Energy and Parallel Transport
(11,
13,
16)
Homotopy theory has a hidden assumption: that strains cost energy.
Are uniform solutions n=n0
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 second-to-last term involving K24 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 90-94
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Entertaining Science done at
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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).