The Kinetic Theory of Gases

The following sections are included:

• CONTENTS

• Preface

• Acknowledgments

The following sections are included:

• THE CALORIC THEORY
  • Laplace
  • Herapath
  • Waterston

• THE CONSERVATION OF ENERGY

• REVIVAL OF THE KINETIC THEORY
  • Clausius
  • Maxwell

• THE VIRIAL THEOREM

• ATOMIC MAGNITUDES
  • Addendum to Volume 1

IN ORDER to help the reader to locate the major works mentioned in the introduction, detailed citations have been collected here in alphabetical order by author. We have attempted in all cases to give complete details of the original publications, together with recent reprints and English translations where they are available. While the average university library may not have some of the older scientific books and periodicals cited here, it should possess the collected works of all of the major scientists, so that there should be no difficulty in finding most of the items listed here in some form…

A qualitative atomic theory of the "spring" of the air—i.e. its property of resisting compression by exerting pressure on a surface in contact with it—is proposed. Boyle conceives the air to be similar to "a heap of little bodies, lying one upon another, as may be resembled to a fleece of wool". Each body is like a little spring, which can be easily bent or rolled up, but also tries to stretch itself out again. This tendency to expand is characteristic not only of air that has been compressed, but also of the ordinary air in the atmosphere, which has to support the weight of a column of air many miles in height above it. In support of this theory, the experiment of Pascal is mentioned, in which it was shown that the height of mercury in a barometer is less at the top of a mountain than at the bottom. (Boyle assumes that the mercury is supported by air pressure rather than by nature's abhorrence of a vacuum.)

The purpose of natural philosophy is to develop mathematical principles so that from the observed phenomena of motions one may investigate the forces of nature, and then from these forces demonstrate other phenomena. Newton suspects that all the phenomena of nature may depend on "… certain forces by which the particles of bodies… are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another." One example of this kind of investigation is a demonstration that "… particles fleeing from each other, with forces that are inversely proportional to the distances of their centres, compose an elastic fluid, whose density is as the compression." (In this way Boyle's experiments on the spring of the air may be explained by assuming that each atom exerts repulsive forces on its neighbors.).

The properties of elastic fluids (i.e. gases) depend especially on the facts that: (1) they possess weight; (2) they expand in all directions unless restrained; and (3) they allow themselves to be more and more compressed as the force of compression increases. These properties can be explained if we assume a fluid to consist of a very large number of small particles in rapid motion. If such a fluid is placed in a cylindrical container with a movable piston on which there rests a weight P, then the particles will strike against the piston and hold it up by their impacts; the fluid will expand as the weight P is moved or reduced, and will become denser if the weight is increased. We can find the relation between P and the density of the fluid by computing the effect of changing the volume underneath the piston. If the volume is decreased, the fluid will exert a greater force on the piston, first, because the number of particles is now greater in proportion to the smaller space in which they are confined, and secondly because any given particle makes more frequent impacts. If in addition we assume that the particles themselves are of negligible size, then it follows that the force of compression is approximately inversely proportional to the volume occupied by the air, if the temperature is fixed; and, if the temperature changes, the force of compression will also change, so that it will be proportional to the square of the velocity of the particles. This result enables us to measure the temperature of a gas by the pressure it exerts; the standard temperature from which the rest are measured should be obtained from boiling rainwater, because this no doubt has very nearly the same temperature all over the earth. Experiments to measure temperature and pressure are described. The theory is applied to atmospheric phenomena and to the firing of projectiles from cannons.

The arguments advanced by theorists who believe that heat is nothing more than molecular motion are summarized. However, it is asserted that "… the hypothesis which assigns existence to the principle of fire (heat), as a distinct elementary principle, is supported by more numerous facts, and by more decisive reasons; it accounts better for all the phenomena of nature, and even for those very phenomena which are adduced in support of the contrary opinion." After stating the various arguments in favor of the materiality of heat, Gregory gives a brief account of the properties of this substance and its effects on matter.

