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Majorization

2013-10-28T22:06:06Z

67.211.125.174: /* Generalizations */

In [[physics]], the '''Fermi–Pasta–Ulam problem''' or '''FPU problem''' was the apparent [[physical paradox|paradox]] in [[chaos theory]] that many complicated enough physical systems exhibited almost exactly [[Frequency|periodic]] behavior – called '''Fermi–Pasta–Ulam recurrence''' – instead of [[ergodic theory|ergodic]] behavior. One of the resolutions of the paradox includes the insight that many non-linear equations are [[integrable systems|exactly integrable]]. Another may be that ergodic behavior may depend on the initial energy of the system.

==The FPU experiment==

In the Summer of 1953 [[Enrico Fermi|Fermi]], [[John Pasta|Pasta]], [[Stanislaw Ulam|Ulam]] and [[Mary Tsingou]] conducted numerical experiments (i.e. computer simulations) of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition would have led them to expect. Fermi thought that after many iterations, the system would exhibit [[thermalization]], an ergodic behavior in which the influence of the initial modes of vibration fade and the system becomes more or less random with [[equipartition theorem|all modes excited more or less equally]]. Instead, the system exhibited a very complicated quasi-periodic behavior. They published their results in a [[Los Alamos National Laboratory|Los Alamos]] technical report in 1955. ([[Enrico Fermi]] died in 1954 and so this technical report was published after Fermi's death.)

The FPU experiment was important both in showing the complexity of nonlinear system behavior and the value of computer simulation in analyzing systems.

Physics Today - January 2008 published additional information regarding the development of FPU http://perso.ens-lyon.fr/thierry.dauxois/PAPERS/pt61_55.2008.pdf

==The FPU lattice system==
Fermi, Pasta and Ulam (FPU) simulated the vibrating string by solving the following discrete system of nearest-neighbor coupled oscillators. We follow the explanation as given in Palais's article. Let there be ''N'' oscillators representing a string of length ''l'' with equilibrium positions <math>p_j=jh,j=0,\dots,N-1</math> where <math>h=l/(N-1)</math> is the lattice spacing. Then the position of the jth oscillator as a function of time is <math>X_j(t)=p_j+x_j(t)</math> so that <math>x_j(t)</math> gives the displacement from equilibrium. FPU used the following equations of motion:

:<math>m\ddot{x}_j=k(x_{j+1}+x_{j-1}-2x_j)[1+\alpha(x_{j+1}-x_{j-1})]</math>

(Note: this equation is not equivalent to the classical one given in the French version of the article)

This is just [[Newton's laws of motion|Newton's second law]] for the ''j''th particle. The first factor <math>k(x_{j+1}+x_{j-1}-2x_j)</math> is just the usual [[Hooke's law]] form for the force. The factor with <math>\alpha</math> is the nonlinear force. We can rewrite this in terms of continuum quantities by defining <math>c=\sqrt{\kappa/\rho}</math> to be the wave speed, where <math>\kappa=k/h</math> is the [[Young's modulus]] for the string and <math>\rho=m/h^3</math> is the density:

:<math>\ddot{x}_j=\frac{c^2}{h^2}(x_{j+1}+x_{j-1}-2x_j)[1+\alpha(x_{j+1}-x_{j-1})]</math>

==Connection to the KdV equation==
The continuum limit of the governing equations for the string (with the quadratic force term) is the [[Korteweg–de Vries equation]] (KdV equation.) The discovery of this relationship and of the [[soliton]] solutions of the KdV equation by [[Martin David Kruskal|Kruskal]] and [[Norman Zabusky|Zabusky]] in 1965 was an important step forward in nonlinear system research. We reproduce below a derivation of this limit, which is rather tricky, as found in Palais's article. Beginning from the "continuum form" of the lattice equations above, we first define ''u''(''x'',''t'') to be the displacement of the string at position ''x'' and time ''t''. We'll then want a correspondence so that <math>u(p_j,t)</math> is <math>x_j(t)</math>.

:<math>\ddot{x}_j=c^2\left(\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\right)[1+\alpha(x_{j+1}-x_{j-1})]</math>

We can use [[Taylor's theorem]] to rewrite the second factor for small <math>h</math>, (subscripts of u denote partial derivatives):

:<math>\begin{align}
\left(\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\right)&=\frac{u(x+h,t)+u(x-h,t)-2u(x,t)}{h^2}\\
&=u_{xx}(x,t)+\left(\frac{h^2}{12}\right)u_{xxxx}(x,t)+O(h^4).\end{align}</math>

Similarly, the second term in the third factor is:

:<math>\alpha(x_{j+1}-x_{j-1})=2\alpha hu_x(x,t)+\left(\frac{\alpha h^3}{3}\right)u_{xxx}(x,t)+O(h^5).</math>

Thus, the FPU system is:

:<math>\frac{1}{c^2}u_{tt}-u_{xx}=(2\alpha h)u_xu_{xx}+\left(\frac{h^2}{12}\right)u_{xxxx}+O(\alpha h^2,h^4).</math>

