From 7c2d27ca89c5b096694b950c766e50df2dc87001 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 8 Jan 2022 14:09:13 +0100 Subject: Minor fixes, rename "Ion Sound Wave" and "Ito Process" --- config.toml | 3 + .../electric-dipole-approximation/index.pdc | 12 +- .../know/concept/guiding-center-theory/index.pdc | 2 +- .../concept/hellmann-feynman-theorem/index.pdc | 2 +- content/know/concept/ion-sound-wave/index.pdc | 265 +++++++++++++++ content/know/concept/ion-sound-waves/index.pdc | 265 --------------- content/know/concept/ito-calculus/index.pdc | 367 --------------------- content/know/concept/ito-process/index.pdc | 367 +++++++++++++++++++++ content/know/concept/langmuir-waves/index.pdc | 8 +- .../know/concept/magnetohydrodynamics/index.pdc | 4 +- .../time-dependent-perturbation-theory/index.pdc | 6 +- 11 files changed, 652 insertions(+), 649 deletions(-) create mode 100644 content/know/concept/ion-sound-wave/index.pdc delete mode 100644 content/know/concept/ion-sound-waves/index.pdc delete mode 100644 content/know/concept/ito-calculus/index.pdc create mode 100644 content/know/concept/ito-process/index.pdc diff --git a/config.toml b/config.toml index d5a8d6d..3463bda 100644 --- a/config.toml +++ b/config.toml @@ -3,3 +3,6 @@ languageCode = "en-us" title = "Prefetch" disableKinds = ["taxonomy", "term"] + +[security.exec] +allow = ['^dart-sass-embedded$', '^go$', '^npx$', '^postcss$', '^pandoc$'] diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc index 96b4fed..265babf 100644 --- a/content/know/concept/electric-dipole-approximation/index.pdc +++ b/content/know/concept/electric-dipole-approximation/index.pdc @@ -46,7 +46,7 @@ In an electromagnetic wave, $\vec{A}$ is oscillating sinusoidally in time and space as follows: $$\begin{aligned} - \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) + \vec{A}(\vec{r}, t) = - i \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) \end{aligned}$$ The corresponding perturbative @@ -59,7 +59,7 @@ $$\begin{aligned} = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) \end{aligned}$$ -Where $\vec{E}_0 = i \omega \vec{A}_0$. +Where $\vec{E}_0 = \omega \vec{A}_0$. Let us restrict ourselves to visible light, whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$. Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$, @@ -82,7 +82,7 @@ and the electron quantum-mechanically. Next, we want to convert $\hat{H}_1$ to use the electric field $\vec{E}$ instead of the potential $\vec{A}$. To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$ -and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/): +and evaluate this in the [interaction picture](/know/concept/interaction-picture/): $$\begin{aligned} \matrixel{2}{\dv*{\vec{r}}{t}}{1} @@ -95,15 +95,15 @@ $$\begin{aligned} Therefore, $\vec{P} / m = i \omega_0 \vec{r}$, where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition, -close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating. +close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating, i.e. $\omega \approx \omega_0$. We thus get: $$\begin{aligned} \hat{H}_1(t) &= - \frac{q}{m} \vec{P} \cdot \vec{A} - = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t) + = - (- i i) q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t) \\ - &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t) + &\approx - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t) = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t) \end{aligned}$$ diff --git a/content/know/concept/guiding-center-theory/index.pdc b/content/know/concept/guiding-center-theory/index.pdc index 12abac0..dee10ba 100644 --- a/content/know/concept/guiding-center-theory/index.pdc +++ b/content/know/concept/guiding-center-theory/index.pdc @@ -174,7 +174,7 @@ $$\begin{aligned} m \dv{\vb{u}_{gc}}{t} = q \big( \vb{E} + \vb{u}_{gc} \cross \vb{B} + \vb{u}_{gc} \cross (\vb{x}_L \cdot \nabla) \vb{B} - + + \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} \big) + + \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} \big) \end{aligned}$$ We approximate this by taking the average over a single gyration, diff --git a/content/know/concept/hellmann-feynman-theorem/index.pdc b/content/know/concept/hellmann-feynman-theorem/index.pdc index 3b88cd8..1d2fe82 100644 --- a/content/know/concept/hellmann-feynman-theorem/index.pdc +++ b/content/know/concept/hellmann-feynman-theorem/index.pdc @@ -74,7 +74,7 @@ to minimize energies with respect to $\lambda$: $$\begin{aligned} \boxed{ \nabla_\lambda E_n - = \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} + = \matrixel{\psi_n}{\nabla_\lambda \hat{H}}{\psi_n} } \end{aligned}$$ diff --git a/content/know/concept/ion-sound-wave/index.pdc b/content/know/concept/ion-sound-wave/index.pdc new file mode 100644 index 0000000..5cba1d0 --- /dev/null +++ b/content/know/concept/ion-sound-wave/index.pdc @@ -0,0 +1,265 @@ +--- +title: "Ion sound wave" +firstLetter: "I" +publishDate: 2021-10-31 +categories: +- Physics +- Plasma physics + +date: 2021-10-31T09:38:14+01:00 +draft: false +markup: pandoc +--- + +# Ion sound wave + +In a plasma, electromagnetic interactions allow +compressional longitudinal waves to propagate +at lower temperatures and pressures +than would be possible in a neutral gas. + +We start from the [two-fluid model's](/know/concept/two-fluid-equations/) momentum equations, +rewriting the [electric field](/know/concept/electric-field/) $\vb{E} = - \nabla \phi$ +and the pressure gradient $\nabla p = \gamma k_B T \nabla n$, +and arguing that $m_e \approx 0$ because $m_e \ll m_i$: + +$$\begin{aligned} + m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} + &= - q_i n_i \nabla \phi - \gamma_i k_B T_i \nabla n_i + \\ + 0 + &= - q_e n_e \nabla \phi - \gamma_e k_B T_e \nabla n_e +\end{aligned}$$ + +Note that we neglect ion-electron collisions, +and allow for separate values of $\gamma$. +We split $n_i$, $n_e$, $\vb{u}_i$ and $\phi$ into an equilibrium +(subscript $0$) and a perturbation (subscript $1$): + +$$\begin{aligned} + n_i + = n_{i0} + n_{i1} + \qquad + n_e + = n_{e0} + n_{e1} + \qquad + \vb{u}_i + = \vb{u}_{i0} + \vb{u}_{i1} + \qquad + \phi + = \phi_0 + \phi_1 +\end{aligned}$$ + +Where the perturbations $n_{i1}$, $n_{e1}$, $\vb{u}_{i1}$ and $\phi_1$ are tiny, +and the equilibrium components $n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$ +by definition satisfy: + +$$\begin{aligned} + \pdv{n_{i0}}{t} = 0 + \qquad + \frac{\mathrm{D} \vb{u}_{i0}}{\mathrm{D} t} = 0 + \qquad + \nabla n_{i0} = \nabla n_{e0} = 0 + \qquad + \vb{u}_{i0} = 0 + \qquad + \phi_0 = 0 +\end{aligned}$$ + +Inserting this decomposition into the momentum equations +yields new equations. +Note that we will implicitly use $\vb{u}_{i0} = 0$ +to pretend that the [material derivative](/know/concept/material-derivative/) +$\mathrm{D}/\mathrm{D} t$ is linear: + +$$\begin{aligned} + m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t} + &= - q_i (n_{i0} \!+\! n_{i1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_i k_B T_i \nabla (n_{i0} \!+\! n_{i1}) + \\ + 0 + &= - q_e (n_{e0} \!+\! n_{e1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_e k_B T_e \nabla (n_{e0} \!+\! n_{e1}) +\end{aligned}$$ + +Using the defined properties of the equilibrium components +$n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$, +and neglecting all products of perturbations for being small, +this reduces to: + +$$\begin{aligned} + m_i n_{i0} \pdv{\vb{u}_{i1}}{t} + &= - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1} + \\ + 0 + &= - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1} +\end{aligned}$$ + +Because we are interested in linear waves, +we make the following plane-wave ansatz: + +$$\begin{aligned} + n_{i1}(\vb{r}, t) + &= n_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \\ + n_{e1}(\vb{r}, t) + &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \\ + \vb{u}_{i1}(\vb{r}, t) + &= \vb{u}_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \\ + \phi_1(\vb{r}, t) + &= \phi_1 \,\,\exp\!(i \vb{k} \cdot \vb{r} - i \omega t) +\end{aligned}$$ + +Which we then insert into the momentum equations for the ions and electrons: + +$$\begin{aligned} + - i \omega m_i n_{i0} \vb{u}_{i1} + &= - i \vb{k} q_i n_{i0} \phi_1 - i \vb{k} \gamma_i k_B T_i n_{i1} + \\ + 0 + &= - i \vb{k} q_e n_{e0} \phi_1 - i \vb{k} \gamma_e k_B T_e n_{e1} +\end{aligned}$$ + +The electron equation can easily be rearranged +to get a relation between $n_{e1}$ and $n_{e0}$: + +$$\begin{aligned} + i \vb{k} \gamma_e k_B T_e n_{e1} + = - i \vb{k} q_e n_{e0} \phi_1 + \quad \implies \quad + n_{e1} + = - \frac{q_e \phi_1}{\gamma_e k_B T_e} n_{e0} +\end{aligned}$$ + +Due to their low mass, the electrons' heat conductivity +can be regarded as infinite compared to the ions'. +In that case, all electron gas compression is isothermal, +meaning it obeys the ideal gas law $p_e = n_e k_B T_e$, so that $\gamma_e = 1$. +Note that this yields the first-order term of a Taylor expansion +of the [Boltzmann relation](/know/concept/boltzmann-relation/). + +At equilibrium, quasi-neutrality demands that $n_{i0} = n_{e0} = n_0$, +so we can rearrange the above relation to $n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$, +which we insert into the ion equation to get: + +$$\begin{gathered} + i \omega m_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \vb{u}_{i1} + = - i q_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \phi_1 \vb{k} - i \gamma_i k_B T_i n_{i1} \vb{k} + \\ + \implies \qquad + \omega m_i \frac{T_e n_{e1}}{q_e \phi_1} \vb{k} \cdot \vb{u}_{i1} + = T_e n_{e1} |\vb{k}|^2 - \gamma_i T_i n_{i1} |\vb{k}|^2 +\end{gathered}$$ + +Where we have taken the dot product with $\vb{k}$, +and used that $q_i / q_e = -1$. +In order to simplify this equation, +we turn to the two-fluid ion continuity relation: + +$$\begin{aligned} + 0 + &= \pdv{(n_{i0} \!