diff options
| author | Prefetch | 2022-11-08 18:14:21 +0100 |
|---|---|---|
| committer | Prefetch | 2022-11-08 18:14:21 +0100 |
| commit | 5ed7553b723a9724f55e75261efe2666e75df725 (patch) | |
| tree | 2d893dbe47b11a569a4de12dba05b9eac35f6350 | |
| parent | 70006b2c540543a96e54254823f95348e9f0ed7a (diff) | |
The tweaks and fixes never stop
19 files changed, 62 insertions, 50 deletions
diff --git a/source/_includes/preamble.html b/source/_includes/preamble.html index 4c1838e..6953524 100644 --- a/source/_includes/preamble.html +++ b/source/_includes/preamble.html @@ -12,12 +12,12 @@ <meta name="theme-color" content="#ffffff"> {% if page.layout == "blog" or page.syntax %} -<link rel="stylesheet" href="/infra/css/syntax.css?v=20221027"> +<link rel="stylesheet" href="/infra/css/syntax.css?v=20221108"> {% endif %} {% if page.layout == "concept" or page.maths %} <link rel="stylesheet" href="/infra/css/katex.min.css?v=20221008"> {% endif %} -<link rel="stylesheet" href="/infra/css/main.css?v=20221106"> +<link rel="stylesheet" href="/infra/css/main.css?v=20221108"> {% if jekyll.environment != "development" %} <script data-goatcounter="https://prefetch.goatcounter.com/count" async src="/infra/js/count.js?v=20221103"></script> diff --git a/source/_layouts/default.html b/source/_layouts/default.html index 2f1cac2..66a70a2 100644 --- a/source/_layouts/default.html +++ b/source/_layouts/default.html @@ -2,7 +2,7 @@ <html> <head> {% include preamble.html %} -<title>{{ page.title }} | {{ site.title }}</title> +<title>{{ page.title | replace: "<br>", " " }} | {{ site.title }}</title> </head> <body> {% include navigation.html %} diff --git a/source/blog/2022/things-i-use/index.md b/source/blog/2022/things-i-use/index.md index 2f4a2bc..65247c5 100644 --- a/source/blog/2022/things-i-use/index.md +++ b/source/blog/2022/things-i-use/index.md @@ -1,5 +1,5 @@ --- -title: "Things I use and recommend" +title: "Software and hardware<br>I use and recommend" date: 2022-09-28 layout: "blog" toc: true diff --git a/source/blog/index.md b/source/blog/index.md index 5a2cf54..7ec566d 100644 --- a/source/blog/index.md +++ b/source/blog/index.md @@ -22,7 +22,7 @@ Just a bunch of random ramblings. %} {% for post in in_year %} <li> - <a href="{{ post.url }}">{{ post.title }}</a> + <a href="{{ post.url }}">{{ post.title | replace: "<br>", " " }}</a> </li> {% endfor %} </ul> diff --git a/source/infra/css/main.css b/source/infra/css/main.css index cdf71e4..242c08d 100644 --- a/source/infra/css/main.css +++ b/source/infra/css/main.css @@ -37,10 +37,14 @@ body { font-family: sans-serif; } -h1, h2, h3 {text-align: center;} -h1 {font-size: 2.0rem;} -h2 {font-size: 1.6rem;} -h3 {font-size: 1.2rem;} +h1, h2, h3, h4 {text-align: center;} +h1 {font-size: 2.1rem;} +h2 {font-size: 1.8rem;} +h3 {font-size: 1.5rem;} +h4 { + font-size: 1.2rem; + font-style: italic; +} a { background-color: transparent; text-decoration: none; diff --git a/source/infra/css/syntax.css b/source/infra/css/syntax.css index 50191d2..08e24b4 100644 --- a/source/infra/css/syntax.css +++ b/source/infra/css/syntax.css @@ -1,4 +1,5 @@ .highlighter-rouge { + padding: 0 2px; border: 1px solid; border-color: #999988; background-color: #f8f8f8; diff --git a/source/know/concept/boltzmann-relation/index.md b/source/know/concept/boltzmann-relation/index.md index d5409d2..b528adf 100644 --- a/source/know/concept/boltzmann-relation/index.md +++ b/source/know/concept/boltzmann-relation/index.md @@ -71,15 +71,16 @@ $$\begin{aligned} } \end{aligned}$$ -However, due to their larger mass, -ions are