If one were to keep terms up to O(''h'') only and assume that <math>2\alpha h</math> approaches a limit, the resulting equation is one which develops [[shock wave|shocks]], which is not observed. Thus one keeps the O(''h''<sup>2</sup>) term as well:

:<math>\frac{1}{c^2}u_{tt}-u_{xx}=(2\alpha h)u_xu_{xx}+\left(\frac{h^2}{12}\right)u_{xxxx}.</math>

We now make the following substitutions, motivated by the decomposition of traveling wave solutions (of the ordinary [[wave equation]], to which this reduces when <math>\alpha,h</math> vanish) into left- and right-moving waves, so that we only consider a right-moving wave. Let <math>\xi=x-ct,\tau=(\alpha h)ct, y(\xi,\tau)=u(x,t)</math>. Under this change of coordinates, the equation becomes:

:<math>y_{\xi\tau}-\left(\frac{\alpha h}{2}\right)y_{\tau\tau}=-y_\xi y_{\xi\xi}-\left(\frac{h}{24\alpha}\right)y_{\xi\xi\xi\xi}.</math>

To take the continuum limit, assume that <math>\alpha/h</math> tends to a constant and <math>\alpha,h</math> tend to zero. If we take <math>\delta=\lim_{h\rightarrow0}\sqrt{h/(24\alpha)}</math>, then:

:<math>y_{\xi\tau}=-y_\xi y_{\xi\xi}-\delta^2y_{\xi\xi\xi\xi}.</math>

Taking <math>v=y_{\xi}</math> results in the KdV equation:

:<math>v_{\tau}+v v_{\xi}-\delta^2v_{\xi\xi\xi}=0.</math>

Zabusky and Kruskal argued that it was the fact that soliton solutions of the KdV equation can pass through one another without affecting the asymptotic shapes that explained the quasi-periodicity of the waves in the FPU experiment. In short, thermalization could not occur because of a certain "soliton symmetry" in the system which broke ergodicity.

==References==
* {{cite document
|last1=Fermi |first1=E.
|last2=Pasta |first2=J.
|last3=Ulam |first3=S.
|year=1955
|title=Studies of Nonlinear Problems
|url=http://www.physics.utah.edu/~detar/phys6720/handouts/fpu/FermiCollectedPapers1965.pdf
|id=Document LA-1940
}}
* {{cite journal
|last1=Zabusky |first1=N. J.
|last2=Kruskal |first2=M. D.
|year=1965
|title=Interactions of solitons in a collisionless plasma and the recurrence of initial states
|journal=[[Physical Review Letters]]
|volume=15 |issue=6 |pages=240–243
|bibcode=1965PhRvL..15..240Z
|doi=10.1103/PhysRevLett.15.240
}}
* {{cite journal
|last1=Palais |first1=R.
|year=1997
|title=The Symmetries of Solitons
|url=http://www.ams.org/journals/bull/1997-34-04/S0273-0979-97-00732-5/S0273-0979-97-00732-5.pdf
|journal=[[Bulletin of the American Mathematical Society]]
|volume=34 |issue=4 |pages=339–403
|doi=10.1090/S0273-0979-97-00732-5
|mr=1462745
}}
* {{cite journal
|last1=Duxois |first1=T.
|last2=Ruffo |first2=S.
|year=2008
|title=Fermi–Pasta–Ulam nonlinear lattice oscillations
|url=http://www.scholarpedia.org/article/Fermi-Pasta-Ulam_nonlinear_lattice_oscillations
|journal=[[Scholarpedia]]
|volume=3 |issue=8 |pages=5538
|doi=10.4249/scholarpedia.5538
|accessdate=2012-04-13
}}
*{{cite book
|editor1-last=Gallavotti |editor1-first=G.
|year=2008
|title=The Fermi–Pasta–Ulam Problem: A Status Report
|url=http://www.springer.com/physics/theoretical%2C+mathematical+%26+computational+physics/book/978-3-540-72994-5
|series=[[Lecture Notes in Physics]]
|volume=728
|publisher=[[Springer Science+Business Media|Springer]]
|isbn=978-3-540-72994-5
}}
* {{cite journal
|last1=Porter |first1=M. A.
|last2=Zabusky |first2=N. J.
|last3=Hu |first3=B.
|last4=Campbell |first4=D. K.
|year=2009
|title=Fermi, Pasta, Ulam and the Birth of Experimental Mathematics
|url=http://people.maths.ox.ac.uk/porterm/papers/fpupop_final.pdf
|journal=[[American Scientist]]
|volume=97 |issue=3 |pages=214–221
|doi=10.1511/2009.78.214
}}

{{DEFAULTSORT:Fermi-Pasta-Ulam problem}}
[[Category:Nonlinear systems]]
[[Category:Ergodic theory]]
[[Category:History of physics]]

Flow distribution in manifolds

2013-08-20T14:35:33Z

67.211.126.51: fixed a misspelling of "the"

