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| this is the sandbox. | | this is the sandbox. |
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| | == Testing MathJax == |
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| | | $$ |
| == Duality in perturbation theory ==
| | \int_a^b f'(s) ds = f(b) - f(a). |
| | | $$ |
| Duality in perturbation theory has been introduced in [[Fra1998]]. It can be formulated by saying that a solution series with a large parameter is dual to a solution series with a small parameter as it can be obtained by interchanging the choice of the perturbation term in the given equation.
| | It works! |
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| A typical perturbation problem can be formulated with the equation
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| <math>
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| \partial_t u = L(u) + \lambda V(u)
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| </math>
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| being <math>\lambda</math> an arbitrary ordering parameter. A solution series with a small parameter <math>\lambda\rightarrow 0</math> can be computed taking
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| <math>
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| u = \sum_{n=0}^{\infty} \lambda^n u_n
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| </math>
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| giving the following equations to be solved
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| <math>
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| \partial_t u_0 = L(u_0)
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| </math>
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| <math>
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| \partial_t u_1 = L'(u_0)u_1 + V(u_0)
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| </math>
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| <math>
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| \vdots
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| </math>
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| where a derivative with respect to the ordering parameter is indicated by a prime. The choice of the ordering parameter is just a conventional matter and one can choice to consider <math>L(u)</math> as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations
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| <math>
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| \partial_t v_0 = V(v_0)
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| </math>
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| <math>
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| \partial_t v_1 = V'(v_0)v_1 + L(v_0)
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| </math>
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| <math>
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| \vdots
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| </math>
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| where <math>L(u)</math> and <math>V(u)</math> are interchanged with the new solution <math>v</math>. In order to undertsand the expansion parameter we rescale the time variable as <math>\tau = \lambda t</math> into the equation to be solved obtaining
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| <math>
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| \lambda\partial_{\tau} u = L(u) + \lambda V(u)
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| </math>
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| and we introduce the small parameter <math>\epsilon=\frac{1}{\lambda}</math>. One sees that applying again the small perturbation theory to the parameter <math>\epsilon\rightarrow 0</math> we get the required set of equations but now the time is scaled as <math>t/\epsilon</math>, that is, at the leading order the development parameter of the series will enter into the scale of the time evolution producing a proper slowing down ruled by the equation
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| <math>
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| \epsilon\partial_t v_0 = V(v_0)
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| </math>
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| that is an equation for adiabatic evolution that in the proper limit <math>\epsilon\rightarrow 0</math> will give the static solution <math>V(v_0)=0</math>. So, the dual series
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| <math>
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| v = \sum_{n=0}^{\infty}\frac{1}{\lambda}v_n
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| </math>
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| is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit <math>\lambda\rightarrow\infty</math> dual to the small perturbation theory <math>\lambda\rightarrow 0</math> we started with and having an adiabatic equation at the leading order.
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Welcome to the sandbox! Please feel free to edit this page as you please by clicking on the "edit" tab at the top of this page. Terry 14:58, 30 July 2006 (EDT)
Some basic editing examples
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this is the sandbox.
Testing MathJax
$$
\int_a^b f'(s) ds = f(b) - f(a).
$$
It works!