+\! n_{i1})}{t} + \nabla \cdot \Big( (n_{i0} \!+\! n_{i1}) (\vb{u}_{i0} \!+\! \vb{u}_{i1}) \Big) + \approx \pdv{n_{i1}}{t} + n_{i0} \nabla \cdot \vb{u}_{i1} +\end{aligned}$$ + +Then we insert our plane-wave ansatz, +and substitute $n_{i0} = n_0$ as before, yielding: + +$$\begin{aligned} + 0 + = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1} + \quad \implies \quad + \vb{k} \cdot \vb{u}_{i1} + = \omega \frac{n_{i1}}{n_{i0}} + = \omega \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}} +\end{aligned}$$ + +Substituting this in the ion momentum equation +leads us to a dispersion relation $\omega(\vb{k})$: + +$$\begin{gathered} + \omega^2 m_i \frac{T_e n_{e1}}{q_e \phi_1} \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}} + = \omega^2 m_i \frac{n_{i1}}{k_B} + = |\vb{k}|^2 \big( T_e n_{e1} - \gamma_i T_i n_{i1} \big) + \\ + \implies \qquad + \omega^2 + = \frac{|\vb{k}|^2}{m_i} \Big( k_B T_e \frac{n_{e1}}{n_{i1}} - \gamma_i k_B T_i \Big) +\end{gathered}$$ + +Finally, we would like to find an expression for $n_{e1} / n_{i1}$. +It cannot be $1$, because then $\phi_1$ could not be nonzero, +according to [Gauss' law](/know/concept/maxwells-equations/). +Nevertheless, authors often ignore this fact, +thereby making the so-called **plasma approximation**. +We will not, and therefore turn to Gauss' law: + +$$\begin{aligned} + \varepsilon_0 \nabla \cdot \vb{E} + = - \varepsilon_0 \nabla^2 \phi_1 + = q_i n_i - q_e n_e + = - q_e (n_{i1} - n_{e1}) +\end{aligned}$$ + +One final time, we insert our plane-wave ansatz, +and use our Boltzmann-like relation between $n_{e1}$ and $n_{e0}$ +to substitute $\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$: + +$$\begin{gathered} + q_e (n_{e1} - n_{i1}) + = |\vb{k}|^2 \varepsilon_0 \phi_1 + = - |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e n_{e0}} + \\ + \implies \qquad + n_{i1} + = n_{e1} + |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e^2 n_{e0}} + = n_{e1} \big( 1 + |\vb{k}|^2 \lambda_{De}^2 \big) +\end{gathered}$$ + +Where $\lambda_{De}$ is the electron [Debye length](/know/concept/debye-length/). +We thus reach the following dispersion relation, +which governs **ion sound waves** or **ion acoustic waves**: + +$$\begin{aligned} + \boxed{ + \omega^2 + = \frac{|\vb{k}|^2}{m_i} \bigg( \frac{k_B T_e}{1 + |\vb{k}|^2 \lambda_{De}^2} + \gamma_i k_B T_i \bigg) + } +\end{aligned}$$ + +The aforementioned plasma approximation is valid if $|\vb{k}| \lambda_{De} \ll 1$, +which is often reasonable, +in which case this dispersion relation reduces to: + +$$\begin{aligned} + \omega^2 + = \frac{|\vb{k}|^2}{m_i} \bigg( k_B T_e + \gamma_i k_B T_i \bigg) +\end{aligned}$$ + +The phase velocity $v_s$ of these waves, +i.e. the speed of sound, is then given by: + +$$\begin{aligned} + \boxed{ + v_s + = \frac{\omega}{k} + = \sqrt{\frac{k_B T_e}{m_i} + \frac{\gamma_i k_B T_i}{m_i}} + } +\end{aligned}$$ + +Curiously, unlike a neutral gas, +this velocity is nonzero even if $T_i = 0$, +meaning that the waves still exist then. +In fact, usually the electron temperature $T_e$ dominates $T_e \gg T_i$, +even though the main feature of these waves +is that they involve ion density fluctuations $n_{i1}$. + + + +## References +1. F.F. Chen, + *Introduction to plasma physics and controlled fusion*, + 3rd edition, Springer. +2. M. Salewski, A.H. Nielsen, + *Plasma physics: lecture notes*, + 2021, unpublished. diff --git a/content/know/concept/ion-sound-waves/index.pdc b/content/know/concept/ion-sound-waves/index.pdc deleted file mode 100644 index c56188b..0000000 --- a/content/know/concept/ion-sound-waves/index.pdc +++ /dev/null @@ -1,265 +0,0 @@ ---- -title: "Ion sound waves" -firstLetter: "I" -publishDate: 2021-10-31 -categories: -- Physics -- Plasma physics - -date: 2021-10-31T09:38:14+01:00 -draft: false -markup: pandoc ---- - -# Ion sound waves - -In a plasma, electromagnetic interactions allow -compressional longitudinal waves to propagate -at lower temperatures and pressures -than would be possible in a neutral gas. - -We start from the [two-fluid model's](/know/concept/two-fluid-equations/) momentum equations, -rewriting the [electric field](/know/concept/electric-field/) $\vb{E} = - \nabla \phi$ -and the pressure gradient $\nabla p = \gamma k_B T \nabla n$, -and arguing that $m_e \approx 0$ because $m_e \ll m_i$: - -$$\begin{aligned} - m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} - &= - q_i n_i \nabla \phi - \gamma_i k_B T_i \nabla n_i - \\ - 0 - &= - q_e n_e \nabla \phi - \gamma_e k_B T_e \nabla n_e -\end{aligned}$$ - -Note that we neglect ion-electron collisions, -and allow for separate values of $\gamma$. -We split $n_i$, $n_e$, $\vb{u}_i$ and $\phi$ into an equilibrium -(subscript $0$) and a perturbation (subscript $1$): - -$$\begin{aligned} - n_i - = n_{i0} + n_{i1} - \qquad - n_e - = n_{e0} + n_{e1} - \qquad - \vb{u}_i - = \vb{u}_{i0} + \vb{u}_{i1} - \qquad - \phi - = \phi_0 + \phi_1 -\end{aligned}$$ - -Where the perturbations $n_{i1}$, $n_{e1}$, $\vb{u}_{i1}$ and $\phi_1$ are tiny, -and the equilibrium components $n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$ -by definition satisfy: - -$$\begin{aligned} - \pdv{n_{i0}}{t} = 0 - \qquad - \frac{\mathrm{D} \vb{u}_{i0}}{\mathrm{D} t} = 0 - \qquad - \nabla n_{i0} = \nabla n_{e0} = 0 - \qquad - \vb{u}_{i0} = 0 - \qquad - \phi_0 = 0 -\end{aligned}$$ - -Inserting this decomposition into the momentum equations -yields new equations. -Note that we will implicitly use $\vb{u}_{i0} = 0$ -to pretend that the [material derivative](/know/concept/material-derivative/) -$\mathrm{D}/\mathrm{D} t$ is linear: - -$$\begin{aligned} - m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t} - &= - q_i (n_{i0} \!+\! n_{i1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_i k_B T_i \nabla (n_{i0} \!+\! n_{i1}) - \\ - 0 - &= - q_e (n_{e0} \!+\! n_{e1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_e k_B T_e \nabla (n_{e0} \!+\! n_{e1}) -\end{aligned}$$ - -Using the defined properties of the equilibrium components -$n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$, -and neglecting all products of perturbations for being small, -this reduces to: - -$$\begin{aligned} - m_i n_{i0} \pdv{\vb{u}_{i1}}{t} - &= - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1} - \\ - 0 - &= - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1} -\end{aligned}$$ - -Because we are interested in linear waves, -we make the following plane-wave ansatz: - -$$\begin{aligned} - n_{i1}(\vb{r}, t) - &= n_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) - \\ - n_{e1}(\vb{r}, t) - &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) - \\ - \vb{u}_{i1}(\vb{r}, t) - &= \vb{u}_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) - \\ - \phi_1(\vb{r}, t) - &= \phi_1 \,\,\exp\!(i \vb{k} \cdot \vb{r} - i \omega t) -\end{aligned}$$ - -Which we then insert into the momentum equations for the ions and electrons: - -$$\begin{aligned} - - i \omega m_i n_{i0} \vb{u}_{i1} - &= - i \vb{k} q_i n_{i0} \phi_1 - i \vb{k} \gamma_i k_B T_i n_{i1} - \\ - 0 - &= - i \vb{k} q_e n_{e0} \phi_1 - i \vb{k} \gamma_e k_B T_e n_{e1} -\end{aligned}$$ - -The electron equation can easily be rearranged -to get a relation between $n_{e1}$ and $n_{e0}$: - -$$\begin{aligned} - i \vb{k} \gamma_e k_B T_e n_{e1} - = - i \vb{k} q_e n_{e0} \phi_1 - \quad \implies \quad - n_{e1} - = - \frac{q_e \phi_1}{\gamma_e k_B T_e} n_{e0} -\end{aligned}$$ - -Due to their low mass, the electrons' heat conductivity -can be regarded as infinite compared to the ions'. -In that case, all electron gas compression is isothermal, -meaning it obeys the ideal gas law $p_e = n_e k_B T_e$, so that $\gamma_e = 1$. -Note that this yields the first-order term of a Taylor expansion -of the [Boltzmann relation](/know/concept/boltzmann-relation/). - -At equilibrium, quasi-neutrality demands that $n_{i0} = n_{e0} = n_0$, -so we can rearrange the above relation to $n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$, -which we insert into the ion equation to get: - -$$\begin{gathered} - i \omega m_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \vb{u}_{i1} - = - i q_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \phi_1 \vb{k} - i \gamma_i k_B T_i n_{i1} \vb{k} - \\ - \implies \qquad - \omega m_i \frac{T_e n_{e1}}{q_e \phi_1} \vb{k} \cdot \vb{u}_{i1} - = T_e n_{e1} |\vb{k}|^2 - \gamma_i T_i n_{i1} |\vb{k}|^2 -\end{gathered}$$ - -Where we have taken the dot product with $\vb{k}$, -and used that $q_i / q_e = -1$. -In order to simplify this equation, -we turn to the two-fluid ion continuity relation: - -$$\begin{aligned} - 0 - &= \pdv{(n_{i0} \!+\! n_{i1})}{t} + \nabla \cdot \Big( (n_{i0} \!+\! n_{i1}) (\vb{u}_{i0} \!+\! \vb{u}_{i1}) \Big) - \approx \pdv{n_{i1}}{t} + n_{i0} \nabla \cdot \vb{u}_{i1} -\end{aligned}$$ - -Then we insert our plane-wave ansatz, -and substitute $n_{i0} = n_0$ as before, yielding: - -$$\begin{aligned} - 0 - = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1} - \quad \implies \quad - \vb{k} \cdot \vb{u}_{i1} - = \omega \frac{n_{i1}}{n_{i0}} - = \omega \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}} -\end{aligned}$$ - -Substituting this in the ion momentum equation -leads us to a dispersion relation $\omega(\vb{k})$: - -$$\begin{gathered} - \omega^2 m_i \frac{T_e n_{e1}}{q_e \phi_1} \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}} - = \omega^2 m_i \frac{n_{i1}}{k_B} - = |\vb{k}|^2 \big( T_e n_{e1} - \gamma_i T_i n_{i1} \big) - \\ - \implies \qquad - \omega^2 - = \frac{|\vb{k}|^2}{m_i} \Big( k_B T_e \frac{n_{e1}}{n_{i1}} - \gamma_i k_B T_i \Big) -\end{gathered}$$ - -Finally, we would like to find