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Traces of Primitive Man in North of Ireland
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Some Ancient British Remains on Clifton Bristol Nat. Soc.
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Report on the Present State of our Knowledge of Thermodynamics. By G. H. BRYAN.

PART II. THE LAWS OF DISTRIBUTION OF ENERGY AND THEIR LIMITATIONS. (With an Appendix by Prof. LUDWIG BOLTZMANN.)

INTRODUCTION.

1. THIS Report deals primarily with the Boltzmann-Maxwell Law and Maxwell's Law of Partition of Kinetic Energy, which form the basis of the Kinetic Theory of Gases. One of the main points kept in view has been to show where to draw the line between dynamical systems which do and dynamical systems which do not satisfy the laws in question.

1

Since the appearance of the first Report several papers have appeared which have thrown a somewhat different light on certain aspects of the subject, and have thus materially assisted in crystallising our knowledge of this branch of Thermodynamics into a definite form. In order to prevent unnecessary controversy, I have, as far as possible, avoided drawing conclusions from arguments of a vague and theoretical nature. Where, however, results are based on purely mathematical calculations they must be understood to be liable to modification should further examination

show the calculations to be faulty or inaccurate. It is necessary to mention this, as one of the investigations cited in Part I. has subsequently been found to be incorrect, with the result of very materially altering our views on the question at issue.2

A great advance in the present subject is due to the extension of the use of generalised co-ordinates, by which greater generality has been given to results and the analysis much simplified, as a comparison of Boltzmann's early papers with modern writings abundantly testifies. A further simplification has been effected by the use of the Jacobian notation.

For convenience I have in places written exp-hE for exp(-hE) or e-hE. The present Report is divided into three sections. In Section I. Maxwell's Law of Partition of Energy is regarded in the aspect of a general dynamical theorem, without reference to any particular applications, and without taking into account the effect of collisions. Section II. treats of the Boltzmann-Maxwell Law for a system of bodies colliding with one another indiscriminately, and partaking of the nature of gas molecules. Section III. deals briefly with certain researches connecting the BoltzmannMaxwell Law with the Theory of Probability, the Virial Equation, and the Second Law of Thermodynamics.

SECTION I.-NON-COLLIDING SYSTEMS.

Clerk Maxwell's Investigations.

2. Clerk Maxwell's investigations3 have played such a prominent part in the literature of the Kinetic Theory that I think it desirable to recapitulate his paper briefly, so as to show more clearly what assumptions he made and how much he actually proved.

1 Cardiff Report, 1891, pp. 85–122.

2 The results stated in the first twelve lines of Part I. Section III. § 44 are now known to be erroneous. See also §§ 36, 37 below.

On Boltzmann's Theorem of the Average Distribution of Energy in a System of Material Points,' Tran. Camb. Phil. Soc., xii. 1879.

It may be safely asserted that a large portion of our progress in the present subject has been made, first, by showing that Maxwell's demonstrations are faulty and unsatisfactory, and subsequently by discovering fresh methods of proof, which, while leading to the same general conclusions, show more clearly the limitations and conditions under which these conclusions hold good. In this process of destruction and reconstruction a large amount of literature has accumulated, and I shall endeavour in the present Report to unearth from the general mass the main results to which these papers tend.

3. Maxwell claims that his theorem is applicable to any dynamical system whatever. 'The material points may act on each other at all distances and according to any law which is consistent with the equation of energy, and they may also be acted on by any forces external to the system, provided these also are consistent with that law. The only assumption which is necessary for the direct proof is that the system, if left to itself in its actual state of motion, will sooner or later every phase which is consistent with the equation of energy.'

pass through 4. Instead, however, of a single system, Maxwell considers a large number of independent dynamical systems, similar in every respect, each defined by its n co-ordinates (91,... q) and the corresponding n momenta (P,... Pn). Each system is capable of passing through every phase which is consistent with the equation of energy, and it is thus assumed that all the systems have the same energy. In the case of a free system unacted on by external forces, the six components of linear and angular momentum remain constant, and Maxwell assumes that these are the same for all systems.

5. Taking the 'action' of the system during any period of the motion, he employs this function to establish the determinantal relation between the multiple differentials of the co-ordinates and momenta at the beginning. and end of any interval, and thus establishes the relation

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from which he deduces that, if the energy E be kept constant, so that p can be expressed as a function of the n-1 other p's, then

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6. Hence it follows that, if the systems are so distributed that the number which initially have their co-ordinates and momenta within the limits of the multiple differential dp2 . . . dpn dq. . . . dq, be

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their total energies being all equal and C a constant, then the same expression gives the law of distribution at any subsequent time. Maxwell says: We have found one solution of the problem of finding a steady distribution; whether there may be other solutions remains to be inves tigated.'