Various properties of forces are stated and illustrated. (By a "force" Mayer means a particular form of what we would now call energy.) Forces are causes, and must therefore be equal to their effects; in other words, they are indestructible; when one force disappears, it must be replaced by another that is equal to it. In contrast to matter, forces have no weight. One should not confuse a force with a property, since a property lacks the essential attribute of a force, the union of indestructibility and convertibility. (Thus gravity is a property, not a force; but the separation in space of ponderable objects is a force, which Mayer calls "falling force".)

We often see motion cease without having caused another motion or the lifting of a weight. Since a force once in existence cannot be annihilated but can only change its form, we are led to ask whether the motion has been converted into heat. If so, then the heat produced must be equivalent to the motion lost, and we have to determine, for example, how much heat corresponds to a given quantity of motion or "falling force". It is found that the heat needed to warm a given weight of water from 0° to 1° C corresponds to the fall of an equal weight from the height of about 365 metres, the latter phrase being the measure of a quantity of falling force.

Matter may be said to be defined by the attributes of impenetrability and extension; it also has a number of properties, such as gravity or weight, repulsion, inertia, and living force (kinetic energy). The last property is simply the force of bodies in motion, and is proportional to their weight and to the square of their velocity. Living force may be transferred from one body to another in collisions, or by attraction through a certain distance. It would seem absurd that this living force can be destroyed, or even lessened, without producing the equivalent of attraction through a given distance or heat, despite the fact that until very recently a contrary opinion has been generally held. We have reason to believe that the manifestations of living force on our globe are, at the present time, as extensive as those which have ever existed, and that the motions of air and water are not annihilated by friction but only converted to heat. The quantity of heat produced is always in proportion to the amount of living force absorbed; the living force possessed by a body of 817 lb when moving with the velocity of eight feet per second is convertible into the quantity of heat which can raise the temperature of 1 lb of water by one degree Fahrenheit.

Until recently it was thought that heat is a substance; but we have shown that it can be converted into living force and into attraction through space, so that it must therefore consist of either living force or of attraction through space. It seems to me that both hypotheses are probably true. There is reason to suppose that the particles of all bodies are in a state of rapid motion; the velocity of the atoms of water, for instance, is at least a mile per second. Similarly, the heat required to melt ice must be that needed to overcome the attraction of the particles for each other.

The basic principles on which are based the propositions in this memoir may be taken to be either of the following: (1) it is not possible by any means to produce an unlimited amount of mechanical forces; (2) all actions in nature can be explained in terms of attractive or repulsive forces between points, depending on the distances between those points. The two principles are essentially equivalent.

The objective of theoretical physics is to explain all natural phenomena in terms of motions of material particles exerting forces on each other. The first principle to be used is that of the conservation of vis viva (living force). For example, the work required to raise a body of weight m to a height h against the force of gravity g is mgh; a body falling through the same height attains the velocity υ = √2gh, hence its vis viva is . The same is valid in general when the particles act on each other with any kind of forces, depending only on their distance. In general, the sum of the "tensions" (potential energies) and vires vivae (kinetic energies) of any system of particles remains constant, and this we may call the principle of conservation of force. The application of the principle in mechanical theorems is discussed; for example, the motions of perfectly elastic bodies and the transmission of waves are considered.

Can we consider heat to be an equivalent of force? The experiments of Rumford, Joule and others indicate that heat is produced by motion, and we conclude that heat is really some kind of molecular motion, so that the same principles may be applied to it.

Krönig has recently proposed that the molecules of a gas do not oscillate about definite positions of equilibrium, but that they move with constant velocity in straight lines until they strike against other molecules, or against the surface of the container. I share this view, but believe that rotary as well as rectilineal motions are present, since the vis viva of the translatory motion alone is too small to represent the whole heat present in the gas. I assert that the translatory motion of the molecules will also be in a constant ratio to the motions of the constituents of the molecules among themselves, because the transfer of vis viva from translatory to internal motions, and conversely, by collisions, must eventually reach an equilibrium.

