Kinetic Theory

1-1 Historical Sketch
1-2 The Krönig-Clausius Model
1-3 The Maxwell Distribution
1-4 The Boltzmann Equation
1-5 Boltzmann's H Theorem, Entropy, and Information
1-6 The Maxwell–Boltzmann Distribution
1-7 Time Reversal, Poincaré Cycles, and the Paradoxes of Loschmidt and Zermelo
1-8 Validity of the Boltzmann Equation
1-9 The Kac Ring Model and the Stosszahlansatz
1-10 Concluding Remarks
PROBLEMS

1-1 Historical Sketch

Among the earliest recorded speculations on the nature of matter and the material world were those of Thales of Miletus (about 640–547 B.C.), many of whose ideas may have had their origin in ancient Egyptian times. He suggested that everything was composed of water and substances derived from water by physical transformation. About 500 B.C. Heraclitus suggested that the fundamental elements were earth, air, fire, and water and that none could be obtained from another by physical means. A little later, Democritus (460–370 B.C. claimed that matter was composed of "atoms," or minute hard particles, moving as separate units in empty space and that there were as many kinds of particles as there were substances. Unfortunately, only second-hand accounts of these writings are available. Similar ideas were put forward by Epicurus (341–276 B.C.) but were disputed by Aristotle (384–322 B.C.), which no doubt accounts for the long delay in further discussions on kinetic theory.

Apart from a poem by Lucretius (A.D. 55), expounding the ideas of Epicurus, nothing further was done essentially until the seventeenth century, when Gassandi examined some of the physical consequences of the atomic view of Democritus. He was able to explain a number of physical phenomena, including the three states of matter and the transition from one state to another; in a sense he fathered modern kinetic theory.

Several years later, Hooke (1635–1705) advanced similar ideas and suggested that the elasticity of a gas resulted from the impact of hard independent particles on the enclosure. He even attempted on this basis to explain Boyle's law (after Robert Boyle, 1627–1691), which states that if the volume of a vessel is allowed to change while the molecules and their energy of motion are kept the same, the pressure is inversely proportional to the volume. The first real contribution to modern kinetic theory, however, came from Daniel Bernoulli (1700–1782). Although he is often incorrectly credited with many of Gassandi's and Hooke's discoveries, he was the first to deduce Boyle's law from the hypothesis that gas pressure results from the impacts of the particles on the boundaries. He also foresaw the equivalence of heat and energy (about 100 years before anyone else realized it), so it is perhaps only just that Daniel Bernoulli has been given the title "father of kinetic theory." His work on kinetic theory is contained in his famous book Hydrodynamica, which was published in 1738. The interested reader can examine a short excerpt of this work in Volume 2 of The World of Mathematics [Newman (B1956)] or in Volume 1 of a three-volume work, Kinetic Theory, by S. G. Brush (B1965), where many other papers of historical interest can be found.

After Bernoulli there is little to record for almost a century. We then find Herapath (1821), Waterston (1845), Joule (1848), Krönig (1856), Clausius (1857), and Maxwell (1860) taking up the. subject in rapid succession. From this point on we shall depart from a full history of development and concentrate on only the significant contributions from Clausius, Maxwell, and Boltzmann, who together paved the way for statistical mechanics.

1-2 The Krönig–Clausius Model

The Krönig-Clausius model [Krönig (1856) and Clausius (1857)] is perhaps the simplest possible model of an ideal gas and is described simply as follows.

Consider a cube of side l containing N molecules each of mass m. We denote by V = l3 the volume of the cube and by A = l2 area of each of the six faces (see Figure 1.1), and we make the following

Assumption. The molecules are evenly divided into six uniform beams moving with velocity c in each of the six coordinate directions.

The problem is to find the pressure the molecules exert on each of the six faces of the cube. Since pressure is force per unit area and force is momentum per unit time, we have

pressure = momentum/unit area × unit time. (2.1)

In time δt, only the fraction

[cδt/l] N (2.2)

of the molecules can strike a face. Only one sixth of these are moving toward any given face, and each molecule imparts momentum 2mc to the face it strikes, so the pressure P exerted on a face is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.3)

If we now identify ½ Nmc2 with the total kinetic energy E we obtain Boyle's law:

PV = 2/3 E. (2.4)

We stress that from the mathematical point of view this model is irreproachable. Moreover, it yields the "correct" relation between P and V. From the physical point of view, however, the assumption that all molecules move with uniform speed parallel to the coordinate axes is clearly absurd. A more realistic assumption would allow the velocities to be described by a probability density. This leads us to Maxwell's model.

1-3 The Maxwell Distribution

In 1859 Maxwell [Maxwell (1860)] proposed the following model: Assume that the velocity components of N molecules, enclosed in a cube with side l, along each of the three coordinate axes are independently and identically distributed according to the density f(α) = F(- α), i.e.,

f(vi) dvi = the probability that the ith velocity component is between vi and vi + dvi, (3-1)

where i = 1, 2, 3 refer to the three coordinate axes. As before, only the fraction

[viδt/l] N (3.2)

of the molecules whose ith velocity component is vi can strike a face perpendicular to the i axis in time δt. Hence the pressure exerted on such a face is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.3)

where <...> denotes the probabilistic average with respect to the distribution f(α), i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

Now, since we are assuming that each component has the same distribution,

[MATHEMATICAL...