Energy and Parallel Transport
(11,
13,
16)
Homotopy theory has a hidden assumption: that strains cost energy.
Are uniform solutions n=n_{0}
preferred?
We answer this by considering the form of the continuum free energy density
. It must have the properties:
 Since gradients are small on molecular lengths, only the low
order gradient terms are included.
 Since the free energy is rotation invariant, we write it in terms
of scalars under rotation: dot and cross products.
 If the material has inversion symmetry, then
when
and
: this implies the final term vanishes
and q=0.
 The secondtolast term involving K_{24} is a total divergence
term. It will only be important if there is lots of surface area or
many internal defect lines...
For a material with inversion symmetry, the only good gradient is no gradient.
(There is a unique parallel transport on a manifold compatible with the
metric and with vanishing torsion...)
This research was paid for by THE US GOVERNMENT
by the NSF.
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Sethna's Research 9094

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Last modified: November 2, 1995
James P. Sethna,
sethna@lassp.cornell.edu
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).