DispersiveWiki:Sandbox

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)

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Duality in perturbation theory

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.

A typical perturbation problem can be formulated with the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u = L(u) + \lambda V(u) }

being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} an arbitrary ordering parameter. A solution series with a small parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda\rightarrow 0} can be computed taking

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u = \sum_{n=0}^{\infty} \lambda^n u_n }

giving the following equations to be solved

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u_0 = L(u_0) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u_1 = L'(u_0)u_1 + V(u_0) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \vdots }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(u)} as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t v_0 = V(v_0) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t v_1 = V'(v_0)v_1 + L(v_0) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \vdots }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(u)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V(u)} are interchanged with the new solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v} . In order to undertsand the expansion parameter we rescale the time variable as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau = \lambda t} into the equation to be solved obtaining

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda\partial_{\tau} u = L(u) + \lambda V(u) }

and we introduce the small parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon=\frac{1}{\lambda}} . One sees that applying again the small perturbation theory to the parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon\rightarrow 0} we get the required set of equations but now the time is scaled as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t/\epsilon} , 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon\partial_t v_0 = V(v_0) }

that is an equation for adiabatic evolution that in the proper limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon\rightarrow 0} will give the static solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V(v_0)=0} . So, the dual series

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}v_n }

is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda\rightarrow\infty} dual to the small perturbation theory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda\rightarrow 0} we started with and having an adiabatic equation at the leading order.

The main mathematical problem of this kind of methods is the existence of the solution series. For the most interesting cases this series are not converging and represent asymptotic approximations to the true solution.

Finally, the success of this method relies on the possibility to obtain a proper analytical solution to the leading order equation.

Nonlinear PDE and Perturbation Methods

The application of the perturbation methods described above to PDE gives an interesting result, i.e. the dual series to the small parameter solution series is a gradient expansion Fra2006.

This can be seen by considering a NLKG equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box\phi+\lambda V(\phi)=0.}

The choice of the perturbation term to compute a small or a large parameter series depends also on the way the derivatives of the field are managed.

In order to see this we apply the computation given in the previous section by rescaling time as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t\rightarrow \lambda t} and take a solution series with a large parameter

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}\phi_n.}

The non trivial set of equations is so obtained

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_\tau^2\phi_0+V(\phi_0)=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_\tau^2\phi_1+V'(\phi_0)\phi_1=\Delta^2\phi_0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \vdots}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau=\lambda t} . Indeed, this is a gradient expansion.

An interesting problem that applies to a given PDE is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box\phi+\lambda V(\phi)=j}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j} is a driving term. When a small parameter series has to be computed we obtain that at the leading order one has generally to solve

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box\phi_0=j}

that is very easy to solve by the Green function method

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_0=\phi_{H}+\int d^Dx'G(x-x')j(x')}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box\phi_{H}=0}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box G=\delta^D}

being $\delta^D$ a Dirac distribution of the given dimensionality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D} .

A similar result holds also for the large parameter series FraA2007,FraB2007. We note that the leading order of the gradient expansion is now

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_\tau^2\phi_0+V(\phi_0)=j.}

A small time series holds that has the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi=\sum_{n=0}^{\infty}a_n\int d\tau'G(\tau-\tau')(\tau-\tau')^nj(\tau')}

being

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_\tau^2 G+V(G)=\delta}

and the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_n} are computed by deriving the equation we started from and with the initial conditions and will generally depend on the values of the source and its derivatives at the intial time. The success of the method relies on the ability to obtain analitically the Green function.