On the Theory of Isothermal-Isobaric Ensemble. I : Mathematical Treatment of the Partition Function
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The isothermal-isobaric ensemble, representing a system which is in thermal and mechanical contact with a bath of temperature T and pressure p, is discussed by a rigorous method of mathematical analysis. The isothermal-isobaric partition function for a system of interacting, N molecules is expanded in terms of the cluster integrals b_l. The b_l are assumed to be volume-independent and positive. With the use of a function l^*(N) satisfying certain conditions, discrimination is made between "large"l[l6gt;(N)] and "small" l[l≦l^*(N)], l being the number of molecules composing a cluster. The terms in the isothermal-isobaric partition function are rearranged according to the total number, M, of "small" clusters and, then, according to the total number, N_2, of molecules composing "large" clusters. The "large" cluster part of the partition function, though it is rather complicated in structure, is expressed by a simple formula containing b_0(=lim[l→∞]b_<l^1/1>, i.e. the contribut molecule to b_l for infinitely large l), with as small an error as one pleases if N is large enough. The sum. H_N(M;α), of those terms in the partition function for which the total number of "small" clusters has a prescribed value M=[Nx][x(0≦x≦1) being a real number and the brackets denoting the integral part] is rigorously calculated in the limit N→∞. It is deduced that as N→∞, (1/N) ln H_N(M ; α) converges to a function h(x;α) uniformly for 0≦x≦1, where h(x;α) consists of analytically different parts, namely : h(x;α)=xlnΣ^∞_<l=1>ab_lξ_<0^l>-lnξ_0(for 0≦x≦x_0),h(x;α)=x lnΣ^∞_<l=1>ab_lξ^l-lnξ(for x_0≦x<1), h(x;α)=ln(ab_1)(for x=1), where α=kT/p,ξ_0=b_<0^<-1>>, and ξ is defined as a function of x by the equation Σ^∞_<l=1b_lξ^l/Σ^∞_<l=1lb_lξ^l=X, and x_0 is the value of x when ξ=ξ_0 It is shown that the isothermal-isobaric partition function, and hence the activity, of the system for N→∞ can be calculated by maximizing h(x ; α) with respect to x (0≦x≦1). Discussion of the system from the physical viewpoint, on the basis of the results obtained in this paper, is left to the next paper II.
- 理論物理学刊行会の論文
- 1967-09-25
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