diff options
Diffstat (limited to 'source/know/concept')
23 files changed, 471 insertions, 300 deletions
diff --git a/source/know/concept/alfven-waves/index.md b/source/know/concept/alfven-waves/index.md index 31576f3..0396c7a 100644 --- a/source/know/concept/alfven-waves/index.md +++ b/source/know/concept/alfven-waves/index.md @@ -61,12 +61,12 @@ $$\begin{aligned} = \frac{1}{\mu_0} \nabla \cross \vb{B}_1 \end{aligned}$$ -Substituting this into the momentum equation, +Substituting this into the above momentum equation, and differentiating with respect to $$t$$: $$\begin{aligned} \rho \pdvn{2}{\vb{u}_1}{t} - = \frac{1}{\mu_0} \bigg( \Big( \nabla \cross \pdv{}{\vb{B}1}{t} \Big) \cross \vb{B}_0 \bigg) + = \frac{1}{\mu_0} \bigg( \Big( \nabla \cross \pdv{\vb{B}_1}{t} \Big) \cross \vb{B}_0 \bigg) \end{aligned}$$ For which we can use Faraday's law to rewrite $$\ipdv{\vb{B}_1}{t}$$, @@ -78,7 +78,7 @@ $$\begin{aligned} = \nabla \cross (\vb{u}_1 \cross \vb{B}_0) \end{aligned}$$ -Inserting this into the momentum equation for $$\vb{u}_1$$ +Inserting this back into the momentum equation for $$\vb{u}_1$$ thus yields its final form: $$\begin{aligned} diff --git a/source/know/concept/binomial-distribution/index.md b/source/know/concept/binomial-distribution/index.md index dc75221..9bb32d3 100644 --- a/source/know/concept/binomial-distribution/index.md +++ b/source/know/concept/binomial-distribution/index.md @@ -46,19 +46,25 @@ $$\begin{aligned} {% include proof/start.html id="proof-mean" -%} -The trick is to treat $$p$$ and $$q$$ as independent until the last moment: +The trick is to treat $$p$$ and $$q$$ as independent and introduce a derivative: + +$$\begin{aligned} + \mu + &= \sum_{n = 0}^N n P_N(n) + = \sum_{n = 0}^N n \binom{N}{n} p^n q^{N - n} + = \sum_{n = 0}^N \binom{N}{n} \bigg( p \pdv{(p^n)}{p} \bigg) q^{N - n} +\end{aligned}$$ + +Then, using the fact that the binomial coefficients appear when writing out $$(p + q)^N$$: $$\begin{aligned} \mu - &= \sum_{n = 0}^N n \binom{N}{n} p^n q^{N - n} - = \sum_{n = 0}^N \binom{N}{n} \Big( p \pdv{(p^n)}{p} \Big) q^{N - n} - \\ &= p \pdv{}{p}\sum_{n = 0}^N \binom{N}{n} p^n q^{N - n} = p \pdv{}{p}(p + q)^N = N p (p + q)^{N - 1} \end{aligned}$$ -Inserting $$q = 1 - p$$ then gives the desired result. +Finally, inserting $$q = 1 - p$$ gives the desired result. {% include proof/end.html id="proof-mean" %} @@ -73,18 +79,21 @@ $$\begin{aligned} {% include proof/start.html id="proof-var" -%} +We reuse the previous trick to find $$\overline{n^2}$$ (the mean squared number of successes): $$\begin{aligned} \overline{n^2} &= \sum_{n = 0}^N n^2 \binom{N}{n} p^n q^{N - n} - = \sum_{n = 0}^N n \binom{N}{n} \Big( p \pdv{}{p}\Big)^2 p^n q^{N - n} + = \sum_{n = 0}^N n \binom{N}{n} \bigg( p \pdv{}{p} \bigg) p^n q^{N - n} + \\ + &= \sum_{n = 0}^N \binom{N}{n} \bigg( p \pdv{}{p} \bigg)^2 p^n q^{N - n} + = \bigg( p \pdv{}{p} \bigg)^2 \sum_{n = 0}^N \binom{N}{n} p^n q^{N - n} \\ - &= \Big( p \pdv{}{p}\Big)^2 \sum_{n = 0}^N \binom{N}{n} p^n q^{N - n} - = \Big( p \pdv{}{p}\Big)^2 (p + q)^N + &= \bigg( p \pdv{}{p} \bigg)^2 (p + q)^N + = N p \pdv{}{p}p (p + q)^{N - 1} \\ - &= N p \pdv{}{p}p (p + q)^{N - 1} - = N p \big( (p + q)^{N - 1} + (N - 1) p (p + q)^{N - 2} \big) + &= N p \big( (p + q)^{N - 1} + (N - 1) p (p + q)^{N - 2} \big) \\ &= N p + N^2 p^2 - N p^2 \end{aligned}$$ @@ -108,7 +117,7 @@ a fact that is sometimes called the **de Moivre-Laplace theorem**: $$\begin{aligned} \boxed{ - \lim_{N \to \infty} P_N(n) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\!\Big(\!-\!\frac{(n - \mu)^2}{2 \sigma^2} \Big) + \lim_{N \to \infty} P_N(n) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\!\bigg(\!-\!\frac{(n - \mu)^2}{2 \sigma^2} \bigg) } \end{aligned}$$ @@ -121,73 +130,94 @@ $$\begin{aligned} \ln\!\big(P_N(n)\big) &= \sum_{m = 0}^\infty \frac{(n - \mu)^m}{m!} D_m(\mu) \quad \mathrm{where} \quad - D_m(n) = \dvn{m}{\ln\!\big(P_N(n)\big)}{n} + D_m(n) + \equiv \dvn{m}{\ln\!\big(P_N(n)\big)}{n} \end{aligned}$$ -We use Stirling's approximation to calculate the factorials in $$D_m$$: +For future convenience while calculating the $$D_m$$, we write out $$\ln(P_N)$$ now: $$\begin{aligned} \ln\!\big(P_N(n)\big) - &= \ln(N!) - \ln(n!) - \ln\!\big((N - n)!\big) + n \ln(p) + (N - n) \ln(q) - \\ - &\approx \ln(N!) - n \big( \ln(n)\!-\!\ln(p)\!-\!1 \big) - (N\!-\!n) \big( \ln(N\!-\!n)\!-\!\ln(q)\!-\!1 \big) + &= \ln(N!) - \ln(n!) - \ln\!\big((N \!-\! n)!\big) + n \ln(p) + (N \!-\! n) \ln(q) \end{aligned}$$ -For $$D_0(\mu)$$, we need to use a stronger version of Stirling's approximation -to get a non-zero result. We take advantage of $$N - N p = N q$$: +For $$D_0(\mu)$$ specifically, +we need to use a strong version of *Stirling's approximation* +to arrive at a nonzero result in the end. +We know that $$N - N p = N q$$: $$\begin{aligned} D_0(\mu) + &= \ln\!\big(P_N(n)\big) \big|_{n = \mu} + \\ + &= \ln(N!) - \ln(\mu!) - \ln\!\big((N \!-\! \mu)!\big) + \mu \ln(p) + (N \!-\! \mu) \ln(q) + \\ &= \ln(N!) - \ln\!\big((N p)!\big) - \ln\!\big((N q)!\big) + N p \ln(p) + N q \ln(q) \\ - &= \Big( N \ln(N) - N + \frac{1}{2} \ln(2\pi N) \Big) + &\approx \Big( N \ln(N) - N + \frac{1}{2} \ln(2\pi N) \Big) - \Big( N p \ln(N p) - N p + \frac{1}{2} \ln(2\pi N p) \Big) \\ &\qquad - \Big( N q \ln(N q) - N q + \frac{1}{2} \ln(2\pi N q) \Big) + N p \ln(p) + N q \ln(q) \\ - &= N \ln(N) - N (p + q) \ln(N) + N (p + q) - N - \frac{1}{2} \ln(2\pi N p q) + &= N \ln(N) - N (p \!+\! q) \ln(N) + N (p \!+\! q) - N - \frac{1}{2} \ln(2\pi N p q) \\ &= - \frac{1}{2} \ln(2\pi N p q) - = \ln\!