an expression for $n_{e1} / n_{i1}$. -It cannot be $1$, because then $\phi_1$ could not be nonzero, -according to [Gauss' law](/know/concept/maxwells-equations/). -Nevertheless, authors often ignore this fact, -thereby making the so-called **plasma approximation**. -We will not, and therefore turn to Gauss' law: - -$$\begin{aligned} - \varepsilon_0 \nabla \cdot \vb{E} - = - \varepsilon_0 \nabla^2 \phi_1 - = q_i n_i - q_e n_e - = - q_e (n_{i1} - n_{e1}) -\end{aligned}$$ - -One final time, we insert our plane-wave ansatz, -and use our Boltzmann-like relation between $n_{e1}$ and $n_{e0}$ -to substitute $\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$: - -$$\begin{gathered} - q_e (n_{e1} - n_{i1}) - = |\vb{k}|^2 \varepsilon_0 \phi_1 - = - |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e n_{e0}} - \\ - \implies \qquad - n_{i1} - = n_{e1} + |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e^2 n_{e0}} - = n_{e1} \big( 1 + |\vb{k}|^2 \lambda_{De}^2 \big) -\end{gathered}$$ - -Where $\lambda_{De}$ is the electron [Debye length](/know/concept/debye-length/). -We thus reach the following dispersion relation, -which governs **ion sound waves** or **ion acoustic waves**: - -$$\begin{aligned} - \boxed{ - \omega^2 - = \frac{|\vb{k}|^2}{m_i} \bigg( \frac{k_B T_e}{1 + |\vb{k}|^2 \lambda_{De}^2} + \gamma_i k_B T_i \bigg) - } -\end{aligned}$$ - -The aforementioned plasma approximation is valid if $|\vb{k}| \lambda_{De} \ll 1$, -which is often reasonable, -in which case this dispersion relation reduces to: - -$$\begin{aligned} - \omega^2 - = \frac{|\vb{k}|^2}{m_i} \bigg( k_B T_e + \gamma_i k_B T_i \bigg) -\end{aligned}$$ - -The phase velocity $v_s$ of these waves, -i.e. the speed of sound, is then given by: - -$$\begin{aligned} - \boxed{ - v_s - = \frac{\omega}{k} - = \sqrt{\frac{k_B T_e}{m_i} + \frac{\gamma_i k_B T_i}{m_i}} - } -\end{aligned}$$ - -Curiously, unlike a neutral gas, -this velocity is nonzero even if $T_i = 0$, -meaning that the waves still exist then. -In fact, usually the electron temperature $T_e$ dominates $T_e \gg T_i$, -even though the main feature of these waves -is that they involve ion density fluctuations $n_{i1}$. - - - -## References -1. F.F. Chen, - *Introduction to plasma physics and controlled fusion*, - 3rd edition, Springer. -2. M. Salewski, A.H. Nielsen, - *Plasma physics: lecture notes*, - 2021, unpublished. diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-calculus/index.pdc deleted file mode 100644 index 3d4dd67..0000000 --- a/content/know/concept/ito-calculus/index.pdc +++ /dev/null @@ -1,367 +0,0 @@ ---- -title: "Itō calculus" -firstLetter: "I" -publishDate: 2021-11-06 -categories: -- Mathematics -- Stochastic analysis - -date: 2021-11-06T14:34:00+01:00 -draft: false -markup: pandoc ---- - -# Itō calculus - -Given two [stochastic processes](/know/concept/stochastic-process/) -$F_t$ and $G_t$, consider the following random variable $X_t$, -where $B_t$ is the [Wiener process](/know/concept/wiener-process/), -i.e. Brownian motion: - -$$\begin{aligned} - X_t - = X_0 + \int_0^t F_s \dd{s} + \int_0^t G_s \dd{B_s} -\end{aligned}$$ - -Where the latter is an [Itō integral](/know/concept/ito-integral/), -assuming $G_t$ is Itō-integrable. -We call $X_t$ an **Itō process** if $F_t$ is locally integrable, -and the initial condition $X_0$ is known, -i.e. $X_0$ is $\mathcal{F}_0$-measurable, -where $\mathcal{F}_t$ is the filtration -to which $F_t$, $G_t$ and $B_t$ are adapted. -The above definition of $X_t$ is often abbreviated as follows, -where $X_0$ is implicit: - -$$\begin{aligned} - \dd{X_t} - = F_t \dd{t} + G_t \dd{B_t} -\end{aligned}$$ - -Typically, $F_t$ is referred to as the **drift** of $X_t$, -and $G_t$ as its **intensity**. -Because the Itō integral of $G_t$ is a -[martingale](/know/concept/martingale/), -it does not contribute to the mean of $X_t$: - -$$\begin{aligned} - \mathbf{E}[X_t] - = \int_0^t \mathbf{E}[F_s] \dd{s} -\end{aligned}$$ - -Now, consider the following **Itō stochastic differential equation** (SDE), -where $\xi_t = \dv*{B_t}{t}$ is white noise, -informally treated as the $t$-derivative of $B_t$: - -$$\begin{aligned} - \dv{X_t}{t} - = f(X_t, t) + g(X_t, t) \: \xi_t -\end{aligned}$$ - -An Itō process $X_t$ is said to satisfy this equation -if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$, -in which case $X_t$ is also called an **Itō diffusion**. -All Itō diffusions are [Markov processes](/know/concept/markov-process/), -since only the current value of $X_t$ determines the future, -and $B_t$ is also a Markov process. - - -## Itō's lemma - -Classically, given $y \equiv h(x(t), t)$, -the chain rule of differentiation states that: - -$$\begin{aligned} - \dd{y} - = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{x} -\end{aligned}$$ - -However, for a stochastic process $Y_t \equiv h(X_t, t)$, -where $X_t$ is an Itō process, -the chain rule is modified to the following, -known as **Itō's lemma**: - -$$\begin{aligned} - \boxed{ - \dd{Y_t} - = \bigg( \pdv{h}{t} + \pdv{h}{x} F_t + \frac{1}{2} \pdv[2]{h}{x} G_t^2 \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} - } -\end{aligned}$$ - -