The molecular conditions that must be satisfied in order that a gas may behave as an ideal gas are indicated. Types of molecular motion occurring in the solid, liquid, and gaseous states are described, and a qualitative theory of evaporation is given.

In order to explain the relations of volumes of gases combining by chemical reactions, it is assumed that in simple gases two or more atoms are combined to form one molecule.

By considering collisions of molecules of mass m against the wall of a container of volume v, assuming that all n molecules move with velocity u, Clausius shows that the pressure of the gas is equal to (mnu²/3v). The absolute temperature is proportional to . The actual velocities of gas molecules can be calculated in this way; for example, at the temperature of melting ice, the velocity of an oxygen molecule is, on the average, 461 metres per second.

The ratio of the vis viva of translatory motion to the total vis viva is found to be equal to 3(γ' - γ)/2γ, where γ is the specific heat of the gas at constant volume (for unit volume) and γ' is specific heat at constant pressure. For air, γ'/γ = 1·421, and hence this ratio is 0·6315.

The theory of gases proposed by Joule, Krönig and Clausius (see preceding selection) was criticized on the ground that if the molecules really move great distances in straight lines, then two gases in contact with each other would rapidly mix, in contrast to experience. The objection may be answered by taking account of the fact that real gases are not "ideal" and therefore the portion of the time during which molecular forces act is not vanishingly small compared with the entire path of a molecule. A quantitative theory may be constructed if one assumes the molecules to be characterized by a certain distance ρ, such that if the centres of gravity of two molecules pass by each other at a distance greater than ρ the intermolecular repulsive forces do not come into play and there is only a slight deflection resulting from attractive forces, while if the distance is less than ρ the repulsive forces cause the molecules to rebound from each other. The spherical volume of radius ρ around the molecule is called the sphere of action of the molecule. We ask: how far on an average can a molecule move before its centre of gravity comes into the sphere of action of another molecule? This average distance is called the mean free path l.

It is shown that the mean length of path of a molecule is in the same proportion to the radius of the sphere of action as the entire space occupied by the gas, to that portion of the space which is actually filled up by the spheres of action of the molecules: . (It is assumed that all the molecules move with the same velocity.) For example, if this ratio is taken to be 1000: 1, and the average distance between molecules is λ = V^1/3, we find that l = 62λ. Although the number of molecules in a given volume of gas is as yet unknown, we must assume that it is quite large, and that λ is very small compared to our usual units of length. Thus it is plausible that the mean path is really quite small, and the objection cited is not valid.

In view of the current interest in the theory of gases proposed by Bernoulli (Selection 3), Joule, Krönig, Clausius (Selections 8 and 9) and others, a mathematical investigation of the laws of motion of a large number of small, hard, and perfectly elastic spheres acting on one another only during impact seems desirable.

It is shown that the number of spheres whose velocity lies between υ and υ + dυ is

where N is the total number of spheres, and α is a constant related to the average velocity:

If two systems of particles move in the same vessel, it is proved that the mean kinetic energy of each particle will be the same in the two systems.

Known results pertaining to the mean free path and pressure on the surface of the container are rederived, taking account of the fact that the velocities are distributed according to the above law.

The internal friction (viscosity) of a system of particles is predicted to be independent of density, and proportional to the square root of the absolute temperature; there is apparently no experimental evidence to confirm this prediction for real gases.

A discussion of collisions between perfectly elastic bodies of any form leads to the conclusion that the final equilibrium state of any number of systems of moving particles of any form is that in which the average kinetic energy of translation along each of the three axes is the same in all the systems, and equal to the average kinetic energy of rotation about each of the three principal axes of each particle (equipartition theorem). This mathematical result appears to be in conflict with known experimental values for the specific heats of gases.

