Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 15.5-1

THE CANONICAL FORMALISM; STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION

Physics — Statistical Mechanics · hypervolume-diagonal-hyperplane

Problem

To establish equation 15.29 let Ωn\Omega_n be the hypervolume subtended by the diagonal hyperplane in nn dimensions. Draw appropriate figures for n=1,2n = 1, 2, and 3 and show that if LL is the intercept on each of the coordinate axes Ω1=L\Omega_1 = L Ω2=Ω1∫0L(1−xL)dx=L22!\Omega_2 = \Omega_1 \int_0^L \left(1 - \frac{x}{L}\right)dx = \frac{L^2}{2!} Ω3=Ω2∫0L(1−xL)2dx=L33!\Omega_3 = \Omega_2 \int_0^L \left(1 - \frac{x}{L}\right)^2 dx = \frac{L^3}{3!} and by mathematical induction Ωn=Ωn−1∫0L(1−xL)n−1dx=Lnn!\Omega_n = \Omega_{n-1}\int_0^L \left(1 - \frac{x}{L}\right)^{n-1}dx = \frac{L^n}{n!}

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