1894.

1 Loc. cit., p. 518

...

7. He next assumes that the momenta (now denoted by a1, may be so chosen as to reduce the kinetic energy to a sum of squares, or

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With this assumption, he integrates (3) with respect to the momenta, and finds by Dirichlet's method

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where b11a. Hence he infers that if ka,2 is the part of the kinetic energy arising out of the momentum a,, then the number of systems in a given configuration, in which k, lies within limits differing by dk,, is г(n) (E-V-k)(-3), T({})г[}(n−1)] (E—V)

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the maximum value being of course equal to T, because the portions of the kinetic energy due to the other momenta cannot be negative. Hence Maxwell infers that the average kinetic energy corresponding to any one of the variables is the same for every one of the variables of the system.' This result is commonly called Clerk Maxwell's Theorem.

8. In Part II. of the paper Maxwell deals with a free system, consisting of n particles not acted on by external forces. For such a system not only the energy but also the velocity-components of the centre of mass and the components of angular momentum round this point in any three fixed directions will be constant throughout the motion. Maxwell therefore assumes them the same for every system. Under these circumstances the 3n momentum-components of the system are not all independent, but seven of them can be expressed in terms of the rest by means of the seven equations of condition, and the law of permanent distribution is expressible in terms of the multiple differential of the 3n co-ordinates and 3n-7 of the velocity components of the particles. The algebra is very long and laborious, and need not be examined in detail here. The objections to Maxwell's investigations can be much more easily discussed and criticised with reference to the simpler case considered in Part I., and the law of distribution in a free system can be treated more simply by alternative methods (vide §§ 16-18, § 45, and appendices A, B below).

The Assumption that the System passes through every Phase consistent with the Equation of Energy.

9. This assumption probably presents greater difficulties than any other part of the Kinetic Theory, and it is therefore advisable to commence by stating under what circumstances it requires to be made

2

The whole of Maxwell's demonstration, and most of the investigations of Boltzmann,' Watson, and other writers on the same subject, are based on the consideration of an infinitely large number of independent systems, similar in every respect, whose co-ordinates and momenta at any instant are distributed according to a fixed law, and the object is to find what this distribution must be in order that it may be independent of the time and unaffected by the motions of the systems. I cannot see that these investigations anywhere assume that each individual system passes through every possible phase. At each instant there must be some systems in every possible phase; but a distribution would obviously be permanent and very much so indeed-in which each system always remained in the same phase, and never passed into any other phase.

10. The assumption first confronts us when we attempt to pass from the consideration of a large number of systems to that of a single system, i.e., if, having investigated the result of averaging the distributions of energy at a given instant over the different systems, we wish to infer similar properties for the corresponding time-averages for any one of the systems.

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It is easy enough to suggest systems to which the assumption is inapplicable. Most of the test cases' which have been suggested as disproving the law, and which will be considered later on, are instances of such systems. Lord Rayleigh has suggested as another instance an elastic ball moving on a table having a circular boundary, at which it is reflected. If, instead of taking a single particle, Lord Rayleigh had supposed the table covered with such particles initially distributed uniformly over its area, and projected in such a manner that at any point as many particles were moving in one direction as in another, he would find these same conditions satisfied at any subsequent time, and this is, to my mind, all that Maxwell proves.

11. It is far less easy to suggest any simple system which does satisfy the assumption. The tracing point of a Lissajous' pendulum curve-tracer, considered by Boltzmann, or in other words a particle whose equations of motion are

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possesses when a, b are incommensurable the property of passing sooner or later through every point within a certain rectangle, but it does not possess the other necessary property of passing through any point in every possible direction in succession. This may be easily seen for the simplest case when a is nearly but not quite equal to b: here the path is nearly elliptical, and there are only two possible directions at any point. Hence, in order to satisfy the assumption, Boltzmann requires a thin elastic cylinder to be placed perpendicularly to the plane of motion, so that the particle may have its direction of motion changed each time it strikes and rebounds from the cylinder. And this introduces collisions into the

problem.

Ueber die Eigenschaften monocyklischer und anderer damit verwandter Systeme, Journal für die reine und angewandte Mathematik, xcviii. p. 68.

⚫ Analogien des zweiten Hauptsatzes,' ibid., c. pp. 206, 207, and other papers. 2 Kinetic Theory of Gases, new edition, p. 23.

Phil. Mag., April 1892, p. 357.

Ueber die mechanischen Analogien des zweiten Hauptsatzes der Thermodynamik,' Journal für die Mathematik, c. p. 203.

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