The following theorem (virial theorem) is proved: let there be any system of material points in stationary motion. (Stationary motion means that the points are constrained to move in a limited region of space, and the velocities only fluctuate within certain limits, with no preferential direction.) Denote by X, Y, Z the components of the force acting on a particle at position x, y, z in space, and form the quantity

where the sum is extended over all points in the system. The mean value of this quantity is called the virial of the system. Then: the mean kinetic energy of the system is equal to its virial.

In the particular case when the points are confined to a volume v by an external pressure p, and the force ϕ(r) between two points depends only on their distance, r, the theorem indicates that the kinetic energy of the internal motions (which we call heat) is given by the equation

(This equation permits one to calculate the relation between pressure, volume, and temperature—the "equation of state"—if the force law ϕ(r) is given.)

The following sections are included:

• Boltzmann

• Dissipation of Energy

• Cultural Influence of the Dissipation of Energy

• The Statistical Nature of the Second Law

• The Eternal Return and the Recurrence Paradox

Please refer to full text.

The theory of transport processes in gases—such as diffusion, heat conduction, and viscosity—is developed on the basis of the assumption that the molecules behave like point-centres of force. The method of investigation consists in calculating mean values of various functions of the velocity of all the molecules of a given kind within an element of volume, and the variations of these mean values due, first, to the encounters of the molecules with others of the same or a different kind; second, to the action of external forces such as gravity; and third, to the passage of molecules through the boundary of the element of volume.

The encounters are analysed of molecules repelling each other with forces inversely as the nth power of the distance. In general the variation of mean values of functions of the velocity due to encounters depends on the relative velocity of the two colliding molecules, and unless the gas is in thermal equilibrium the velocity distribution is unknown so that these variations cannot be calculated directly. However, in the case of inverse fifth-power forces the relative velocity drops out, and the calculations can be carried out. It is found that in this special case the viscosity coefficient is proportional to the absolute temperature, in agreement with experimental results of the author. An expression for the diffusion coefficient is also derived, and compared with experimental results published by Graham.

A new derivation is given of the velocity–distribution law for a gas in thermal equilibrium. The theory is also applied to give an explanation of the Law of Equivalent Volumes, the conduction of heat through gases, the hydrodynamic equations of motion corrected for viscosity (Navier–Stokes equation), the relaxation of inequalities of pressure, and the final equilibrium of temperature in a column of gas under the influence of gravity.

According to the mechanical theory of heat, the thermal properties of gases and other substances obey perfectly definite laws in spite of the fact that these substances are composed of large numbers of molecules in states of rapid irregular motion. The explanation of these properties must be based on probability theory, and for this purpose it is necessary to know the distribution function which determines the number of molecules in each state at every time. In order to determine this distribution function, f(x, t) = number of molecules having energy x at time t, a partial differential equation for f is derived by considering how it changes during a small time interval as a result of collisions among molecules. If there are no external forces, and conditions are uniform throughout the gas, this equation takes the form (equation (16)):

where the variables x and x' denote the energies of two molecules before a collision, and ξ and (x + x' - ξ) denote their energies after the collision; ψ(x, x', ξ) is a function which depends on the nature of the forces between the molecules.

If the velocity distribution is given by Maxwell's formula

then the expression in brackets in the above equation will vanish, and the time-derivative of f(x, t) will be zero. This is essentially the result already obtained in another way by Maxwell: once this velocity distribution has been reached, it will not be disturbed by collisions.

With the aid of the partial differential equation for f, we are able to go further and prove that if the distribution of states is not Maxwellian, it will tend toward the Maxwellian distribution as time goes on. This proof consists in showing that a quantity defined in terms of f,

can never increase but must always decrease or remain constant, if f satisfies the above differential equation. [This statement is now known as the H-theorem.] E must approach a minimum value and remain constant thereafter, and the corresponding final value of f will be the Maxwell distribution. Since E is closely related to the thermodynamic entropy in the final equilibrium state, our result is equivalent to a proof that the entropy must always increase or remain constant, and thus provides a microscopic interpretation of the second law of thermodynamics.