much slower to respond to fluctuations in the above equilibrium. +But due to their large mass, +ions respond much slower to fluctuations in the above equilibrium. Consequently, after a perturbation, -the ions spend much more time in a transient non-equilibrium state +the ions spend more time in a transient non-equilibrium state than the electrons, so this formula for $$n_i$$ is only valid if the perturbation is sufficiently slow, -allowing the ions to keep up. +such that the ions can keep up. Usually, electrons do not suffer the same issue, -thanks to their small mass and fast response. +thanks to their small mass and hence fast response. + ## References diff --git a/source/know/concept/coulomb-logarithm/index.md b/source/know/concept/coulomb-logarithm/index.md index b843eb3..b3be5ac 100644 --- a/source/know/concept/coulomb-logarithm/index.md +++ b/source/know/concept/coulomb-logarithm/index.md @@ -147,12 +147,12 @@ We thus find: $$\begin{aligned} \boxed{ \sigma_\mathrm{small} - = 8 \ln(\Lambda) \sigma_\mathrm{large} - = \frac{q_1^2 q_2^2 \ln\!(\Lambda)}{2 \pi \varepsilon_0^2 |\vb{v}|^4 \mu^2} + = 8 \sigma_\mathrm{large} \ln(\Lambda) + = \frac{q_1^2 q_2^2 \ln(\Lambda)}{2 \pi \varepsilon_0^2 |\vb{v}|^4 \mu^2} } \end{aligned}$$ -Here, $$\ln\!(\Lambda)$$ is known as the **Coulomb logarithm**, +Here, $$\ln(\Lambda)$$ is known as the **Coulomb logarithm**, with the **plasma parameter** $$\Lambda$$ defined below, equal to $$9/2$$ times the number of particles in a sphere with radius $$\lambda_D$$: @@ -168,7 +168,7 @@ $$\begin{aligned} The above relation between $$\sigma_\mathrm{small}$$ and $$\sigma_\mathrm{large}$$ gives us an estimate of how much more often small deflections occur, compared to large ones. -In a typical plasma, $$\ln\!(\Lambda)$$ is between 6 and 25, +In a typical plasma, $$\ln(\Lambda)$$ is between 6 and 25, such that $$\sigma_\mathrm{small}$$ is 2-3 orders of magnitude larger than $$\sigma_\mathrm{large}$$. Note that $$t$$ is now fixed as the period @@ -179,7 +179,7 @@ for significant energy transfer between partices: $$\begin{aligned} \frac{1}{t} = n |\vb{v}| \sigma_\mathrm{small} - = \frac{q_1^2 q_2^2 \ln\!(\Lambda) \: n}{2 \pi \varepsilon_0^2 \mu^2 |\vb{v}|^3} + = \frac{q_1^2 q_2^2 \ln(\Lambda) \: n}{2 \pi \varepsilon_0^2 \mu^2 |\vb{v}|^3} \sim \frac{n}{T^{3/2}} \end{aligned}$$ diff --git a/source/know/concept/fabry-perot-cavity/index.md b/source/know/concept/fabry-perot-cavity/index.md index 6eefc6e..c013f1d 100644 --- a/source/know/concept/fabry-perot-cavity/index.md +++ b/source/know/concept/fabry-perot-cavity/index.md @@ -50,6 +50,7 @@ $$\begin{aligned} A_4 e^{i k_m n_R \ell/2} &= A_2 e^{- i k_m n_C \ell/2} + A_3 e^{i k_m n_C \ell/2} \end{aligned}$$ + $$\begin{aligned} - i k_m n_L A_1 e^{i k_m n_L \ell/2} &= - i k_m n_C A_2 e^{i k_m n_C \ell/2} + i k_m n_C A_3 e^{- i k_m n_C \ell/2} diff --git a/source/know/concept/ion-sound-wave/index.md b/source/know/concept/ion-sound-wave/index.md index cb86c04..8749f1a 100644 --- a/source/know/concept/ion-sound-wave/index.md +++ b/source/know/concept/ion-sound-wave/index.md @@ -95,16 +95,16 @@ we make the following plane-wave ansatz: $$\begin{aligned} n_{i1}(\vb{r}, t) - &= n_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + &= n_{i1} \exp(i \vb{k} \cdot \vb{r} - i \omega t) \\ n_{e1}(\vb{r}, t) - &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + &= n_{e1} \exp(i \vb{k} \cdot \vb{r} - i \omega t) \\ \vb{u}_{i1}(\vb{r}, t) - &= \vb{u}_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + &= \vb{u}_{i1} \exp(i \vb{k} \cdot \vb{r} - i \omega t) \\ \phi_1(\vb{r}, t) - &= \phi_1 \,\,\exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + &= \phi_1 \,\,\exp(i \vb{k} \cdot \vb{r} - i \omega t) \end{aligned}$$ Which we then insert into the momentum equations for the ions and electrons: diff --git a/source/know/concept/kramers-kronig-relations/index.md b/source/know/concept/kramers-kronig-relations/index.md index 3880113..711023e 100644 --- a/source/know/concept/kramers-kronig-relations/index.md +++ b/source/know/concept/kramers-kronig-relations/index.md @@ -58,7 +58,8 @@ $$\begin{aligned} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}$$ -From the definition of the Fourier transform we know that $$2 \pi A B / |s| = 1$$: +From the definition of the Fourier transform we know that +$$2 \pi A B / |s| = 1$$: $$\begin{aligned} \tilde{\chi}(\omega) diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md index 7dc5dbf..be47567 100644 --- a/source/know/concept/langmuir-waves/index.md +++ b/source/know/concept/langmuir-waves/index.md @@ -130,8 +130,8 @@ where we neglect $$(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1}$$ because $$\vb{u}_{e1}$$ is so small by assumption: $$\begin{gathered} - m_e (n_{e0} \!+\! n_{e1}) \Big( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t} - + \big( (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \cdot \nabla \big) (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \Big) + m_e (n_{e0} \!+\! n_{e1}) \bigg( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t} + + \big( (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \cdot \nabla \big) (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \bigg) = q_e \big( n_{e0} \!+\! n_{e1} \big) \big( \vb{E}_0 \!+\! \vb{E}_1 \big) \\ \implies \qquad diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md index a84d274..1f42f73 100644 --- a/source/know/concept/laser-rate-equations/index.md +++ b/source/know/concept/laser-rate-equations/index.md @@ -187,7 +187,8 @@ $$\begin{aligned} } \end{aligned}$$ -To rewrite this, we replace $$|\vb{E}|^2$$ with the photon number $$N_p$$ as follows, +To rewrite this, we replace $$|\vb{E}|^2$$ +with the photon number $$N_p$$ as follows, with $$U = \varepsilon_0 n^2 |\vb{E}|^2 / 2$$ being the energy density of the light: $$\begin{aligned} diff --git a/source/know/concept/lehmann-representation/index.md b/source/know/concept/lehmann-representation/index.md index 74bd457..0dec718 100644 --- a/source/know/concept/lehmann-representation/index.md +++ b/source/know/concept/lehmann-representation/index.md @@ -40,7 +40,7 @@ $$\begin{aligned} \Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar} \hat{c}_{\nu'}^\dagger}{n} \end{aligned}$$ -Where we used that the trace $$\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$$ +Where we used that the trace $$\Tr(x) = \sum_{n} \matrixel{n}{x}{n}$$ is invariant under cyclic permutations of $$x$$. The $$\Ket{n}$$ form a basis of eigenstates of $$\hat{H}$$, so we insert an identity operator $$\sum_{n'} \Ket{n'} \Bra{n'}$$: diff --git a/source/know/concept/martingale/index.md b/source/know/concept/martingale/index.md index 9d3c6b4..53a346a 100644 --- a/source/know/concept/martingale/index.md +++ b/source/know/concept/martingale/index.md @@ -45,12 +45,14 @@ Modifying property (3) leads to two common generalizations. The stochastic process $$M_t$$ above is a **submartingale** if the current value is a lower bound for the expectation: -3. For $$0 \le s \le t$$, the conditional expectation $$\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$$. +3. For $$0 \le s \le t$$, the