Foundations of statistical mechanics by Oliver Penrose

By Oliver Penrose

Beginning with fundamental actual assumptions and their idealization within the type of postulates, this article examines the implications of those postulates, culminating within the derivation of the basic formulation for calculating percentages when it comes to dynamical amounts and concluding with an research of the proposal of entropy.Solutions. 1970 variation.

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I f the permuted functions, ^^Ψ^^^ are different and mutually orthogonal, the normahzation factor ^ is i V ! , which, however, does not concern us here. / / , however, all the Ν! 6), since different numbered molecules exchange quantum numbers. Out of these Ν! different functions we can construct only one totally symmetric function, or only one totally antisymmetric one. , since all the Ν! 6) κ Kt where the sum over K'f includes all the states found as solutions of the Schrödinger equation without the hmitation on selecting totallysymmetric or antisymmetric functions in permutations of the identical molecules.

The classical state of the system is given by one point in λ-space, or for Ν molecules of the same species by Ν points in /i-space. I f several molecular species are present in the system the /^-spaces of the different species usually have differing numbers of degrees of freedom, and a different //-space must be used for each species. If, however, as is sometimes the case, one is describing only the posi­ tions and momenta of the centers of mass of the molecules, a single /^-space of three momenta and three coordinates may be used for all species, and the state of the system described by giving points of type a, N^, of type etc.

97ΓΚ for Hg. Except for Hg the value of ξ is very much less than unity at temperatures for which the molecules are normally gaseous. There is no analytical expression for the sum of eq. (3). If, however, I < < 1, the terms of low j values lie close together in value, and one may approximate. 7) which leads to -kT In Q,,, = kT In ξ. 4. By use of the Euler-Maclaurin summation formula one finds the somewhat better approximation. Orot = ^ ' [ 1 + (1/3)1 + (1/15)|2 + . . 8) which can be used when ξ is smaU but not neghgible compared with unity.

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