To clarify the reasoning involved in these proofs, we replace the continuous energy variable x by a discrete variable which can take only the values ε, 2∊, 3∊, …; we then show that the same result can be derived by taking the limit ∊ → 0.

The differential equation for f is also given for the case when the velocity distribution may vary from one place to another, and all directions of velocity are not equivalent (equation (44)). For the special case of intermolecular forces varying inversely as the fifth power of the distance, this equation has a simple exact solution, and the coefficients of viscosity, heat conduction, and diffusion may be calculated. The results are essentially the same as those found by Maxwell [Selection 1].

The above results are generalized to gases composed of polyatomic molecules.

The equations of motion in abstract dynamics are perfectly reversible; any solution of these equations remains valid when the time variable t is replaced by -t. Physical processes, on the other hand, are irreversible: for example, the friction of solids, conduction of heat, and diffusion. Nevertheless, the principle of dissipation of energy is compatible with a molecular theory in which each particle is subject to the laws of abstract dynamics.

Dissipation of energy, such as that due to heat conduction in a gas, might be entirely prevented by a suitable arrangement of Maxwell demons, operating in conformity with the conservation of energy and momentum. If no demons are present, the average result of the free motions of molecules will be to equalize temperature-differences. If we allowed this equalization to proceed for a certain time, and then reversed the motions of all the molecules, we would observe a disequalization. However, if the number of molecules is very large, as it is in a gas, any slight deviation from absolute precision in the reversal will greatly shorten the time during which disequalization occurs. In other words, the probability of occurrence of a distribution of velocities which will lead to disequalization of temperature for any perceptible length of time is very small. Furthermore, if we take account of the fact that no physical system can be completely isolated from its surroundings but is in principle interacting with all other molecules in the universe, and if we believe that the number of these latter molecules is infinite, then we may conclude that it is impossible for temperature-differences to arise spontaneously. A numerical calculation is given to illustrate this conclusion.

Loschmidt has pointed out that according to the laws of mechanics, a system of particles interacting with any force law, which has gone through a sequence of states starting from some specified initial conditions, will go through the same sequence in reverse and return to its initial state if one reverses the velocities of all the particles. This fact seems to cast doubt on the possibility of giving a purely mechanical proof of the second law of thermodynamics, which asserts that for any such sequence of states the entropy must always increase.

Since the entropy would decrease as the system goes through this sequence in reverse, we see that the fact that entropy actually increases in all physical processes in our own world cannot be deduced solely from the nature of the forces acting between the particles, but must be a consequence of the initial conditions. Nevertheless, we do not have to assume a special type of initial condition in order to give a mechanical proof of the second law, if we are willing to accept a statistical viewpoint. While any individual non-uniform state (corresponding to low entropy) has the same probability as any individual uniform state (corresponding to high entropy), there are many more uniform states than non-uniform states. Consequently, if the initial state is chosen at random, the system is almost certain to evolve into a uniform state, and entropy is almost certain to increase.

Poisson attempted to show that a mechanical system is stable, in the sense that it will eventually return to a configuration very close to its initial one. He did not think that all solutions of the equations of dynamics would be stable: stability may depend on the initial conditions.

It is proved that there are infinitely many ways of choosing the initial conditions such that the system will return infinitely many times as close as one wishes to its initial position. There are also an infinite number of solutions that do not have this property, but it is shown that these unstable solutions can be regarded as "exceptional" and may be said to have zero probability.

The advocates of the mechanistic conception of the universe have met with several obstacles in their attempts to reconcile mechanism with the facts of experience. In the mechanistic hypothesis, all phenomena must be reversible, while experience shows that many phenomena are irreversible. It has been suggested that the apparent irreversibility of natural phenomena is due merely to the fact that molecules are too small and too numerous for our gross senses to deal with them, although a "Maxwell demon" could do so and would thereby be able to prevent irreversibility.