- - - -
- -The most important application of Itō's lemma -is to perform coordinate transformations, -to make the solution of a given Itō SDE easier. - - -## Coordinate transformations - -The simplest coordinate transformation is a scaling of the time axis. -Defining $s \equiv \alpha t$, the goal is to keep the Itō process. -We know how to scale $B_t$, be setting $W_s \equiv \sqrt{\alpha} B_{s / \alpha}$. -Let $Y_s \equiv X_t$ be the new variable on the rescaled axis, then: - -$$\begin{aligned} - \dd{Y_s} - = \dd{X_t} - &= f(X_t) \dd{t} + g(X_t) \dd{B_t} - \\ - &= \frac{1}{\alpha} f(Y_s) \dd{s} + \frac{1}{\sqrt{\alpha}} g(Y_s) \dd{W_s} -\end{aligned}$$ - -$W_s$ is a valid Wiener process, -and the other changes are small, -so this is still an Itō process. - -To solve SDEs analytically, it is usually best -to have additive noise, i.e. $g = 1$. -This can be achieved using the **Lamperti transform**: -define $Y_t \equiv h(X_t)$, where $h$ is given by: - -$$\begin{aligned} - \boxed{ - h(x) - = \int_{x_0}^x \frac{1}{g(y)} \dd{y} - } -\end{aligned}$$ - -Then, using Itō's lemma, it is straightforward -to show that the intensity becomes $1$. -Note that the lower integration limit $x_0$ does not enter: - -$$\begin{aligned} - \dd{Y_t} - &= \bigg( f(X_t) \: h'(X_t) + \frac{1}{2} g^2(X_t) \: h''(X_t) \bigg) \dd{t} + g(X_t) \: h'(X_t) \dd{B_t} - \\ - &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g^2(X_t) \frac{g'(X_t)}{g^2(X_t)} \bigg) \dd{t} + \frac{g(X_t)}{g(X_t)} \dd{B_t} - \\ - &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g'(X_t) \bigg) \dd{t} + \dd{B_t} -\end{aligned}$$ - -Similarly, we can eliminate the drift $f = 0$, -thereby making the Itō process a martingale. -This is done by defining $Y_t \equiv h(X_t)$, with $h(x)$ given by: - -$$\begin{aligned} - \boxed{ - h(x) - = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg) - } -\end{aligned}$$ - -The goal is to make the parenthesized first term (see above) -of Itō's lemma disappear, which this $h(x)$ does indeed do. -Note that $x_0$ and $x_1$ do not enter: - -$$\begin{aligned} - 0 - &= f(x) \: h'(x) + \frac{1}{2} g^2(x) \: h''(x) - \\ - &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g^2(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg) -\end{aligned}$$ - - -## Existence and uniqueness - -It is worth knowing under what condition a solution to a given SDE exists, -in the sense that it is finite on the entire time axis. -Suppose the drift $f$ and intensity $g$ satisfy these inequalities, -for some known constant $K$ and for all $x$: - -$$\begin{aligned} - x f(x) \le K (1 + x^2) - \qquad \quad - g^2(x) \le K (1 + x^2) -\end{aligned}$$ - -When this is satisfied, we can find the following upper bound -on an Itō process $X_t$, -which clearly implies that $X_t$ is finite for all $t$: - -$$\begin{aligned} - \boxed{ - \mathbf{E}[X_t^2] - \le \big(X_0^2 + 3 K t\big) \exp\!\big(3 K t\big) - } -\end{aligned}$$ - -
- - - -
- -If a solution exists, it is also worth knowing whether it is unique. -Suppose that $f$ and $g$ satisfy the following inequalities, -for some constant $K$ and for all $x$ and $y$: - -$$\begin{aligned} - \big| f(x) - f(y) \big| \le K \big| x - y \big| - \qquad \quad - \big| g(x) - g(y) \big| \le K \big| x - y \big| -\end{aligned}$$ - -Let $X_t$ and $Y_t$ both be solutions to a given SDE, -but the initial conditions need not be the same, -such that the difference is initially $X_0 \!-\! Y_0$. -Then the difference $X_t \!-\! Y_t$ is bounded by: - -$$\begin{aligned} - \boxed{ - \mathbf{E}\big[ (X_t - Y_t)^2 \big] - \le (X_0 - Y_0)^2 \exp\!\Big( \big(2 K \!+\! K^2 \big) t \Big) - } -\end{aligned}$$ - -
- - - -
- -Using these properties, it can then be shown -that if all of the above conditions are satisfied, -then the SDE has a unique solution, -which is $\mathcal{F}_t$-adapted, continuous, and exists for all times. - - - -## References -1. U.H. Thygesen, - *Lecture notes on diffusions and stochastic differential equations*, - 2021, Polyteknisk Kompendie. diff --git a/content/know/concept/ito-process/index.pdc b/content/know/concept/ito-process/index.pdc new file mode 100644 index 0000000..d27a2fb --- /dev/null +++ b/content/know/concept/ito-process/index.pdc @@ -0,0 +1,367 @@ +--- +title: "Itō process" +firstLetter: "I" +publishDate: 2021-11-06 +categories: +- Mathematics +- Stochastic analysis + +date: 2021-11-06T14:34:00+01:00 +draft: false +markup: pandoc +--- + +# Itō process + +Given two [stochastic processes](/know/concept/stochastic-process/) +$F_t$ and $G_t$, consider the following random variable $X_t$, +where $B_t$ is the [Wiener process](/know/concept/wiener-process/), +i.e. Brownian motion: + +$$\begin{aligned} + X_t + = X_0 + \int_0^t F_s \dd{s} + \int_0^t G_s \dd{B_s} +\end{aligned}$$ + +Where the latter is an [Itō integral](/know/concept/ito-integral/), +assuming $G_t$ is Itō-integrable. +We call $X_t$ an **Itō process** if $F_t$ is locally integrable, +and the initial condition $X_0$ is known, +i.e. $X_0$ is $\mathcal{F}_0$-measurable, +where $\mathcal{F}_t$ is the filtration +to which $F_t$, $G_t$ and $B_t$ are adapted. +The above definition of $X_t$ is often abbreviated as follows, +where $X_0$ is implicit: + +$$\begin{aligned} + \dd{X_t} + = F_t \dd{t} + G_t \dd{B_t} +\end{aligned}$$ + +Typically, $F_t$ is referred to as the **drift** of $X_t$, +and $G_t$ as its **intensity**. +Because the Itō integral of $G_t$ is a +[martingale](/know/concept/martingale/), +it does not contribute to the mean of $X_t$: + +$$\begin{aligned} + \mathbf{E}[X_t] + = \int_0^t \mathbf{E}[F_s] \dd{s} +\end{aligned}$$ + +Now, consider the following **Itō stochastic differential equation** (SDE), +where $\xi_t = \dv*{B_t}{t}$ is white noise, +informally treated as the $t$-derivative of $B_t$: + +$$\begin{aligned} + \dv{X_t}{t} + = f(X_t, t) + g(X_t, t) \: \xi_t +\end{aligned}$$ + +An Itō process $X_t$ is said to satisfy this equation +if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$, +in which case $X_t$ is also called an **Itō diffusion**. +All Itō diffusions are [Markov processes](/know/concept/markov-process/), +since only the current value of $X_t$ determines the future, +and $B_t$ is also a Markov process. + + +## Itō's lemma + +Classically, given $y \equiv h(x(t), t)$, +the chain rule of differentiation states that: + +$$\begin{aligned} + \dd{y} + = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{x} +\end{aligned}$$ + +However, for a stochastic process $Y_t \equiv h(X_t, t)$, +where $X_t$ is an Itō process, +the chain rule is modified to the following, +known as **Itō's lemma**: + +$$\begin{aligned} + \boxed{ + \dd{Y_t} + = \bigg( \pdv{h}{t} + \pdv{h}{x} F_t + \frac{1}{2} \pdv[2]{h}{x} G_t^2 \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} + } +\end{aligned}$$ + +
+ + + +
+ +The most important application of Itō's lemma +is to perform coordinate transformations, +to make the solution of a given Itō SDE easier. + + +## Coordinate transformations + +The simplest coordinate transformation is a scaling of the time axis. +Defining $s \equiv \alpha t$, the goal is to keep the Itō process. +We know how to scale $B_t$, be setting $W_s \equiv \sqrt{\alpha} B_{s / \alpha}$. +Let $Y_s \equiv X_t$ be the new variable on the rescaled axis, then: + +$$\begin{aligned} + \dd{Y_s} + = \dd{X_t} + &= f(X_t) \dd{t} + g(X_t) \dd{B_t} + \\ + &= \frac{1}{\alpha} f(Y_s) \dd{s} + \frac{1}{\sqrt{\alpha}} g(Y_s) \dd{W_s} +\end{aligned}$$ + +$W_s$ is a valid Wiener process, +and the other changes are small, +so this is still an Itō process. + +To solve SDEs analytically, it is usually best +to have additive noise, i.e. $g = 1$. +This can be achieved using the **Lamperti transform**: +define $Y_t \equiv h(X_t)$, where $h$ is given by: + +$$\begin{aligned} + \boxed{ + h(x) + = \int_{x_0}^x \frac{1}{g(y)} \dd{y} + } +\end{aligned}$$ + +Then, using Itō's lemma, it is straightforward +to show that the intensity becomes $1$. +Note that the lower integration limit $x_0$ does not enter: + +$$\begin{aligned} + \dd{Y_t} + &= \bigg( f(X_t) \: h'(X_t) + \frac{1}{2} g^2(X_t) \: h''(X_t) \bigg) \dd{t} + g(X_t) \: h'(X_t) \dd{B_t} + \\ + &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g^2(X_t) \frac{g'(X_t)}{g^2(X_t)} \bigg) \dd{t} + \frac{g(X_t)}{g(X_t)} \dd{B_t} + \\ + &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g'(X_t) \bigg) \dd{t} + \dd{B_t} +\end{aligned}$$ + +Similarly, we can eliminate the drift $f = 0$, +thereby making the Itō process a martingale. +This is done by defining $Y_t \equiv h(X_t)$, with $h(x)$ given by: + +$$\begin{aligned} + \boxed{ + h(x) + = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^y \frac{2 f(z)}{g^2(z)} \dd{z} \bigg) \dd{y} + } +\end{aligned}$$ + +The goal is to make the parenthesized first term (see above) +of Itō's lemma disappear, which this $h(x)$ does indeed do. +Note that $x_0$ and $x_1$ do not enter: + +$$\begin{aligned} + 0 + &= f(x) \: h'(x) + \frac{1}{2} g^2(x) \: h''(x) + \\ + &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g^2(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg) +\end{aligned}$$ + + +## Existence and uniqueness + +It is worth knowing under what condition a solution to a given SDE exists, +in the sense that it is finite on the entire time axis. +Suppose the drift $f$ and intensity $g$ satisfy these inequalities, +for some known constant $K$ and for all $x$: + +$$\begin{aligned} + x f(x) \le K (1 + x^2) + \qquad \quad + g^2(x) \le K (1 + x^2) +\end{aligned}$$ + +When this is satisfied, we can find the following upper bound +on an Itō process $X_t$, +which clearly implies that $X_t$ is finite for all $t$: + +$$\begin{aligned} + \boxed{ + \mathbf{E}[X_t^2] + \le \big(X_0^2 + 3 K t\big) \exp\!\big(3 K t\big) + } +\end{aligned}$$ + +
