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| == Gradient Expansion == | | == Testing MathJax == |
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| | | $$ |
| Given a nonlinear equation as
| | \int_a^b f'(s) ds = f(b) - f(a). |
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| <math> \Box \phi+\lambda\phi^3=0 </math>
| | It works! |
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| one can always build a gradient expansion by assuming <math>\lambda\rightarrow\infty</math> as
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| == Duality in perturbation theory ==
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| In this section we will show how a duality principle holds in perturbation theory
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| showing how to derive a strong coupling expansion with the leading order ruled by
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| an adiabatic dynamics in order to study the evolution of a physical system.
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| We consider the following perturbation problem
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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 $\lambda$ an arbitrary ordering parameter: As is well known
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| an expansion parameter is obtained by the computation of the series itself. The standard
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| approach assume the limit $\lambda\rightarrow 0$ and putting
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| <math>
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| u = u_0 + \lambda u_1 +\ldots
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| </math>
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| one gets the equations for the series
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| <math>
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| \partial_t u_0 &=& L(u_0) \\ \nonumber
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| \partial_t u_1 &=& L'(u_0)u_1 + V(u_0) \\ \nonumber
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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. We recognize here a conventional
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| small perturbation theory as it should be. But the ordering parameter is just a conventional matter and so one
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| may ask what does it mean to consider $L(u)$ as a perturbation instead with respect to the same parameter.
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| 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) \\ \nonumber
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| \partial_t v_1 &=& V'(v_0)v_1 + L(v_0) \\ \nonumber
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| &\vdots&
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| </math>
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| where we have interchanged $L(u)$ and $V(u)$ and renamed the solution as $v$. The question to be answered is
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| what is the expansion parameter now and what derivative the prime means. To answer this question we rescale the
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| time variable as $\tau = \lambda t$ into eq.(\ref{eq:eq1}) obtaining the equation
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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 let us introduce the small parameter $\epsilon=\frac{1}{\lambda}$. It easy to see that applying again the
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| small perturbation theory to the parameter $\epsilon\rightarrow 0$ we get the set of equations (\ref{eq:set})
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| but now the time is scaled as $t/\epsilon$, that is, at the leading order the development parameter of the
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| 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 we can recognize as an equation for adiabatic evolution that in the proper limit $\epsilon\rightarrow 0$
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| will give the static solution $V(u_0)=0$. We never assume this latter solution but rather we will study
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| the evolution of eq.(\ref{eq:lead}). Finally, the proof is complete as we have obtained a dual series
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| <math>
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| u = v_0 + \frac{1}{\lambda} v_1 +\ldots
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| </math>
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| by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion
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| holding in the limit $\lambda\rightarrow\infty$ dual to the small perturbation theory $\lambda\rightarrow 0$
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| we started with and having an adiabatic equation at the leading order.
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| It is interesting to note that, for a partial differential equation, | |
| we can be forced into a homogeneous equation because, generally, if we require
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| also a scaling into space variables we gain no knowledge at all on the evolution of a
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| physical system. On the other side, requiring a scaling on the space variables and not on
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| the time variable will wash away any evolution of the system. So, on most physical systems
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| a strong perturbation means also a homogeneous solution but this is not a general rule. As
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| an example one should consider fluid dynamics where two regimes dual each other can be found
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| depending if it is the Eulerian or the Navier-Stokes term to prevail. In general relativity
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| things stay in a way to get a homogeneous equation at the leading order. The reason for this
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| is that products of derivatives or second order derivatives in space coordinates are
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| the only elements forming the Einstein tensor beside time dependence.
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