An Introduction to the Statistical Theory of Classical by G.H. A. Cole

By G.H. A. Cole

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THE ROLE OF THE PARTITION FUNCTION The link between the microscopic statistical theory and macroscopic thermodynamics is made by considering the entropy, S. The microscopic phase function here is — k In f(iV) so that S= -Jfc>. 11) for f(JV) leads to the expression S = k\nZ^9 where (E) is the phase average of the Hamiltonian function. 19) 46 STATISTICAL THEORY OF CLASSICAL FLUIDS provided the Helmholtz free energy F is defined by F= -kTlnZ(V,T). 20) This relation between F a n d Z is basic to statistical thermodynamics.

For example, the macroscopic energy of a physical system, possibly interpreted through its temperature, is to be linked to the mean energy of the microscopic model representing the gross system. Trajectories in certain regions of phase space will contribute more strongly to this average than those in other phase regions due to variations in y-phase density. In determining the mean energy, therefore, MICROSCOPIC REPRESENTATION 35 independent knowledge of the distribution functions is necessary for linking the micro- and macro-physical works: this is true for the derivation of any mean value of physical significance.

The distribution in position) provides a means of distinguishing between the three main phases of matter. 8a) 43 FLUID STATISTICAL THERMODYNAMICS (where h ^ N), g(h)(rh) = W=H) / ^h+1Krh+1) dr ft+1 . 8b) It follows immediately that the pair and triplet configuration distributions are given by *(2)(Γι' Γζ) = W^2) j g<3>(ri Γ2 Γ3> dF3 ' ' ' j^<»(r lf r2) = ^ ) ] j ... j gW(rN) Sdr,, nY*\ri, r2, r3) = ^ ^ j ... Π dr3·. 9c) These expressions will be of great importance in the later development of the theory.

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