T-I-2draft

Linear response theory and Kramers-Kronig relations

The goal of this homework is to introduce the theory of linear response and, among the related properties, such as causality, the so-called Kramers-Kronig relations. We will consider the damped harmonic oscillator as a prototypical example to discuss this topic.

Reminder on the Fourier transform

Throughout this assignment, we will use the symmetric convention of Fourier transform. The Fourier transform of a function $f(t)\in L^{1}(\mathbb {R} )$ is defined as:

${\widehat {f}}(\omega ):={\frac {1}{\sqrt {2\pi }}}\int dt\,f(t)\,e^{i\omega t}.$

With this convention, the Fourier transform of a product results in a convolution, with the following pre-factor: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2\pi}\,\widehat{(f\, g)}=\widehat f \ast \widehat g .}

The repeated application of the transform gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\widehat f}(t)=f(-t).}

The conjugate of a Fourier transform is equal to the transform of the conjugate, with a change of sign in the argument: ${\overline {\widehat {f}}}(\omega )={\widehat {\overline {f}}}(-\omega ).$

Furthermore, we can exchange the Fourier transform under the integral sign: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int _\mathbb{R}dt \, \hat f(t)\, g(t)=\int _\mathbb{R}dt \, f(t)\,\hat g(t) . }

The last property implies the result Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int _\mathbb{R}d\omega \, \overline{\hat f}(\omega)\,\hat g(\omega)=\int _\mathbb{R}dt \, \overline{f(t)}\,\hat g(t) , } which gives directly the Plancherel theorem (see lecture notes, page 45).

Example: damped harmonic oscillator in one dimension

We consider a damped harmonic oscillator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)} , which is described by the following equation of motion:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot x + 2 \gamma \,\dot x + \omega _0^2\,x=f(t). }

First, we will compute its response to an harmonic perturbation, and then its response to a generic perturbation by introducing its Green function.

Response to an harmonic perturbation

We assume that the oscillator is excited by an harmonic perturbation of the form: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=F_\omega\,e^{-i\omega t}. } We look for a solution of the form: $x(t)=X_{\omega }\,e^{-i\omega t}.$

Q1: Show that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_\omega=\hat g(\omega)\, F_\omega, }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g(\omega)=-\frac{1}{(\omega-\omega_1)(\omega-\omega_2)}. }

Write the explicit form of the poles that we have indicated as $\omega _{1}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2} .

Response to a generic perturbation

We consider a generic perturbation represented by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} , whose Fourier transform is given by ${\hat {f}}(\omega )$.

Q2: Write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)} as a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g(\omega)} and ${\hat {f}}(\omega )$.

Q3: Show that it follows that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d t' \, g(t-t')f(t'), }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)} is the function having ${\hat {g}}(\omega )$ as Fourier transform. The above equation is nothing but the convolution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} : ${\sqrt {2\pi }}x(t)=(f\ast g)(t)$.

Holomorphism of the response and consequences

Q4: Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)} is the response of the system to an impulsive perturbation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=\sqrt{2\pi}\delta(t)} . What is the effect of a perturbation applied at time $s$ on the solution at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t<s} ?

Q5: We have defined Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g (\omega)} for $\omega \in \mathbb {R}$, i.e. on the real axis. Show that its extension to the complex plane is holomorphic everywhere (except at the poles). Using the explicit expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat g (\omega)} derived above, compute explicitly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)} . You will have to treat separately the case of positive time and the case of negative time. Use the residue theorem.

Q6: Deduce the expression of the response:

$x(t)=\int _{-\infty }^{t}dt'\,e^{-\gamma (t-t')}\,{\frac {\sin \left[(\omega _{0}^{2}-\gamma ^{2})^{1/2}(t-t')\right]}{(\omega _{0}^{2}-\gamma ^{2})^{1/2}}}f(t').$

The above expression indicates the Green function of the system. Notice that the response of the system is causal.

It is worth highlighting the importance of the holomorphism in this proof. Indeed, we have seen that we had to apply Jordan's lemma. Therefore, we need a relatively fast decay of the response function at infinity, i.e. a system that does not diverge.

We also point out that, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} was negative, we would have lost causality: we can guess a relation between causality and the dissipation of energy in a passive system.

