Axiomatics of classical statistical mechanics by Rudolf Kurth, I. N. Sneddon, S. Ulam, M. Stark

By Rudolf Kurth, I. N. Sneddon, S. Ulam, M. Stark

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8 still holds when the partial 00 sums of ΣΛ(^) a r e bounded by any function integrable over L. 12. I n the following the indefinite Lebesgue integral is to be considered. Let A be an additive class of subsets of Rn. Then a correspondence of real numbers F(S) to the sets 8 of A is called a set function defined in A. ,Sk of A, no two of which have common points, DEFINITIONS. +F{Sk). ) is finite, then the function F(L) is called absolutely additive. A set function F(L) defined for all Lebesgue sets L is said to be absolutely continuous if, for every positive constant €, there is a positive constant 8(e) such t h a t \F(L)\

The "total moment of momentum" of the system) are First Integrals of the equations of motion. ) There is a potential V(q, t) for the system, namely, V{q,t)AÍ where ÍV*»(\q*-q»\,t) V*"(r, t)=- f f*v(r, t) dr. 52 AXIOMATICS OF STATISTICAL MECHANICS I t contains an arbitrary additive constant. / / all the functions f*9 are independent of the time variable t, the system lias the energy integral 1 n nv Ύίν 1 n ΟΣ^Γ+ΟΣ n Σ^(|ί*-3Ί). ^„=i rnr ^λ=1ϊ,β1 Note. ,7i exert on mass-point 1; each single force always has one of the two directions of the straight line passing through both the mass-points concerned, and its magnitude depends only on the mutual distance of the particles (and, eventually, on the time variable).

Now it will be shown that f—φ* is orthogonal to each element φ of L. With φ, φ* + αφ also belongs to L for each value of a. ) Therefore, and, finally, (φ* - / , φ) = 0, as has been stated. Now put g = Φ*> h'f-φ*, and the proof of the theorem is complete. , Mengenlehre, Berlin, 1947. There is an English translation, published in 1950 in New York. , Einleitung in die Mengenlehre (Springer). , 1928 and later. , Abstract Set Theory, Amsterdam, 1953. This book, however, is not known to me through my own study.

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