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# Stabilization scheme
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Some of the terms in the EDF introduce the time-dependent propagation of the high-momenta components. Examples are terms that contain division by density and may lead to noise generation in regions where density vanishes.
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These modes can amplify during the time-dependent propagation and destabilize the integration scheme. To avoid this, we introduced the filtering scheme.
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1. compute mean-field $`V_\sigma(\vec{r})`$,
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2. go to Fourier space $`V_\sigma(\vec{k})`$,
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3. apply filter function $`\tilde{V}_\sigma(\vec{k})=V_\sigma(\vec{k})\cdot FD(\frac{k^2}{2m},\mu, T)`$,
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4. go back to coordinate space $`\tilde{V}_\sigma(\vec{r})`$ and use it during the time-propagation.
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As the filter function, we use the Fermi-Dirac function:
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```math
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FD(e_k,\mu, T)=\frac{1}{\exp[\frac{e_k-\mu}{T}]+1}
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```
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The same procedure can be used to folder noise that is generated in the pairing potential $`\Delta(\vec{r})`$.
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# Testing script
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You can use the attached script [tools/high-frequency-filter.py](https://gitlab.fizyka.pw.edu.pl/wtools/w-bsk/-/blob/devel/tools/high-frequency-filter.py) to test the impact of the filtering scheme on the input signal. Below is an example of the script output.
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# Controlling the filter
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The filter can be controlled via the input file:
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```bash
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# -------------- HIGH K-WAVES FILTER ----------------
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# See: Wiki -> Stabilization of the time-dependent code
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# hkf_mode 1 # 0 - no noise filtering (default)
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# 1 - noise filtering for mean-fields only
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# 2 - noise filtering of mean-fields and pairing field
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# hkf_mu 0.9 # mu parameter of the Fermi-Dirac (filtering) function, in Ec units, default=0.9
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# hkf_T 0.02 # T parameter of the Fermi-Dirac (filtering) function, in Ec units, default=0.02
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``` |
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\ No newline at end of file |
