Majorization

In physics, the Fermi–Pasta–Ulam problem or FPU problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called Fermi–Pasta–Ulam recurrence – instead of ergodic behavior. One of the resolutions of the paradox includes the insight that many non-linear equations are 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 Fermi, Pasta, 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 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 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 $p_j=jh,j=0,\dots ,N-1$ where $h=l/(N-1)$ is the lattice spacing. Then the position of the jth oscillator as a function of time is $X_j(t)=p_j+x_j(t)$ so that $x_j(t)$ gives the displacement from equilibrium. FPU used the following equations of motion:

$m{\ddot{x}}_j=k(x_{j+1}+x_{j-1}-2x_j)[1+\alpha (x_{j+1}-x_{j-1})]$

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

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

${\ddot{x}}_j=\frac{c^2}{h^2}(x_{j+1}+x_{j-1}-2x_j)[1+\alpha (x_{j+1}-x_{j-1})]$

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 Kruskal and 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 $u(p_j,t)$ is $x_j(t)$.

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

We can use Taylor's theorem to rewrite the second factor for small $h$, (subscripts of u denote partial derivatives):

$\begin{array}{rl}\left(\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\right)& =\frac{u(x+h,t)+u(x-h,t)-2u(x,t)}{h^2}\\ & =u_{xx}(x,t)+\left(\frac{h^2}{12}\right)u_{xxxx}(x,t)+O(h^4).\end{array}$

Similarly, the second term in the third factor is:

$\alpha (x_{j+1}-x_{j-1})=2\alpha h{u}_x(x,t)+\left(\frac{\alpha h^3}{3}\right)u_{xxx}(x,t)+O(h^5).$

Thus, the FPU system is:

$\frac{1}{c^2}{u}_{tt}-u_{xx}=(2\alpha h)u_x{u}_{xx}+\left(\frac{h^2}{12}\right)u_{xxxx}+O(\alpha h^2,h^4).$

If one were to keep terms up to O(h) only and assume that $2\alpha h$ approaches a limit, the resulting equation is one which develops shocks, which is not observed. Thus one keeps the O(h²) term as well:

$\frac{1}{c^2}{u}_{tt}-u_{xx}=(2\alpha h)u_x{u}_{xx}+\left(\frac{h^2}{12}\right)u_{xxxx}.$

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

$y_{\xi \tau }-\left(\frac{\alpha h}{2}\right)y_{\tau \tau }=-y_{\xi }{y}_{\xi \xi }-\left(\frac{h}{24\alpha }\right)y_{\xi \xi \xi \xi }.$

To take the continuum limit, assume that $\alpha /h$ tends to a constant and $\alpha ,h$ tend to zero. If we take $\delta =\underset{h\to 0}{\lim }\sqrt{h/(24\alpha )}$, then:

$y_{\xi \tau }=-y_{\xi }{y}_{\xi \xi }-{\delta }^2{y}_{\xi \xi \xi \xi }.$

Taking $v=y_{\xi }$ results in the KdV equation:

$v_{\tau }+v{v}_{\xi }-{\delta }^2{v}_{\xi \xi \xi }=0.$

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

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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