References
- Ehrenfest , P. and Ehrenfest , T. 1959 . “ Begriffliche Grundlager der Statistischen Auffassung in der Mechanik ” . In The Conceptual Foundations of the Statistical Approach in Mechanics , Edited by: Moravcsik , M. J. Cornell University Press . Encykl. d. math. Wiss. 4, 2II, Heft 6, No. 10a. (19)English translation by
- The instantaneous state of a system is represented by a point of 2γN-dimensional phase space, if the gas consists of N particles, each having γ degrees of freedom. Moreover it will be assumed that the external force field does not change in the course of time; cf. P and T. Ehrenfest, loc. cit. No. 9a and b
- Plancherel , M. Prof. 1912 . Archives des sc. phys. et nat. (4) , 33 : 254 – 55 . contains no indication of the method of proof. Herr Prof. Plancherel was so kind as to communicate to me an outline of his proof. It appears that his ideas move in part in the same direction as mine, but they are different in some essential points. [See translator's note following footnote 19]The present work had already been edited when I learned from a notice in Fortschritte der Physik that the same matter was the subject of a report of Herr (Freiburg, Switzerland) to the Berne meeting of the Swiss Physical Society (March 9, 1912). The report of the meeting
- In the following it is assumed that γN > 1
- Γ' need not coincide with Γ; for example, if the molecules have finite extension and are impenetrable then Γ' will have gaps
- With non-vanishing kinetic energy
- One can always do this in the case of a gas
- Brouwer , L. E. J. 1911 . Math. Ann. , 70 : 161 For the conservation of the dimension number and of the region under a reversibly unique and continuous mapping see
- 1912 . Math. Ann. , 71 : 305 314
- 1912 . Math. Ann. , 72 : 55
- Moreover, at no point of F(K1) can the tangent plane be perpendicular to Γ-space. For, in this case, at least one of the first derivatives of E must become infinite there, which, according to the above, is forbidden for all points of F(K1
- The boundary of R' will be included in R'
- one chooses, for the following, S and R' not as large as possible but rather such that S is contained in the interior of a larger “cube” S and accordingly R' in the interior of a larger similar region R' (where the smallest distance N of the exterior surfaces of R' and R is greater than zero)
- This can have discontinuities at instants when collisions take place, if these collisions produce discontinuities in the momenta. The parts of C lying in R' or R are still always continuous
- It is explicitly emphasized that for finite but unbounded times the mapping A can generally have only one-sided continuity but not reversible continuity. For: neighboring time moments must to be sure correspond to neighboring points of R', but not conversely. I.e., there corresponds also in this case to each finite accumulation value tW of {ti} the only accumulation point of the corresponding (Pi); but conversely, here, an accumulation point PW of a series {Pi} lying in R' need not correspond to an accumulation point of the corresponding {ti)
- Schoenflies , A. 1908 . Die Entwicklung der Lehre von den Purktmannigfaltig-keiten II . Jahresb. d. Dtsch. Math. -Ver., Erganz. , Bd. 2 : 165 See the work of L.E.J. Brouwer first mentioned in footnote 8; for the present case it is sufficient to use the much simpler consideration given by
- Schoenflies , A. 153 loc. cit
- Poincare , H. 1890 . Acta Math. , 13 : 69
- Cantor , G. 1882 . Math. Ann. , 20 : 117
- Baire , R. 1905 . Lecons Sur les Fonctions discontinues 78 Paris 106 A set of the first category in a region G is a set which can be constructed from denumerably many sets nowhere dense in G. Each set that is not of the first category in G is then a set of the second category in G
- Baire , R. 79 lac. cit. 106
- 1913 . Annalen der Physik [4] , 42 : 1061 Translator's note: The proof of Plancherel mentioned in footnote 3 was published in The translation follows