conditional expectation + $$\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$$. Analogouly, $$M_t$$ is a **supermartingale** if the current value is an upper bound instead: -3. For $$0 \le s \le t$$, the conditional expectation $$\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$$. +3. For $$0 \le s \le t$$, the conditional expectation + $$\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$$. Clearly, submartingales and supermartingales are *biased* random walks, since they will tend to increase and decrease with time, respectively. diff --git a/source/know/concept/maxwell-boltzmann-distribution/index.md b/source/know/concept/maxwell-boltzmann-distribution/index.md index ebb2460..318e659 100644 --- a/source/know/concept/maxwell-boltzmann-distribution/index.md +++ b/source/know/concept/maxwell-boltzmann-distribution/index.md @@ -81,7 +81,7 @@ after normalization: $$\begin{aligned} f(p_x, p_y, p_z) - = \Big( \frac{1}{2 \pi m k_B T} \Big)^{3/2} \exp\!\Big( \!-\!\frac{(p_x^2 + p_y^2 + p_z^2)}{2 m k_B T} \Big) + = \Big( \frac{1}{2 \pi m k_B T} \Big)^{3/2} \exp\!\bigg( \!-\!\frac{(p_x^2 + p_y^2 + p_z^2)}{2 m k_B T} \bigg) \end{aligned}$$ We now rewrite this using the velocities $$v_x = p_x / m$$, @@ -90,7 +90,7 @@ and update the normalization, giving: $$\begin{aligned} \boxed{ f(v_x, v_y, v_z) - = \Big( \frac{m}{2 \pi k_B T} \Big)^{3/2} \exp\!\Big( \!-\!\frac{m (v_x^2 + v_y^2 + v_z^2)}{2 k_B T} \Big) + = \Big( \frac{m}{2 \pi k_B T} \Big)^{3/2} \exp\!\bigg( \!-\!\frac{m (v_x^2 + v_y^2 + v_z^2)}{2 k_B T} \bigg) } \end{aligned}$$ diff --git a/source/know/concept/random-phase-approximation/index.md b/source/know/concept/random-phase-approximation/index.md index 698e1e7..0d0b428 100644 --- a/source/know/concept/random-phase-approximation/index.md +++ b/source/know/concept/random-phase-approximation/index.md @@ -24,7 +24,7 @@ and $$k_F^2$$ from the energy $$1 / \beta$$: $$\begin{aligned} \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty \cdots \:\dd{\vb{k}} - \:\:\sim\:\: + \quad\sim\quad k_F^5 \end{aligned}$$ diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md index f934ab7..9666167 100644 --- a/source/know/concept/self-steepening/index.md +++ b/source/know/concept/self-steepening/index.md @@ -19,7 +19,7 @@ nonlinear Schrödinger equation: $$\begin{aligned} 0 - = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma \Big(1 + \frac{i}{\omega_0} \pdv{}{t} \Big) \big(|A|^2 A\big) + = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma \Big(1 + \frac{i}{\omega_0} \pdv{}{t} \Big) |A|^2 A \end{aligned}$$ Where $$\omega_0$$ is the angular frequency of the pump. diff --git a/source/know/concept/stokes-law/index.md b/source/know/concept/stokes-law/index.md index 3a02a83..40212ef 100644 --- a/source/know/concept/stokes-law/index.md +++ b/source/know/concept/stokes-law/index.md @@ -50,10 +50,10 @@ $$\begin{gathered} \\ v_r = U f(r) \cos\theta - \qquad + \qquad \quad v_\theta = - U g(r) \sin\theta - \qquad + \qquad \quad v_\phi = 0 \end{gathered}$$ @@ -76,7 +76,7 @@ $$\begin{aligned} \\ &= U \dv{f}{r} \cos\theta - \frac{U g}{r} \cos\theta + \frac{2 U f}{r} \cos\theta - \frac{U g}{r} \cos\theta \\ - &= U \cos\theta \Big( \dv{f}{r} + \frac{2}{r} f - \frac{2}{r} g \Big) + &= U \Big( \dv{f}{r} + \frac{2}{r} f - \frac{2}{r} g \Big) \cos\theta \end{aligned}$$ The parenthesized expression must be zero for all $$r$$, @@ -103,16 +103,16 @@ which is as follows for our ansatz $$p(r, \theta)$$: $$\begin{aligned} 0 = \nabla^2 p - &= \frac{1}{r^2} \pdv{}{r}\Big( r^2 \pdv{p}{r} \Big) + \frac{1}{r^2 \sin\theta} \pdv{}{\theta}\Big( \sin\theta \pdv{p}{\theta} \Big) + &= \frac{1}{r^2} \pdv{}{r}\Big( r^2 \pdv{p}{r} \Big) + \frac{1}{r^2 \sin\theta} \pdv{}{\theta}\Big( \pdv{p}{\theta} \sin\theta \Big) \\ 0 &= \frac{\eta U \cos\theta}{r^2} \dv{}{r}\Big( r^2 \dv{q}{r} \Big) - \frac{\eta U q}{r^2 \sin\theta} \pdv{}{\theta}\Big( \sin^2\theta \Big) \\ - &= \frac{\eta U \cos\theta}{r^2} \dv{}{r}\Big( r^2 \dv{q}{r} \Big) + &= \frac{\eta U}{r^2} \dv{}{r}\Big( r^2 \dv{q}{r} \Big) \cos\theta - \frac{2 \eta U q}{r^2 \sin\theta} \sin\theta \cos\theta \\ - &= \eta U \cos\theta \Big( \dvn{2}{q}{r} + \frac{2}{r} \dv{q}{r} - \frac{2}{r^2} q \Big) + &= \eta U \Big( \dvn{2}{q}{r} + \frac{2}{r} \dv{q}{r} - \frac{2}{r^2} q \Big) \cos\theta \end{aligned}$$ Again, the parenthesized expression must be zero for all $$r$$, @@ -129,7 +129,7 @@ The pressure is therefore: $$\begin{aligned} p - = \eta U \cos\theta \Big( \frac{C_3}{r^2} + C_4 r \Big) + = \eta U \Big( \frac{C_3}{r^2} + C_4 r \Big) \cos\theta \end{aligned}$$ Consequently, its gradient $$\nabla p$$ in spherical coordinates is as follows: @@ -137,7 +137,7 @@ Consequently, its gradient $$\nabla p$$ in spherical coordinates is as follows: $$\begin{aligned} \nabla p = \vu{e}_r \pdv{p}{r} + \vu{e}_\theta \frac{1}{r} \pdv{p}{\theta} - = \vu{e}_r \Big( \eta U \cos\theta \dv{q}{r} \Big) - \vu{e}_\theta \Big( \eta U \sin\theta \frac{q}{r} \Big) + = \vu{e}_r \Big( \eta U \dv{q}{r} \cos\theta \Big) - \vu{e}_\theta \Big( \eta U \frac{q}{r} \sin\theta \Big) \end{aligned}$$ According to the Stokes equation, this equals $$\eta \nabla^2 \va{v}$$. @@ -148,24 +148,24 @@ $$\begin{aligned} &= \pdvn{2}{v_r}{r} + \frac{1}{r^2} \pdvn{2}{v_r}{\theta} + \frac{2}{r} \pdv{v_r}{r} + \frac{\cot\theta}{r^2} \pdv{v_r}{\theta} - \frac{2}{r^2} \pdv{v_\theta}{\theta} - \frac{2}{r^2} v_r - \frac{2 \cot\theta}{r^2} v_\theta \\ - &= U \cos\theta \Big( \dvn{2}{f}{r} - \frac{1}{r^2} f + \frac{2}{r} \dv{f}{r} - \frac{1}{r^2} f - + \frac{2}{r^2} g - \frac{2}{r^2} f + \frac{2}{r^2} g \Big) + &= U \Big( \dvn{2}{f}{r} - \frac{1}{r^2} f + \frac{2}{r} \dv{f}{r} - \frac{1}{r^2} f + + \frac{2}{r^2} g - \frac{2}{r^2} f + \frac{2}{r^2} g \Big) \cos\theta \\ - &= U \cos\theta \Big( \dvn{2}{f}{r} + \frac{2}{r} \dv{f}{r} - \frac{4}{r^2} f + \frac{4}{r^2} g \Big) + &= U \Big( \dvn{2}{f}{r} + \frac{2}{r} \dv{f}{r} - \frac{4}{r^2} f + \frac{4}{r^2} g \Big) \cos\theta \end{aligned}$$ Substituting $$g$$ for the expression we found from incompressibility lets us simplify this: $$\begin{aligned} \eta (\nabla^2 \va{v})_r - &= \eta U \cos\theta \Big( \dvn{2}{f}{r} + \frac{4}{r} \dv{f}{r} \Big) + &= \eta U \Big( \dvn{2}{f}{r} + \frac{4}{r} \dv{f}{r} \Big) \cos\theta \end{aligned}$$ The Stokes equation says that this must be equal to the $$r$$-component of $$\nabla p$$: $$\begin{aligned} - \eta U \cos\theta \Big( \dvn{2}{f}{r} + \frac{4}{r} \dv{f}{r} \Big) - = \eta U \cos\theta \Big( \!-\! \frac{2 C_3}{r^3} + C_4 \Big) + \eta U \Big( \dvn{2}{f}{r} + \frac{4}{r} \dv{f}{r} \Big) \cos\theta + = \eta U \Big( \!-\! \frac{2 C_3}{r^3} + C_4 \Big) \cos\theta \end{aligned}$$ Where we have inserted $$\idv{q}{r}$$. @@ -213,14 +213,15 @@ $$\begin{gathered} \\ \boxed{ v_r - = U \cos\theta \Big( 1 + \frac{a^3}{2 r^3} - \frac{3 a}{2 r} \Big) + = U \Big( 1 + \frac{a^3}{2 r^3} - \frac{3 a}{2 r} \Big) \cos\theta \qquad v_\theta - = - U \sin\theta \Big( 1 - \frac{a^3}{4 r^3} - \frac{3 a}{4 r} \Big) + = - U \Big( 1 - \frac{a^3}{4 r^3} - \frac{3 a}{4 r} \Big) \sin\theta } \end{gathered}$$ + ## Drag force From the definition of [viscosity](/know/concept/viscosity/), |