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 EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.5)

where <E> is the average total kinetic energy of the molecules and υ2 = υ21 + υ22 + υ23. Thus Equation 3.3 gives

PV = 2/3 , (3.6)

which again is Boyle's law.

Maxwell actually went one step further and assumed in addition to Equation 3.1 that the distribution depends only on the magnitude of the velocity; in other words,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.7)

from which it follows immediately that

f(υi) = A exp (-Bυ2i)

or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.8)

which is the famous Maxwell distribution for velocities.

Again, from the mathematical point of view, the derivation of Equations 3.6 and 3.8 is perfectly legitimate, granted the assumptions. However, even though the assumptions represent a vast improvement on Clausius' assumption, they still may be objected to on physical grounds. The main objection is that from the dynamical point of view, the velocities do not enter independently into the equations describing collisions between molecules, so it is reasonable to expect that the velocities of different molecules are correlated in some way. Maxwell himself realized this and briefly returned to the problem in his great memoir entitled "On the Dynamical Theory of Gases" [Maxwell (1867)], in which he proposed: "It is the velocities of two colliding molecules, rather than the velocity components of a single molecule, that are statistically independent." By considering two colliding molecules and using the fact that the total kinetic energy is conserved in a collision, Maxwell was again able to show that his distribution followed from this perhaps more realistic assumption.

At this time Ludwig Boltzmann (1844–1906) was just beginning his career in theoretical physics. Inspired perhaps by Maxwell's memoir, he became immersed in Maxwell's assumptions and the role collisions play in bringing about equilibrium. This work finally culminated in his great work of 1872 [Boltzmann (1872)], which contained the now-famous "Boltzmann equation." A derivation of this equation is given in the following section.

1-4 The Boltzmann Equation

Unlike Maxwell, who presupposed equilibrium and looked for analytical conditions on the distribution function required to maintain stable equilibrium, Boltzmann started out by assuming that the gas was not in equilibrium and attempted to show that equilibrium can result from collisions between molecules. To this end he derived his famous equation.

For simplicity, we consider, as Boltzmann did, a spacially homogeneous gas of N hard-sphere particles (i.e., perfectly elastic billiard balls), enclosed in a volume V, with mass m, diameter a, and velocity distribution function f(v, t) defined by

f(v,t)d3 υ = number of particles at time t with velocity in the volume element d3υ around v. (4.1)

We make the following assumptions:

1. Only binary collisions occur. That is, situations in which three or more particles come together simultaneously are excluded. Physically speaking, this is a reasonable assumption if the gas is sufficiently dilute (i.e., if the mean free path is much larger than the average molecular size).

2. The distribution function for pairs of molecules is given by

f(2) (v1, v2, t) = f(v1, t) f (v2, t). (4.2)

In other words, the number of pairs of molecules at time t, f(2) (v1, v2, t) d3 υ1d3 υ2 with velocities in d3 υ1 around v1 and in d3 υ2 around v2, respectively, is equal to the product of f(v1, t) d3 υ1 and f(v2, t) d3 υ2. This is Boltzmann's famous Stosszahlansatz, or assumption of "molecular chaos," and it is this innocent-looking assumption [originally due to Clausius (1857)] that was, and still is some hundred or so years later, the most widely discussed.

It is obvious that assumption 1 is purely a dynamical assumption and that 2 is basically a statistical assumption. Unfortunately, the latter was not clearly or adequately stressed, particularly by Boltzmann at a time when objections to his equation were based solely on mechanical considerations. We shall discuss this point at length in later sections after first deriving here Boltzmann's equation for f(v, t).

By definition, the rate of change of f(v, t)d3 υ is equal to the net gain of molecules in d3 υ as a result of collisions, i.e.,

[partial derivative]f/[partial derivative]t = nin - nout, (4.3)

where

nin(out)d3 υ = the number of binary collisions at time t in which one of the final (initial) molecules is in d3 υ. (4.4)

Consider now a particular molecule [1] with velocity in d3 υ1 around v1, and all those molecules [2] with velocities in d3 υ2 around v2 which can collide with molecule [1]. It is to be remembered that we are considering now only binary collisions of hard spheres with diameter a.

In the relative coordinate system with molecule [1] at rest, the center of molecule [2] must be in the "collision cylinder," as shown in Figure 1.2, if it is to collide with molecule [1] during the time interval δt. The collision cylinder and appropriate parameters are shown in more detail in Figure 1.3. The collision cylinder has volume b dφ db gδt, and from Figure 1.2 it is

b = a cos (θ/2), (4.5)

and hence that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.6)

where dΩ = sin θ dθ dφis the solid angle element of the "scattered" particle [2].

Now using the definition of nout (Equation 4.4), and the Stosszahlansatz Equation (4.2) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.7)

To determine nin we merely look at the inverse collision (v'1, v2) -> (v1, v2) and use the above results to obtain immediately

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.8)

It is to be noted that the second integral in Equation 4.7 is over v2 and in Equation 4.8 over v'2.

Since we are considering perfectly elastic spheres, energy (½mv2) and momentum (mv) are conserved in a collision; hence since all masses are assumed to be equal,

v1 + v2 = v'1 + v'2 (momentum conservation) (4.9)

and

υ21 + υ22 = υ'21 υ'22 (energy conservation). (4.10)

It follows almost immediately (see Problem 3) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.11)

and that the transformation (v1, v2) -> (v'1, v'2) is orthogonal, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.12)

In fact, it is not difficult (see Problem 3) to derive the following explicit relations from Equations 4.9 and 4.10:

v'1 = v1 + [(v2 – v1) • l]l (4.13)

and

v'2 = v2 – [(v2 – v1) • l]l, (4.14)

where, as in Figure 1.2, 1 is the unit vector in the direction of the line of centers of the two spheres at the time of collision.

Using the results 4.11 and 4.12 combined with Equations 4.7 and 4.8 for nin and nout, the equation of motion (4.3) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.15)

where f1 = f(v1, t), f'2 = f(v'2, t). This is Boltzmann's equation, which can clearly be generalized in a straightforward way: first, for more general interactions (than hard spheres) and, second, for nonuniform systems.