The kinetic theory of gases is up to now the most serious attempt to reconcile mechanism and experience, but it is still faced with the difficulty that a mechanical system cannot tend toward a permanent final state but must always return eventually to a state very close to its initial state [Selection 5]. This difficulty can be overcome only if one is willing to assume that the universe does not tend irreversibly to a final state, as seems to be indicated by experience, but will eventually regenerate itself and reverse the second law of thermodynamics.

Poincaré's recurrence theorem [Selection 5] shows that irreversible processes are impossible in a mechanical system. A simple proof of this theorem is given.

The kinetic theory cannot provide an explanation of irreversible processes unless one makes the implausible assumption that only those initial states that evolve irreversibly are actually realized in nature, while the other states, which from a mathematical viewpoint are more probable, actually do not occur. It is concluded that it is necessary to formulate either the second law of thermodynamics or the mechanical theory of heat in an essentially different way, or else give up the latter theory altogether.

Poincare's theorem, on which Zermelo's remarks are based [Selection 7], is clearly correct, but Zermelo's application of it to the theory of heat is not. The nature of the H-curve (entropy vs. time) which can be deduced from the kinetic theory is such that if an initial state deviates considerably from the Maxwell distribution, it will tend toward that distribution with enormously large probability, and during an enormously long time will deviate from it by only vanishingly small amounts. Of course if one waits long enough, the initial state will eventually recur, but the recurrence time is so long that there is no possibility of ever observing it.

In contradiction to Zermelo's statement, the singular initial states which do not approach the Maxwell distribution are very small in number compared to those that do. Consequently there is no difficulty in explaining irreversible processes by means of the kinetic theory.

According to the molecular-kinetic view, the second law of thermodynamics is merely a theorem of probability theory. The fact that we never observe exceptions does not prove that the statistical viewpoint is wrong, because the theory predicts that the probability of an exception is practically zero when the number of molecules is large.

Boltzmann has conceded [Selection 8] that the commonly accepted version of the second law of thermodynamics is incompatible with the mechanical viewpoint. Whereas the author holds that the former, a principle that summarizes an abundance of established experimental facts, is more reliable than a mathematical theorem based on unverifiable hypotheses, Boltzmann wishes to preserve the mechanical viewpoint by changing the second law into a "mere probability theorem", which need not always be valid.

Boltzmann's assertion, that the statistical formulation of the second law is really equivalent to the usual one, is based on postulated properties of the H-curve which he has not proved, and which seem to be impossible. His argument that any arbitrarily chosen initial state will probably be a maximum on the H-curve, if it were valid, would prove that the H-curve consists entirely of maxima, which is nonsense.

The only way that the mechanical theory can lead to irreversibility is by the introduction of a new physical assumption, to the effect that the initial state always corresponds to a point at or just past the maximum on the H-curve; but this would be assuming what was supposed to be proved.

The second law of thermodynamics can be proved from the mechanical theory if one assumes that the present state of the universe, or at least that part which surrounds us, started to evolve from an improbable state and is still in a relatively improbable state. This is a reasonable assumption to make, since it enables us to explain the facts of experience, and one should not expect to be able to deduce it from anything more fundamental.

The applicability of probability theory to physical situations, which is disputed by Zermelo, cannot by rigorously proved, but the fact that one never observes those events that theoretically should be quite rare is certainly not a valid argument against the theory.

One may speculate that the universe as a whole is in thermal equilibrium and therefore dead, but there will be local deviations from equilibrium which may last for the relatively short time of a few eons. For the universe as a whole, there is no distinction between the "backwards" and "forwards" directions of time, but for the worlds on which living beings exist, and which are there-fore in relatively improbable states, the direction of time will be determined by the direction of increasing entropy, proceeding from less to more probable states.