[[File:Co-citation Proximity Analysis Concept.svg|thumb|400px|right|alt=Documents B and C are cited in closer proximity to each other in the full-text of the citing document, when compared to document A. Hence, according to co-citation proximity analysis, documents B and C are more strongly related than documents A and B or A and C. |Figure visualizing the Co-citation Proximity Analysis (CPA) approach to document similarity computation.]]
'''Co-citation Proximity Analysis''' or '''CPA''' is a document similarity measure that uses [[Citation| citation analysis]] to assess semantic similarity between documents at both the global document level as well as at individual section-level.<ref name="Gipp">Bela Gipp and Joeran Beel, 2009 [http://gipp.com/wp-content/papercite-data/pdf/gipp09a.pdf "Citation Proximity Analysis (CPA) – A new approach for identifying related work based on Co-Citation Analysis"] in Birger Larsen and Jacqueline Leta, editors, Proceedings of the 12th International Conference on Scientometrics and Informetrics (ISSI’09), volume 2, pages 571–575, Rio de Janeiro (Brazil), July 2009.</ref><ref>Bela Gipp and Joeran Beel. "Method and system for detecting a similarity of documents". Patent Application, Oct 27, 2011. [http://www.patentlens.net/patentlens/patents.html?patnums=US_2011_0264672_A1&language=en&|US 2011/0264672 A1].</ref> The similarity measure builds on the [[Co-citation|co-citation analysis]] approach, but differs in that it exploits the information implied in the placement of citations within the full-texts of documents.

Co-citation Proximity Analysis was conceived by Gipp in 2006 <ref>Bela Gipp, 2006. "Doctoral Proposal: (Co-)Citation Proximity Analysis – A Measure to Identify Related Work"</ref> and the description of the document similarity measure was later published by Gipp and Beel in 2009.<ref name="Gipp" /> The similarity measure rests on the assumption that within a document’s full-text, the documents cited in close proximity to each other tend to be more strongly related than those documents cited farther apart. The figure to the right illustrates the concept. The CPA approach to document similarity assumes the documents B and C to be more strongly related than the documents B and A, because the citations to B and C occur within the same sentence, whereas the citations to B and A are separated by several paragraphs.

The advantage of the CPA approach compared to other citation and co-citation analysis approaches is an improvement in precision. Other widely used citation analysis approaches, such as [[Bibliographic coupling|Bibliographic Coupling]], [[Co-citation|Co-Citation]] or the [[Bibliographic coupling#History|Amsler measure]], do not take into account the location or proximity of citations within documents. The CPA approach allows a more granular automatic classification of documents and can also be used to identify not only related documents, but the specific sections within texts that are most related.

==Method of Calculation==
The CPA similarity measure calculates a '''Citation Proximity Index (CPI)''' for each set of documents cited by an examined document.<ref name="Gipp" /> Cited documents are assigned a weight of <math>\frac{1}{2^n}</math>, where ''n'' stands for the number of levels between citations. Beginning at the lowest level, levels may be defined as citation groups, sentences, paragraphs, chapters, and finally the entire document or even journal.

There are several variations of the CPA algorithm.

*''Basic-CPA'' – fundamental concept of CPA as described above
*''Extended-CPA'' – considers the tree structure and order of citations within citation groups
*''Multidimensional-CPA'' – uses additional information such as the impact factor
*''Hybrid-CPA'' – combines the CPI with other similarity measures, for example text-based measures. This boosts performance especially for documents with insufficient citation information.

==Performance==
The CPA similarity measure builds upon the co-citation document similarity approach with the distinctive addition of citation proximity analysis, therefore the CPA approach gathers a more granular picture on document relatedness. CPA has been found to outperform co-citation analysis, especially when documents contain extensive bibliographies and if documents have not been referenced frequently (i.e. have a low co-citation score).<ref name="Gipp" /> Liu and Chen found that sentence-level co-citations are potentially more efficient markers for use in co-citation analysis in comparison to the loosely coupled article-level only co-citations, since sentence-level co-citations tend to preserve the essential structure of the traditional co-citation network and also form a much smaller subset of all co-citation instances <ref>Shengbo Liu and Chaomei Chen, 2001 [http://cluster.ischool.drexel.edu/~cchen/papers/2011/ISSI2011.pdf "The Effects of Co-citation Proximity on Co-citation Analysis"], The 13th Conference of the International Society for Scientometrics and Informetrics (ISSI), July 4–7, 2011 Durban, South Africa.</ref>

==References==
{{reflist}}

==Further reading==
Bela Gipp and Joeran Beel. Identifying Related Documents For Research Paper Recommender By CPA And COA. In S. I. Ao, C. Douglas, W. S. Grundfest, and J. Burgstone, editors, Proceedings of the world congress on engineering and computer science 2009, volume 1 of Lecture Notes in Engineering and Computer Science, pages 636-639, Berkeley (USA), oct 2009. International Association of Engineers (IAENG), Newswood Limited. Available at http://sciplore.org/pub/

Bela Gipp. Measuring Document Relatedness by Citation Proximity Analysis and Citation Order Analysis. In M. Lalmas, J. Jose, A. Rauber, F. Sebastiani, and I. Frommholz, editors, Proceedings of the 14th European conference on digital libraries (ecdl’10): research and advanced technology for digital libraries, volume 6273 of Lecture Notes of Computer Science (LNCS). Springer, sep 2010. Available at http://sciplore.org/pub/

[[Category:Bibliometrics]]