\Big( \frac{1}{\sqrt{2\pi \sigma^2}} \Big) + = \ln\!\bigg( \frac{1}{\sqrt{2\pi \sigma^2}} \bigg) \end{aligned}$$ -Next, we expect that $$D_1(\mu) = 0$$, because $$\mu$$ is the maximum. -This is indeed the case: +Next, for $$D_m(\mu)$$ with $$m \ge 1$$, +we can use a weaker version of Stirling's approximation: + +$$\begin{aligned} + \ln(P_N) + &\approx \ln(N!) - n \big( \ln(n) \!-\! 1 \big) - (N \!-\! n) \big( \ln(N \!-\! n) \!-\! 1 \big) + n \ln(p) + (N \!-\! n) \ln(q) + \\ + &\approx \ln(N!) - n \big( \ln(n) - \ln(p) - 1 \big) - (N\!-\!n) \big( \ln(N\!-\!n) - \ln(q) - 1 \big) +\end{aligned}$$ + +We expect that $$D_1(\mu) = 0$$, because $$P_N$$ is maximized at $$\mu$$. +Indeed it is: $$\begin{aligned} D_1(n) - &= - \big( \ln(n)\!-\!\ln(p)\!-\!1 \big) + \big( \ln(N\!-\!n)\!-\!\ln(q)\!-\!1 \big) - 1 + 1 + &= \dv{}{n} \ln\!\big((P_N(n)\big) \\ - &= - \ln(n) + \ln(N - n) + \ln(p) - \ln(q) + &= - \big( \ln(n) - \ln(p) - 1 \big) + \big( \ln(N\!-\!n) - \ln(q) - 1 \big) - \frac{n}{n} + \frac{N \!-\! n}{N \!-\! n} + \\ + &= - \ln(n) + \ln(N \!-\! n) + \ln(p) - \ln(q) \\ D_1(\mu) - &= \ln(N q) - \ln(N p) + \ln(p) - \ln(q) - = \ln(N p q) - \ln(N p q) - = 0 + &= - \ln(\mu) + \ln(N \!-\! \mu) + \ln(p) - \ln(q) + \\ + &= - \ln(N p q) + \ln(N p q) + \\ + &= 0 \end{aligned}$$ -For the same reason, we expect that $$D_2(\mu)$$ is negative. +For the same reason, we expect $$D_2(\mu)$$ to be negative. We find the following expression: $$\begin{aligned} D_2(n) - &= - \frac{1}{n} - \frac{1}{N - n} - \qquad + &= \dvn{2}{}{n} \ln\!\big((P_N(n)\big) + = \dv{}{n} D_1(n) + = - \frac{1}{n} - \frac{1}{N - n} + \\ D_2(\mu) - = - \frac{1}{Np} - \frac{1}{Nq} + &= - \frac{1}{Np} - \frac{1}{Nq} = - \frac{p + q}{N p q} = - \frac{1}{\sigma^2} \end{aligned}$$ -The higher-order derivatives tend to zero for $$N \to \infty$$, so we discard them: +The higher-order derivatives vanish much faster as $$N \to \infty$$, so we discard them: $$\begin{aligned} D_3(n) = \frac{1}{n^2} - \frac{1}{(N - n)^2} - \qquad + \qquad \quad D_4(n) = - \frac{2}{n^3} - \frac{2}{(N - n)^3} - \qquad + \qquad \quad \cdots \end{aligned}$$ @@ -197,13 +227,14 @@ the Taylor series approximately becomes: $$\begin{aligned} \ln\!\big(P_N(n)\big) \approx D_0(\mu) + \frac{(n - \mu)^2}{2} D_2(\mu) - = \ln\!\Big( \frac{1}{\sqrt{2\pi \sigma^2}} \Big) - \frac{(n - \mu)^2}{2 \sigma^2} + = \ln\!\bigg( \frac{1}{\sqrt{2\pi \sigma^2}} \bigg) - \frac{(n - \mu)^2}{2 \sigma^2} \end{aligned}$$ -Taking $$\exp$$ of this expression then yields a normalized Gaussian distribution. +Raising $$e$$ to this expression then yields a normalized Gaussian distribution. {% include proof/end.html id="proof-normal" %} + ## References 1. H. Gould, J. Tobochnik, *Statistical and thermal physics*, 2nd edition, diff --git a/source/know/concept/central-limit-theorem/index.md