Statistical Physics

The following sections are included:

• Contents

• Preface

The theoretical transition from mechanics to the properties of matter is complex as well as technically difficult. Major parts of it are ever-expanding research subjects. In principle, given the laws of dynamics — classical or quantum-mechanical — and a model of the forces acting between the constituent parts of the material, we can determine all that is fit to be determined about the system. Its properties, i.e. material properties, are subject to computation — if the initial conditions are given as well, of course…

The following sections are included:

• Avogadro's law, or the equation of state of an ideal gas

• Temperature and thermal equilibrium

• Equipartition of energy per molecule and its constituent parts — a fundamental problem

• The density in an isothermal atmosphere — the Boltzmann factor in the potential energy

• The Maxwell–Boltzmann distribution

• Averages and distributions

The following sections are included:

• Historical background

• Characteristic scales of Brownian motion

• Random walk

• Brownian motion, random force and friction: the Langevin equation

• Solving the Langevin equation: approximations and orders of magnitude

• Applications and implications

The following sections are included:

• Introduction

• The mean free path and mean free time

• Self-diffusion

• The mobility coefficient

• The connection between the diffusion coefficient and the mobility

• Viscosity and thermal conductivity

• Appendix: a more detailed calculation of the diffusion coefficient

The following sections are included:

• Self-assessment exercises

• Solutions to exercises in the text

• Solutions to self-assessment exercises

As promised in the Introduction to Part I, we now turn to the treatment of material systems from a more detailed point of view. This means that the system, be it a system of molecules, a system of magnetic moments, or a system of electrons and protons, is described by a dynamical model. The dynamical model has two components: (a) the rules according to which the system evolves in time (the laws of classical mechanics (Newton), or the laws of relativistic mechanics (Einstein), or the laws of quantum mechanics (Schrödinger); (b) the forces governing this evolution (electric, magnetic, gravitational, etc.). Moreover, in such a model we must also define the relevant degrees of freedom (these may be electrons and protons); it may suffice to describe the system as a collection of atoms, or perhaps molecules are sufficiently stable and can serve as building blocks. For instance, the ideal gas discussed in Part I is described by the coordinates and velocities (or momenta) of each of its molecules — a total of 6N variables. Its evolution in time is determined by the laws of classical mechanics (Newton), and to describe its microscopic evolution in time we would have to solve 3N coupled, second order, differential equations…

The following sections are included:

• The first law

• The second law and the entropy

• Thermodynamic potentials

• The third law

The following sections are included:

• Introduction

• The first law in magnetic variables

The following sections are included:

• Magnetic states, angular momentum and paramagnetism

• Microscopic states, observables

• Probabilities and averages

The following sections are included:

• Number of states and probabilities

• Calculating averages and correlations

• Numerical examples and Stirling's formula

The following sections are included:

• Microscopic states and thermodynamic equilibrium

• β and the temperature

• Sharpness of the maximum

• Identification of temperature and entropy

• Negative temperature

• Summary

The following sections are included:

• The canonical ensemble

• The partition function and thermodynamic quantities

• Susceptibility and specific heat of a paramagnet

• Paramagnet with J > ½

It is commonly said that the magnitude of the entropy measures the disorder in a system, and that the tendency of the entropy to increase is the same as the tendency for increasing disorder. These concepts and relationships take on a more definite and quantitative character within information theory, which is related in this way to thermodynamics. Here we exemplify only a few of these concepts in connection with the simple paramagnetic model, without giving detailed definitions and without entering into a formal discussion…

The magnetization of several paramagnetic ionic salts has been measured, from very small fields up to saturation. Figure 2.7.1 describes the experimental results for the ions of chromium, iron and gadolinium which are all triply ionized, composed in salts, whose other components are nonmagnetic. The correspondence between theory and experiment is very impressive. This includes the fact that measurements performed at different temperatures, for the same salt, when plotted as a function of H/T, fall on the same curve…

The following sections are included:

