List of disproved mathematical ideas

2014-01-23T22:10:49Z

81.167.104.3: Removed factual inaccuracy.

The '''topological entanglement entropy'''{{ref|KitaevPreskill}} {{ref|LevinWen}}, usually denoted by ''γ'', is a number characterizing many-body states that possess [[topological order]].
The short form ''topological entropy'' is often used, although the same name in [[ergodic theory]] refers to an unrelated mathematical concept (see [[topological entropy]]).

A non-zero topological entanglement entropy reflects the presence of long range quantum entanglements in a many-body quantum state. So the topological entanglement entropy links [[topological order]] with pattern of
long range quantum entanglements.

Given a [[topological order|topologically ordered]] state, the topological entropy can be extracted from the asymptotic behavior of the [[Von Neumann entropy]] measuring the [[quantum entanglement]] between a spatial block and the rest of the system. The entanglement entropy of a simply connected region of boundary length ''L'', within an infinite two-dimensional topologically ordered state, has the following form for large ''L'':

:<math> S_L \; \longrightarrow \; \alpha L -\gamma +\mathcal{O}(L^{-\nu}) \; , \qquad \nu>0 \,\!</math>

''-γ'' is the topological entanglement entropy.

The topological entanglement entropy is equal to the logarithm of the total [[quantum dimension]] of the quasiparticle excitations of the state.

For example, the simplest fractional quantum Hall states, the Laughlin states at filling fraction 1/''m'', have ''γ'' = ½log(''m''). The ''Z''2 fractionalized states, such as topologically ordered states of
''Z''2 spin-liquid, [[quantum dimer models]] on non-bipartite lattices, and Kitaev's [[toric code]] state, are characterized ''γ'' = log(2).

==See also==

*[[Quantum topology]]
*[[Topological defect]]
*[[Topological order]]
*[[Topological quantum field theory]]
*[[Topological quantum number]]
*[[Topological string theory]]

== References ==
{{reflist}}
{{Nofootnotes|date=April 2008}}
===Introduction of the measure===
#{{note|KitaevPreskill}} Topological Entanglement Entropy, Alexei Kitaev and John Preskill, [http://link.aps.org/abstract/PRL/v96/e110404 Phys. Rev. Lett. '''96''', 110404 (2006)].
#{{note|LevinWen}} Detecting Topological Order in a Ground State Wave Function, Michael Levin and Xiao-Gang Wen, [http://link.aps.org/abstract/PRL/v96/e110405 Phys. Rev. Lett. '''96''', 110405 (2006)].

===Calculations for specific topologically ordered states===

* M. Haque, O. Zozulya and K. Schoutens; Phys. Rev. Lett. '''98''', 060401 (2007).
* S. Furukawa and G. Misguich, Phys. Rev. B '''75''', 214407 (2007).

{{physics-stub}}
[[Category:Condensed matter physics]]
[[Category:Statistical mechanics]]

Minimum polynomial extrapolation

2013-02-18T22:48:21Z

81.167.190.166: Undid revision 476846078 by 132.206.224.62 (talk) Change made the matlab code not correspond to the text.

The '''strain energy release rate''' (or simply energy release rate) is the [[energy]] [[dissipation|dissipated]] during [[fracture]] per unit of newly created fracture surface area. This quantity is central to [[fracture mechanics]] because the energy that must be supplied to a [[Fracture|crack]] tip for it to grow must be balanced by the amount of energy dissipated due to the formation of new surfaces and other dissipative processes such as [[plasticity (physics)|plasticity]].

For the purposes of calculation, the energy release rate is defined as
:<math>
G := -\cfrac{\partial (U-V)}{\partial A}
</math>
where <math>U</math> is the potential energy available for crack growth, <math>V</math> is the work associated with any external forces acting, and <math>A</math> is the crack area (crack length for two-dimensional problems). The units of <math>G</math> are J/m2.

The energy release rate [[material failure theory|failure criterion]] states that a crack will grow when the available energy release rate <math>G</math> is greater than or equal to a critical value <math>G_c</math>
:<math>
G \ge G_c
</math>
The quantity <math>G_c</math> is the '''fracture energy''' and is considered to be a material property which is independent of the applied loads and the geometry of the body.

== Relation to fracture toughness ==
For two-dimensional problems ([[plane stress]], [[plane strain]], [[antiplane shear]]) involving cracks that move in a straight path, the [[mode I]] [[stress intensity factor]] (<math>K_I</math>) is related to the energy release rate (<math>G</math>) by
:<math>
G = \cfrac{K_I^2}{E'}
</math>
where <math>E</math> is the [[Young's modulus]] and <math>E' = E</math> for [[plane stress]] and <math>E' = E/(1-\nu^2)</math> for [[plane strain]].

Therefore the energy release rate failure criterion may also be expressed as
:<math>
K_I \ge K_{Ic}
</math>
where <math>K_{Ic}</math> is the mode I [[fracture toughness]].

== See also ==
{{Continuum mechanics|cTopic=[[Solid mechanics]]}}
*[[Fracture]]
*[[Fracture mechanics]]
*[[Fracture toughness]]
*[[J integral]]
*[[Stress intensity factor]]
*[[Tearing energy]]
*[[Configurational force]]
*[[Crack driving force]]

==References==
<references>
</references>

*Rivlin, R. S., & Thomas, A. G. (1953). Rupture of rubber. I. Characteristic energy for tearing. Journal of Polymer Science, 10(3), 291-318.
*[http://dx.doi.org/10.1016/B978-0-12-394584-6.00010-8 Chapter 10 – Strength of Elastomers], A.N. Gent, W.V. Mars, In: James E. Mark, Burak Erman and Mike Roland, Editor(s), The Science and Technology of Rubber (Fourth Edition), Academic Press, Boston, 2013, Pages 473-516, ISBN 9780123945846, 10.1016/B978-0-12-394584-6.00010-8

[[Category:Mechanical failure]]
[[Category:Fracture mechanics]]
[[Category:Solid mechanics]]
[[Category:Mechanics]]
[[Category:Rubber properties]]

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List of disproved mathematical ideas

Minimum polynomial extrapolation