+ + + +
+ +If a solution exists, it is also worth knowing whether it is unique. +Suppose that $f$ and $g$ satisfy the following inequalities, +for some constant $K$ and for all $x$ and $y$: + +$$\begin{aligned} + \big| f(x) - f(y) \big| \le K \big| x - y \big| + \qquad \quad + \big| g(x) - g(y) \big| \le K \big| x - y \big| +\end{aligned}$$ + +Let $X_t$ and $Y_t$ both be solutions to a given SDE, +but the initial conditions need not be the same, +such that the difference is initially $X_0 \!-\! Y_0$. +Then the difference $X_t \!-\! Y_t$ is bounded by: + +$$\begin{aligned} + \boxed{ + \mathbf{E}\big[ (X_t - Y_t)^2 \big] + \le (X_0 - Y_0)^2 \exp\!\Big( \big(2 K \!+\! K^2 \big) t \Big) + } +\end{aligned}$$ + +
+ + + +
+ +Using these properties, it can then be shown +that if all of the above conditions are satisfied, +then the SDE has a unique solution, +which is $\mathcal{F}_t$-adapted, continuous, and exists for all times. + + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc index cd9b449..3c3f1d2 100644 --- a/content/know/concept/langmuir-waves/index.pdc +++ b/content/know/concept/langmuir-waves/index.pdc @@ -34,7 +34,7 @@ We also need [Gauss' law](/know/concept/maxwells-equations/): $$\begin{aligned} \varepsilon_0 \nabla \cdot \vb{E} - = q_e (n_i - n_e) + = q_e (n_e - n_i) \end{aligned}$$ We split $n_e$, $\vb{u}_e$ and $\vb{E}$ into a base component @@ -85,10 +85,10 @@ and use the plasma's quasi-neutrality $n_i = n_{e0}$ to get: $$\begin{aligned} \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big) - = q_e (n_i - n_{e0} \!-\! n_{e1} ) + = q_e (n_{e0} + n_{e1} - n_i) \quad \implies \quad \varepsilon_0 \nabla \cdot \vb{E}_1 - = - q_e n_{e1} + = q_e n_{e1} \end{aligned}$$ Since we are looking for linear waves, @@ -110,7 +110,7 @@ Inserting this into the continuity equation and Gauss' law yields, respectively: $$\begin{aligned} - i \omega n_{e1} = - i n_{e0} \vb{k} \cdot \vb{u}_{e1} \qquad \quad - i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1} + -\! i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1} \end{aligned}$$ However, there are three unknowns $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$, diff --git a/content/know/concept/magnetohydrodynamics/index.pdc b/content/know/concept/magnetohydrodynamics/index.pdc index 6672a74..89d23db 100644 --- a/content/know/concept/magnetohydrodynamics/index.pdc +++ b/content/know/concept/magnetohydrodynamics/index.pdc @@ -333,7 +333,7 @@ This term can be dropped in any of the following cases: $$\begin{gathered} 1 - \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{U} \cross \vb{B} \big|} + \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{u} \cross \vb{B} \big|} \sim \frac{\rho v_\mathrm{char} / \tau_\mathrm{char}}{v_\mathrm{char} |\vb{B}| q_i n_i} \approx \frac{m_i n_i}{|\vb{B}| q_i n_i \tau_\mathrm{char}} = \frac{1}{\omega_{ci} \tau_\mathrm{char}} @@ -366,7 +366,7 @@ where we have used Ampère's law to find $\vb{J} \sim \vb{B} / \mu_0 \lambda_\ma $$\begin{aligned} 1 - \ll \frac{\big| \vb{U} \cross \vb{B} \big|}{\big| \eta \vb{J} \big|} + \ll \frac{\big| \vb{u} \cross \vb{B} \big|}{\big| \eta \vb{J} \big|} \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta \vb{J}} \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta |\vb{B}| / \mu_0 \lambda_\mathrm{char}} = \mathrm{R_m} diff --git a/content/know/concept/time-dependent-perturbation-theory/index.pdc b/content/know/concept/time-dependent-perturbation-theory/index.pdc index 1fbd9ce..76ab684 100644 --- a/content/know/concept/time-dependent-perturbation-theory/index.pdc +++ b/content/know/concept/time-dependent-perturbation-theory/index.pdc @@ -59,13 +59,13 @@ We then take the inner product with an arbitrary stationary basis state $\ket{m} $$\begin{aligned} 0 - &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \frac{d c_n}{dt} \braket{m}{n} \Big) \exp\!(- i E_n t / \hbar) + &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \dv{c_n}{t} \braket{m}{n} \Big) \exp\!(- i E_n t / \hbar) \end{aligned}$$ Thanks to orthonormality, this removes the latter term from the summation: $$\begin{aligned} - i \hbar \frac{d c_m}{dt} \exp\!(- i E_m t / \hbar) + i \hbar \dv{c_m}{t} \exp\!(- i E_m t / \hbar) &= \lambda \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} \exp\!(- i E_n t / \hbar) \end{aligned}$$ @@ -74,7 +74,7 @@ $\omega_{mn} \equiv (E_m - E_n) / \hbar$ to get: $$\begin{aligned} \boxed{ - i \hbar \frac{d c_m}{dt} + i \hbar \dv{c_m}{t} = \lambda \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} \exp\!(i \omega_{mn} t) } \end{aligned}$$ -- cgit v1.2.3