interfaces.
Isamu Kusaka
7. Statistical Mechanics of Inhomogeneous Fluids
Abstract
While thermodynamics of interfaces provides a theoretical foundation for understanding various interfacial phenomena, its application depends on the availability of the fundamental equation. In the absence of experimental access to this information, this is a task best left to statistical mechanics. In this Chapter, we introduce a powerful method from statistical mechanics that allows us to study interfaces and inhomogeneous systems in general based on underlying molecular level models.
Isamu Kusaka
8. Quantum Formulation
Abstract
In this chapter, we present the mathematical formalism used in quantum mechanics first and then derive expressions for canonical and microcanonical partition functions for quantum mechanical systems. This will help you develop familiarity with the basic ideas of quantum mechanics and the bra–ket notation you may encounter when consulting more advanced textbooks on statistical mechanics. The important conclusion of this chapter is that the distinction between classical and quantum mechanical versions of statistical mechanics stems from the explicit expressions for the density of states these mechanics predict and from the manner in which a given system populates the microstates accessible to it. Many optional sections are included to provide explicit derivations of several key results from quantum mechanics we have already used in earlier Chapters, but should probably be omitted upon the first reading. Keeping with our most immediate goals, we will not concern ourselves with experimental findings that forced the radical departure from classical mechanics and the eventual formulation of quantum mechanics in the early twentieth century. Interested readers can find these accounts in earlier chapters of many textbooks on quantum mechanics.
Isamu Kusaka
Backmatter
Metadata
Title
Statistical Mechanics for Engineers
Author
Isamu Kusaka
Copyright Year
2015
Electronic ISBN
978-3-319-15018-5
Print ISBN
978-3-319-13809-1
DOI
https://doi.org/10.1007/978-3-319-13809-1