The following sections are included:

• INTRODUCTION

• BOYLE AND LINUS

• BERNOULLI: GENIUS WITHOUT A GADFLY

• HERAPATH: GADFLY WITHOUT A GENIUS

• CLAUSIUS AND BUYS-BALLOT

• MAXWELL, LOSCHMIDT, BOLTZMANN, AND GUTHRIE

• BOLTZMANN AND LOSCHMIDT

• BOLTZMANN, CULVERWELL, AND BURBURY

• BOLTZMANN AND ZERMELO

• NOTES

• REFERENCES

A recurrent theme in the physical science of the past three centuries has been provided by the program attributed to ISAAC NEWTON: from the phenomena of nature to find the forces between particles of matter, and from these forces to explain and predict other phenomena. The success or failure of this program as a guide for scientific research can be assessed by considering some of the cases in which it has been applied: NEWTON'S own theory of gas pressure, the BOSCOVICH theory of interatomic forces, the LAPLACE theory (short-range attractive forces and long-range repulsive forces), the billiard-ball model used in the elementary kinetic theory of gases, the MAXWELL r^-5 repulsive force, and the VAN DER WAALS equation. A more detailed examination is presented of the rise and fall of the "Lennard-Jones potential" in relation to calculations and experimental data on virial coefficients and transport properties of gases, solid state properties, and the quantum theory of interatomic forces.

The history of the subject suggests that the hypothetico-deductive model of scientific method has not been followed in practice, since the reasons for adopting or rejecting new interatomic force laws are often not simply related to the success or failure of the force law in calculations of gas properties.

At present there is serious doubt as to whether it is worthwhile trying to establish a single "realistic" force law for the interaction between two atoms or molecules. It may be more fruitful to abandon this program and to choose force laws instead on the basis of their convenience in a particular mathematical theory of the properties of matter.

The First Scientific Revolution, dominated by the physical astronomy of Copernicus, Kepler, Galileo and Newton, established the concept of a "clockwork universe" or "world-machine" in which all changes are cyclic and all motions are in principle determined by causal laws. The Second Scientific Revolution, associated with the theories of Darwin, Maxwell, Planck, Einstein, Heisenberg and Schrodinger, substituted a world of process and chance whose ultimate philosophical meaning still remains obscure. This paper will sketch one aspect of the change in physical theory that occurred after 1800: the acceptance of randomness at the atomic level and its relation to the acceptance of irreversibility as a characteristic property of natural phenomena…

One of the most exciting moments in a scientist's work is the sudden realization that a substantial body of knowledge or technique developed in one discipline can be directly applied to a fundamental problem in another discipline. Such an application often has a marked effect on the growth of the science which can receive the transplanted material; think of Max Born's discovery that Heisenberg's strange arrays of transition amplitudes were in fact matrices for whose manipulation an elaborate mathematical theory already existed. Less common but perhaps just as remarkable are the examples where the donor science gains more from the interaction than the recipient; but this is just what happened when Artur Rosenthal and Michel Plancherel independently found that existing mathematical theory could quickly dispose of a question raised by physicists. Their decisive negative answer to the question: Can a mechanical system eventually pass through every point on the energy surface in its phase space?" provoked little more than politely-stifled yawns from the physicists, but introduced a major new branch of mathematical research, ergodic theory…

The following sections are included:

• Reduction, Statistics, and Irreversibility

• Beginnings of Statistical Mechanics

• Victory

• Entropy and Probability

• Phase Transitions

• Notes

• References

The bibliography below lists articles and books published from 1965 through 2001 (with occasional exceptions) on the history of kinetic theory, statistical mechanics, thermodynamics, the nature of heat, and the physical properties of gases. Reprints and edited collections of original sources are included, as well as historical commentaries. The "target period" during which the original research was done covers a quarter of a millenium: from Robert Boyle's work on gas pressure around 1660 to the beginnings of quantum theory at the end of the 19th century (a century that, in accordance with the practice of historians, is considered to end in 1914)…

The following sections are included:

• Index to Parts I, II, III