• Self-assessment exercises

• Solutions to exercises in the text

• Solutions to self-assessment exercises

In Part II we saw that the canonical ensemble emerges naturally from the ensemble that characterizes an isolated system (the micro canonical ensemble). There it was demonstrated in the special case of a simple system — the paramagnet, but the argument can be generalized in a natural way to any system. The canonical description is the most useful one, as in most practical cases it is possible to control the temperature and not the energy of the system. Indeed we will use the canonical ensemble, as a starting point in the analysis of most systems we shall consider…

The following sections are included:

• The partition function and the internal energy

• Thermodynamic work

• Entropy, free energy, the first and second laws

• The paramagnet — revisited

• On the statistical meaning of the free energy

The following sections are included:

• Microscopic states

• Partition function for oscillators

• Einstein's solid

The following sections are included:

• Statistical mechanics of a single particle

• Statistical mechanics of a classical gas

The following sections are included:

• The ideal gas

• Mixtures of ideal gases — Dalton's law

• Maxwell–Boltzmann distribution and equipartition

• Ideal gas of quantum particles

The following sections are included:

• Two difficulties

• The Gibbs paradox and its resolution

• Remarks on the third law of thermodynamics

• Summary

The following sections are included:

• Paramagnet: fluctuations in the magnetization

• Energy fluctuations and the specific heat

• Summary

The following sections are included:

• Self-assessment exercises

• Solutions to exercises in the text

• Solutions to self-assessment exercises

This part is a continuation and extension of Part III and is based on the implementation of the methods presented in the previous parts. In Chap. 1 the subject of ideal gases of molecules devoid of internal structure, analyzed in Part III, is extended to deal with diatomic molecules. We learn how to take into account the internal structure of the molecules, and its effects on the properties of the gases. We shall dwell upon the heat capacity problem which we encountered in Part I and resolve it here at last…

The following sections are included:

• Center of mass and internal motions

• Kinematics of a diatomic molecule

• Gas of general composite molecules

• Diatomic gas: classical treatment

• Diatomic molecules: vibration and rotation

• The equipartition principle and its violation

• Diatomic gas — quantum calculation

The following sections are included:

• Conditions for chemical equilibrium

• The law of mass action

• Dissociation in a diatomic gas

The following sections are included:

• Sound waves in a crystal

• Vibrational modes, phonons and enumeration of states

• The Debye model

The following sections are included:

• General considerations of radiation at thermal equilibrium

• Radiation density

• Black body radiation

• Absorption and emission of radiation — Kirchhoff's law

• Role of black body radiation in modern physics

The following sections are included:

• Appendix Calculation of Some Integrals

• Self-assessment exercises

• Solutions to exercises in the text

• Solutions to self-assessment exercises

In the preceding parts we analyzed the thermodynamic properties of systems in the canonical ensemble. In this part we will get acquainted with a generalization of the canonical ensemble to systems that can exchange particles with their surroundings. This generalization, the grand canonical ensemble, which we present in Chap. 1, is of double importance. On the one hand it contributes to a deeper understanding of the statistical physics of classical particles. On the other hand, the new ensemble is the natural tool for the development of statistical mechanics of systems of identical particles in quantum conditions. This is not because a system of (nonrelativistic) quantum-mechanical particles does not conserve the number of particles. It is merely of great technical convenience and is analogous to the application of the canonical ensemble to isolated systems, which have a constant energy…

The following sections are included:

• Definitions and motivation

• Connection to thermodynamics

The following sections are included:

• Classification of states — occupation numbers

• Quantum statistics — many-particle states

• Thermodynamics of fermions and bosons

• Average occupation numbers

The following sections are included:

• The Drude model

• A critique of the Drude model

• The Sommerfeld model

• Electrons at high and low temperatures

• Metals at room temperature

• Thermodynamics of the Sommerfeld model

The following sections are included:

• Bose–Einstein distribution

• Chemical potential at low temperatures

• Bose–Einstein condensation

• Superfluidity

• Bose–Einstein condensation in helium

• Viscosity of a superfluid

• Fermi liquid and superconductivity

The following sections are included:

• Appendix Calculation of Some Integrals

• Self-assessment exercises

• Solutions to exercises in the text

• Solutions to self-assessment exercises

The following sections are included:

• Index