510 Prof. P. Ehrenfest on Adiabatic Invariants 



Planck's hypothesis of energy elements and its generaliza- 

 tions destroy this basis ; they introduce, as it may be 

 expressed, a weight depending on q, p, and a 



G(q,p,a); (28) 



all regions of the /jl space have the " weight zero " (are 

 "forbidden") with the exception of: the discontinuously 

 distributed " allowed" regions, the position of which depends 

 on the value of the parameters a *. The latter circumstance 

 is of particular importance. 



So we arrive at the following problem : In what manner 

 must the choice of the "weight-function" Gr(q, p, a) — in other 

 words, the choice of the " allowed " regions — be limited, espe- 

 cially in their dependence on the a, in order that Boltzmann's 

 equation (26) may remain valid? 



I have treated this problem first in a special case *j% then 

 generally J. 



For molecules of one degree of freedom (harmonically 

 and unharmonically vibrating resonators) I could wholly 

 solve the question. The result I arrived at § may be expressed 

 in the language of this article in the following form : 



An ensemble of such-like molecules (resonators) will fulfil 

 Boltzmann's relation between entropy and probability if, and 

 only if, the allowed motions are determined by means of the 

 adiabatically invariant condition 



2T 



v 



= It dqdp = fixed numerical values fl l9 12. . . . (29) || 



Planck's hypothesis on the elements of energy for har- 

 monically vibrating oscillators and its generalization by 

 Debye satisfy this condition ; in this case 1? 12 2 . . . are taken 

 equal to 0, A, 2h, . . . H (comp. § 5, equation (14). 



* The form and dimensions of the " allowed " ellipses in the q-p 

 diagram of a Planck's resonator are altered if the inertia and elasticity 

 of the resonator are changed. In an analagous way the " allowed" 

 ellipses, belonging to the principal modes of vibration of a " Hohlraum " 

 or of the lattice of a crystal, are altered by a compression, 



t Paper A, § 5. } Paper D. 



§ Paper D, § 7, remark. 



|| L. c. § 7, this invariant is noted by i. 



1| That Planck's hypothesis on the energy elements is in harmony with 

 the Second Law (and with the adiabatic hypothesis) has come about in 

 the following way : in the deduction of his theory of radiation Planck 

 at a certain moment puts the elements of energy (which were not yet 

 determined before) equal to hv in order to make his radiation formula 

 correspond with the displacement law of W. Wien (cf. Planck, Vorles- 

 ungen iiber JVarmestrahlung, 1st edition, 1906, p. 153, eq. 226). Com- 

 pare also the other quantum formulas, paper J), § 6. 



