1.2.7. Dispersion Relation and Instabilities Using Green’s Function

1.2.7.1. A Simple System

For a nonlinear system which has a linearized relation

\[\mathscr D S = 0,\]

where \(\mathscr D\) is a derivative operator that contains terms like \(\partial_t\) and \(\boldsymbol\nabla\).

Assuming collective modes \(S = Q e^{i(\mathbf k \cdot \mathbf x - \omega t)}\), the equation becomes

\[D(\omega, \mathbf k) = 0,\]

which is in fact the dispersion relation.

The picture behind these math is that we are treating the field as waves. By “solving” the dispersion relation, we find \(\omega(\mathbf k)\) or \(\mathbf k(\omega)\), which is substituted into the wave form,

\[S = Q e^{i(\mathbf k(\omega) \cdot \mathbf x - \omega t)},\]

or

\[S = Q e^{i(\mathbf k \cdot \mathbf x - \omega(\mathbf k) t)}.\]

However, we need to integrate over all Fourier modes in general. So we have the solution of the field

\[S = \int d\omega Q_\omega e^{i(\mathbf k(\omega) \cdot \mathbf x - \omega t)},\]

or

\[S = \int d\mathbf k Q_{\mathbf k} e^{i(\mathbf k \cdot \mathbf x - \omega(\mathbf k) t)}.\]

Instability of the system is related the convergence of the integrals. This was done by P. A. Sturrock. [Sturrock1958]

For the general equation of coherence

\[\mathscr D S = f(\mathbf x,t),\]

the authors consider the Green’s function of the equations, i.e.,

\[\mathscr D G = \delta(\mathbf x)\delta(t).\]

Fourier transform of \(G\) leads to

\[D(\omega,\mathbf k) \tilde G = \mathrm{Const},\]

where the constant is chosen to be 1. The formal solution is

\[\tilde G = \frac{1}{D(\omega,\mathbf k)}.\]

Inverse Fourier transform of the solution

\[G(t,\mathbf x) = \frac{1}{(2\pi)^2} \iint d\mathbf k d \omega \tilde G e^{i(\mathbf k \cdot \mathbf x - \omega t)} = \frac{1}{(2\pi)^2} \iint d\mathbf k d \omega \frac{1}{D(\omega,\mathbf k)} e^{i(\mathbf k \cdot \mathbf x - \omega t)}.\]

Choice of Coefficient

The Fourier transform is chosen to be

\[\tilde g = \frac{1}{(2\pi)^2} \iint d\omega d \mathbf k g e^{i(\mathbf k \cdot \mathbf x - \omega t)},\]

which has coefficient \(1/(2\pi)^2\). The inverse transform would have no coefficient.

Solution to S

The solution for coherence is

\[S = \iint G f d\mathbf x dt,\]

which indicates that the Fourier mode

\[\tilde S = \tilde G \tilde f.\]
[Sturrock1958]Sturrock, P. A. (1958). Kinematics of Growing Waves. Physical Review, 112(5), 1488–1503.

One Pole Case

I have no idea how is Eqn. 27 derived.

Also have questions about Eqn. 34 for convective instability of 0 group velocity.

1.2.7.2. Neutrinos

The off diagonal elements of the density matrix \(S\) obays the equation of the form

\[i(\partial_t + \mathbf v \cdot \nabla) S = f(\mathbf x, t)\]

in linear regime.

1.2.7.3. Gap


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