General properties of linear response

We look for a generalization of the properties that we have seen in the previous example. We discuss an equation of motion (or possibly another physical equation)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}x(t)=f(t), }

with a linear operator ${\mathcal {L}}$ (of differentiation, multiplication, integration, etc.), and with an inhomogeneity $f(t)$ (external stimulation, perturbation, etc.). The generic solution is unknown since the explicit form of the operator is not specified. However, we know that this solution must have some physical properties:

• Linearity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a function or a linear functional of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , which reads Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)=\int _\mathbb{R}ds \, g(t-s)f(s). }
• Causality: in the sense that the effect cannot precede the cause: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)=0} for $t<0$.
• The total response to a finite perturbation must be finite.
• The response to a real perturbation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)\in \mathbb{R}} , must be real as well, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)\in\mathbb{R}} (at least in the case we are not considering quantum mechanics effects).

Link between odd/even and real/imaginary

Q7: Show the following properties of the Fourier transform:

• if $g\in \mathbb {R}$ is an odd function, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g\in i\mathbb{R} } is also an odd function;
• if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in i\mathbb{R}} is an odd function, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g\in \mathbb{R}} is also an odd function;
• if $g\in \mathbb {R}$ is an even function, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g\in i\mathbb{R} } is also an even function;
• if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in i\mathbb{R} } is an even function, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g\in \mathbb{R} } is also an even function.

Q8: Conclude that, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is real, its Fourier transform is written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g=\hat g ' +i\hat g ''} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g '} odd, and ${\hat {g}}''$ even.

Q9: Derive the same properties for the case of the oscillator.

Preliminaries

We define the sign function as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(t)=\begin{cases} -1 & t<0\\ +1 &t\geq 0 \end{cases}, }

and we want to show that its Fourier transform is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat S(\omega)=\sqrt{\frac{2}{\pi}}\frac{i}{\omega}. }

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is not summable, we have to use an auxiliary function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_\epsilon (t):=e^{-\epsilon |t|}S(t)} .

Q10: Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{\varepsilon\rightarrow 0}\hat S_\epsilon (\omega)=\sqrt{\frac{2}{\pi}}\frac{i}{\omega}} .

Q11: Discuss the possibility of using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat S} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat S_\epsilon} in a computation (e.g., in the example below).

Causality and Kramers-Kronig relations

Q12: Using the expression of the Fourier transform of a convolution and the sign function defined above, show the following property of Fourier transform:

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall t<0} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \quad\hat g(\omega)=\frac{1}{i\pi}\,{\rm PV} \int_\mathbb{R}d\xi \,\frac{\operatorname{Im} \hat g(\xi)}{\xi -\omega}} .

By separating the real and imaginary part of this equation, we obtain Kramers-Kronig relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Re}\hat g(\omega)=\frac{1}{\pi}\, {\rm PV}\int_\mathbb{R}d\xi \,\frac{\operatorname{Im}\hat g(\omega)}{\xi-\omega}, } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Im} \hat g(\omega)=-\frac{1}{\pi} \,{\rm PV}\int_\mathbb{R}d\xi \,\frac{\operatorname{Re}\hat g(\omega)}{\xi-\omega}. }

Q13: Show that Kramers-Kronig relations express the fact that the extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g (\omega + i \nu)} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g (\omega)} on the complex upper-half plane Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\nu>0)} is holomorphic, without any poles, and that it decreases sufficiently fast as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\omega +i\nu|\rightarrow \infty} .

We will start from a weaker result:

Show that the Kramers-Kronig relation is valid for all functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g (\omega +i\nu)} with the aforementioned properties. Apply the residue theorem integrating along the path Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} displayed in the figure below, that can be reproduced via the following code:

gamma = ParametricPlot[{{R Cos[t], R Sin[t]}, {-R*(1 - t/Pi) + (x - r)*t/Pi, 0}, {r Cos[t + Pi] + x, r Sin[t + Pi]}, {(x + r)*(1 - t/Pi) + R*t/Pi, 0}} /. {R -> 2,  r -> 0.5, x -> 1}, {t, 0, Pi}, Ticks -> {{{-R, "-R"}, {x - r, "x-r"}, {x, "x"}, {x + r, "x+r"}, {R, "R"}} /. {R -> 2, r -> 0.5, x -> 1}, None}, TicksStyle -> Directive[Red, 20], AxesLabel -> {None}] /. Line[u_] :> Sequence[Arrowheads[Table[.05, {1}]], Arrow@Line[u]]

Sketch of the integration path Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} .

Integrate the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat g (z)/(z-\omega)} along this path. Compute the contribution of the small half-circle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_r} 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://wikimedia.org/api/rest_v1/":): {\displaystyle r\rightarrow 0} using its Taylor expansion in the vicinity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} and performing the change of variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z'=\omega + r\,e^{i\theta}} .

The last result suggests the existence of a relation with the causality at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall t<0} and the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is holomorphic without poles in the upper-half plane. This was detailed in the Titchmarsch theorem.