By Jerry Marsden
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Useful integration effectively entered physics as course integrals within the 1942 Ph. D. dissertation of Richard P. Feynman, however it made no experience in any respect as a mathematical definition. Cartier and DeWitt-Morette have created, during this e-book, a brand new method of sensible integration. The booklet is self-contained: mathematical rules are brought, constructed generalised and utilized.
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Mayer,'J. Chem. Phys. 1 5 , 1 8 7 ( 1 9 4 7 ) . )]-l, Χ Π /urfÄ. „+j+1) P - (2, ·'·,Η+1/Κ+2, . . , η + 5 + l ) Ο Ν Κ - I) I Μ Κ Ν S Ι Ο Ν Λ L MATHEMATICAL /u«0; P H Y S I C S IN O N E Ru>at DIMENSION (36) 1104 SA LS BW RG. ZWANZIG, Α Χ Γ) KIR Κ WOOD ( where <ι is the diameter of a rigid sphere. Although the K '%v) = e x p [ > ' ] following discussion could be carried out in terms of an 1 arbitrary intennolecular potential, the simplicity of the 14- Γ expressions for a hard sphere potential aids in the clarity of the presentation.
We see that in this region the details of the model affect the energy per spin in leading order. In the third section we compute the energy per spin for the one-dimensional spherical model (and Gaussian model) with exponential interactions between spins. We verify explicitly the results of the second section for this type of interaction by comparing the results of the third section with the previously known results for the Ising model. We also verify explicitly that the behavior below the critical point is different for the spherical and Ising models.
In principle one could solve these equations for a system in which one considers more than only nearest neighbor interactions. Since most non-ionic forces have a range of only a few molecular diameters, the number of terms in the sums of E q . (34) could be taken to be, say, less than six. However, when we restrict our considerations to nearest neighbor interactions, a considerable simplification takes place, for as we have seen in Eq. (27) a fixed molecule effectively divides the system into two independent subsystems.