babilities of all energies are equal for any one molecule. If m is the mass of a molecule, the most probable number of molecules with speeds between u and u+du is This is the Boltzmann-Maxwell distribution of speed for a system of monatomic molecules moving in one plane. 53. To obtain the Boltzmann-Maxwell distribution for molecules moving in three dimensions, Boltzmann finds it necessary to make a different assumption with regard to the à priori probabilities. He assumes, in fact, that if u, v, w be the velocities along the axes of co-ordinates, all values of u, v, w are à priori equally probable. The problem of determining the à posteriori most probable distribution therefore reduces to finding the minimum of For the general case of molecules with r degrees of freedom in a field of force, he assumes that all values of the co-ordinates and momenta p are à priori equally probable. In This is the assumption that would be fulfilled if the values were selected by drawings from an urn containing tickets, each ticket having a set of values of pi q, inscribed on it, and the number of tickets in which these values lie between the limits dp dq, being measured by the product of the multiple differential dp dq, into a constant. the case of a mixture of gases there would have to be a number of urns equal to the number of gases, and the number of tickets drawn from each urn would have to equal the number of molecules of the gas in question in the mixture. The final result is that the à posteriori most probable distribution is that for which the function Ως flog ƒ dp ... dq,. (55) is a minimum, referring to the different kinds of molecules in a mixture of several gases. The expression - differs by a constant from Boltzmann's Minimum Function. We have seen in § 43 that, when there are collisions between the molecules, this function always tends to a minimum until the BoltzmannMaxwell distribution is attained, and the present investigation therefore shows that the gas tends to pass from distributions of lesser probability to distributions of greater probability, until it attains the most probable distribution of all-namely, the Boltzmann-Maxwell distribution." Finally, Boltzmann proves that the function is proportional to the entropy (plus a constant), thus affording a verification of the theorem that the entropy of a system tends to a minimum. The identification of Boltz mann's minimum function with the entropy is established more briefly by Burbury in his recent paper, to be discussed shortly.1 54. The particular assumption as to the law of à priori probability precludes the above investigations from furnishing a complete proof of the Boltzmann-Maxwell Law. In a subsequent paper 2 Boltzmann has removed this restriction, and has considered the à posteriori probabilities corresponding to any assumed law of à priori probability. In other words, we start with a large number (N) of molecules having a given distribution of energy, and from them a smaller number (n) are selected, and their mean energy is found to have a certain value which may be either the same or different from that of the original N. It is required to find the most probable law of distribution in the n selected molecules, or, generally, the probability of any given distribution. Boltzmann first considers the case where the original molecules follow the Boltzmann-Maxwell Law for two dimensional space, and points out the necessary modifications for space of three dimensions. In the general case, supposing f, f2, fp to denote the à priori probabilities of a molecule having energies, 2, . . . pɛ, the à posteriori probability of a combination in which the numbers of molecules having these energies are wow,... w, respectively is proportional to 2, where The approximate expression for o! now gives log =Σw; logfiΣw log w; +constant and Boltzmann finds the following results. Wi If the mean energy of the selected n molecules is equal to the mean energy of the original N, the most probable distribution of energy in the latter is identical with the distribution in the former. If, however, the mean energy of the smaller number is unequal to that of the larger, the most probable distribution is that given by the form Boltzmann's investigation was probably an attempt to arrive at the Boltzmann-Maxwell distribution as the ultimate result of a number of successive processes such as the above, independently of the initial distribution. This has recently been actually accomplished by Burbury by the application of a different method as follows: 3 55. Burbury bases his investigation on a generalisation of the theory of Least Squares, which asserts that if we regard the variations of a series of n quantities x1, x2, . . . x, as being each the result of an infinite number N of independent simultaneous increments divided each by N, then the On the Law of Distribution of Energy,' Phil. Mag., January 1894. 2 Weitere Bemerkungen über einige Probleme der mechanischen Wärmetheorie,' Sitzb. der k. Wiener Akad., lxxviii. (ii.), June 1878. The second part of the paper deals with the equilibrium of a gas under gravity, and is less interesting. 3 Phil. Mag., January 1894. chance that the values of x1, x2 shall lie between c1 and c1 + de 1, c2 and c2+de, &c., is proportional to an expression of the form where S is a certain homogeneous quadratic function of the c's, and T a constant. This result, which for a single variable leads to the well-known error-law, is independent of the original law of distribution of the increments, provided that positive and negative values of these increments are equally probable. Taking S as proportional to the kinetic energy of a system, and supposing the number of such systems to be very great, Burbury next shows that if a redistribution of S among the systems is effected in a certain way, the ultimate result will be the Boltzmann-Maxwell distribution, and this will remain unaffected by any further redistribution. The method of redistribution is such that energy is conserved in the final result, but not in the intermediate processes, and Burbury suggests that the process of redistribution of energy between the molecules may be effected by waves transmitted through the ether. The proof requires us to assume that these waves satisfy the principle of superposition, otherwise the law cannot be permanent. The author, however, claims that the method is applicable to systems in which no group of molecules is ever free from the action of other parts of the system, and for which those proofs of the BoltzmannMaxwell Law treated in Sections I., II. of this Report fail. Burbury then finds the expression for Boltzmann's minimum function, and calling this B he verifies that the entropy of the system is equal to 2B/n (plus a constant). The whole treatment is very powerful and suggestive, and the paper opens up a wide field for discussion and speculation. 