Geometry and Physics:
  • Divide n-space in two subspaces with dimensions n1 and n2=n-n1
  • Define an n-sphere and two spheres in n1 and n2 space, respectively, such that r2 = r12+r22
  • Find that combination of r's which produces the largest product of sphere volumes, V1.V2
  • On a logarithmic scale the volume of the n-sphere equals the product of the two subspheres.
  • But: this product may be understood as the volume of a hypercylinder inscribed in the n-sphere, having its "base area" in space n1 and its "height" in space n2.
Physical interpretation:
Assume that the log-volume is identical to the entropy S. Then:
  • The largest product V1.V2 is achieved when dS/dE is equal in both subspaces (Temperature)
  • In equilibrium, S=S1+S2 (Extensivity of S)
Back        [Code: Entropy1]