New answers tagged differential-equations
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This kind of exponential decay toward "equilibrium" can be derived when one looks at a Markov process.
In this case, if we call $S_t$ the state of the system at time $t$ and $S_{t+1}$ the state at time $t+1$, one has for the evolution:
$S_{t+1} = T S_t $
where $T$ is called the transition matrix. This implies that $S_t = T^t S_0$. The idea is then to ...
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This form of $dE/d\tau$ is valid only when the system is not too far from equilibrium and linear response assumption is valid. The fact that $dE/d\tau$ depends on the difference $E - E(0)$ alone is a consequence of assuming a linear response.
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With a short straightforward calculation, I came to this picture:
That is, if the ellipse semi-major and semi-minor axes are given by vectors $\pmb{a}$ and $\pmb{b}$, then the eigenverctors are proportional to $\pmb{a}\pm i\pmb{b}$ (with maybe some complex factors), and their order would give the direction of rotation: from the ...
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Your intuition about the Helmholtz equation is correct in that it is the right way to handle the spatial dependence. If you are in free space, then you should be tackling this problem using a plane-wave basis by taking the Fourier transform of your equation:
$$
\frac{\partial^2 \tilde\phi(t,\mathbf k)}{\partial t^2} +(c^2k^2+\omega_0^2)\tilde\phi(t,\mathbf ...
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Why do you want to have an understanding of the gapless edge states without using bulk topology? If you allow me to use the bulk topology, an argument is that you can continuously move the edge and consider that as an adiabatic parameter which interpolate two systems. To be more precise, you can consider a sphere with part of it in one topological state A ...
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Time-dependent Schrodinger equation is an elliptic PDE if the Hamiltonian is time-independent.
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