56. The assumption in the first place that each molecule is capable of assuming only a discrete instead of a continuous series of different states, the number of these states being made infinite in the limit, forms the basis of Boltzmann's proof of his Minimum Theorem for polyatomic gas-molecules. Natanson,2 taking Boltzmann's starting-point of a number of systems whose energies can only have one of a series of discrete values and employing equations (43) (44) above, has worked out the final distribution of energy among the molecules on the supposition that interchange of energy takes place by collisions, and he has also determined the rate at which the system approaches the Boltzmann-Maxwell distribution. He gives a complete solution of the problem for the particular case where p=3, and shows that only analytical difficulties prevent the method from being applied to higher values of p. however, p is made infinite, the results agree with those found by Boltzmann and by Tait (§ 44). P. 153. When, Neuer Beweis zweier Sätze,' Sitzber. der k. Wiener Akad., xcv. (ii.), Jan. 1887, 2 Ueber die Geschwindigkeit mit welcher Gase den Maxwell'schen Zustand erreichen,' Annalen der Physik und Chemie, xxxiv. (1888). The Connection between the Virial Equation, the Second Law, and the Boltzmann-Maxwell Law. 57. The virial equation cannot be used to prove the Second Law. When applied to a perfect gas it leads to the equation (59) From this equation, combined with the Second Law, certain properties of perfect gases, with which we are all familiar, are deduced in treatises on thermodynamics. These are that the latent heat of expansion is equal to the pressure, that the difference of specific heats is equal to R, and so on. Hence corresponding conditions must hold good in molecular thermodynamics in order that the Second Law may be consistent with the virial equation. A general condition under which the Second Law is true was originally found by R. C. Nichols,' and has recently been put into a somewhat different form by Burbury.2 If x or U denote the potential energy, this condition may be written 1/d 1 (1 x - dx) = d dx dt dv (60) where the bar denotes average values. 3 Burbury has rightly pointed out that the methods of Clausius and Szily require such a condition to be satisfied in order that they may give the Second Law. The condition comes in when we try to assign a meaning to the quasi-period i,' without which meaning the result is unintelligible. Burbury shows that if we assume i=v1T 1 (i.e., a definite time), then Χ must be a function of v only, so that and this is a special case of Nichols' condition. It is also to be observed that in the Clausius-Szily method the averages are time-averages. This, if not an objection, is at least a disadvantage, since it does not show the relation between the Second Law and the Boltzmann-Maxwell Law, in which averages are taken over a large number of molecules in a 'special state' of permanent distribution. 58. That relation forms the subject of a very recent paper by Burbury,5 which is a development of his second letter to Nature.' The proof of the Second Law on the assumption of the BoltzmannMaxwell Law has long been known, and was given by Boltzmann as early 'On the Proof of the Second Law of Thermodynamics,' Phil. Mag., 1876 (i.), P. 369. 2 Nature, December 14, 1893; January 11, 1894. 3 Ibid., December 14, 1893. Report on Thermodynamics,' Part I. Section 1, Cardiff Report, 1891, p. 88. as 1871, and by Burbury in 1876.2 Burbury now treats the converse problem of determining the law of distribution from the Second Law. He finds that if the law of distribution of the co-ordinates x1, x2, given by the expression ... xn be and if the law of distribution of co-ordinates and velocities be given by Fdo do' where do'=di de then OQ/T will be a complete differential if either (i) F= function of (r—U)/T=${(r−U)/T} say (which does not vanish for infinite values of the variables), or where is the kinetic energy, and U the potential energy of a molecule. Burbury says: 'And since F and ƒ must vanish for all infinite values of the variables, we are led to F=C exp-λ U f=C' exp -λ T where A is some positive numerical quantity. . . T 59. Now this is obviously a solution, but it is not the only solution, and I think the real inferences are slightly different from those he has drawn. They are sufficiently interesting to be treated in detail, and they are intimately connected with another point which at first suggests an objection to the proof, namely, that do d' is the multiple differential of the co-ordinates and velocities, and is therefore not in general an invariant like the multiple differential of the co-ordinates and momenta. In § 15 of his paper Burbury states that this does not matter. 'If they' (i.e., do and do') 'do vary, that is, in effect, if the limits of integration vary, the assumption F= {(U+7)/T} will still make OQ/T a complete differential.' Now if y1, y, denote the generalised momenta corresponding to x, the Jacobian n ¿„) —J(X1, X2 will in general be a function of the co-ordinates x1, . . . x,, and its form will depend on the choice of co-ordinates. Hence, if Burbury's proof be correct, we have really shown that the Second Law will be satisfied if the distribution be determined by any expression of the form where by suitable choice of co-ordinates the form of J may be varied quite arbitrarily. And by § 22 this expression represents, not necessarily the Boltzmann-Maxwell distribution, but a distribution satisfying the ''Analytische Beweise des zweiten Hauptsatzes,' &c., Sitzber. der k. Wiener Akad., lxiii. (ii.), 1871, p. 712. 2 Phil. Mag., January 1876, p. 61. 1894. H |