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Geometry and Physics:
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Divide n-space in two subspaces with dimensions
n1 and n2=n-n1
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Define an n-sphere and two spheres in n1 and
n2 space, respectively, such that
r2 = r12+r22
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Find that combination of r's which produces the largest product
of sphere volumes, V1.V2
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On a logarithmic scale the volume of the n-sphere equals the product
of the two subspheres.
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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:
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The largest product V1.V2 is achieved when
dS/dE is equal in both subspaces (Temperature)
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In equilibrium, S=S1+S2 (Extensivity of S)
Back
[Code: Entropy1]
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