b/source/know/concept/central-limit-theorem/index.md index 595cee7..e933ee7 100644 --- a/source/know/concept/central-limit-theorem/index.md +++ b/source/know/concept/central-limit-theorem/index.md @@ -18,24 +18,24 @@ the resulting means $$\mu_m$$ are normally distributed across the $$M$$ samples if $$N$$ is sufficiently large. More formally, for $$N$$ independent variables $$x_n$$ with probability distributions $$p(x_n)$$, -the central limit theorem states the following, -where we define the sum $$S$$: +we define the following totals of all variables, means and variances: $$\begin{aligned} - S = \sum_{n = 1}^N x_n - \qquad - \mu_S = \sum_{n = 1}^N \mu_n - \qquad - \sigma_S^2 = \sum_{n = 1}^N \sigma_n^2 + t \equiv \sum_{n = 1}^N x_n + \qquad \qquad + \mu_t \equiv \sum_{n = 1}^N \mu_n + \qquad \qquad + \sigma_t^2 \equiv \sum_{n = 1}^N \sigma_n^2 \end{aligned}$$ -And crucially, it states that the probability distribution $$p_N(S)$$ of $$S$$ for $$N$$ variables +The central limit theorem then states that +the probability distribution $$p_N(t)$$ of $$t$$ for $$N$$ variables will become a normal distribution when $$N$$ goes to infinity: $$\begin{aligned} \boxed{ - \lim_{N \to \infty} \!\big(p_N(S)\big) - = \frac{1}{\sigma_S \sqrt{2 \pi}} \exp\!\Big( -\frac{(\mu_S - S)^2}{2 \sigma_S^2} \Big) + \lim_{N \to \infty} \!\big(p_N(t)\big) + = \frac{1}{\sigma_t \sqrt{2 \pi}} \exp\!\bigg( -\frac{(t - \mu_t)^2}{2 \sigma_t^2} \bigg) } \end{aligned}$$ @@ -45,7 +45,8 @@ Given a probability density $$p(x)$$, its [Fourier transform](/know/concept/four is called the **characteristic function** $$\phi(k)$$: $$\begin{aligned} - \phi(k) = \int_{-\infty}^\infty p(x) \exp(i k x) \dd{x} + \phi(k) + \equiv \int_{-\infty}^\infty p(x) \exp(i k x) \dd{x} \end{aligned}$$ Note that $$\phi(k)$$ can be interpreted as the average of $$\exp(i k x)$$. @@ -54,17 +55,19 @@ where an overline denotes the mean: $$\begin{aligned} \phi(k) - = \sum_{n = 0}^\infty \frac{k^n}{n!} \: \phi^{(n)}(0) - \qquad + = \sum_{n = 0}^\infty \frac{k^n}{n!} \bigg( \dvn{n}{\phi}{k} \Big|_{k = 0} \bigg) + \qquad \qquad \phi(k) - = \overline{\exp(i k x)} = \sum_{n = 0}^\infty \frac{(ik)^n}{n!} \overline{x^n} + = \overline{\exp(i k x)} + = \sum_{n = 0}^\infty \frac{(ik)^n}{n!} \overline{x^n} \end{aligned}$$ By comparing the coefficients of these two power series, we get a useful relation: $$\begin{aligned} - \phi^{(n)}(0) = i^n \: \overline{x^n} + \dvn{n}{\phi}{k} \Big|_{k = 0} + = i^n \: \overline{x^n} \end{aligned}$$ Next, the **cumulants** $$C^{(n)}$$ are defined from the Taylor expansion of $$\ln\!\big(\phi(k)\big)$$: @@ -73,73 +76,82 @@ $$\begin{aligned} \ln\!\big( \phi(k) \big) = \sum_{n = 1}^\infty \frac{(ik)^n}{n!} C^{(n)} \quad \mathrm{where} \quad - C^{(n)} = \frac{1}{i^n} \: \dvn{n}{}{k} \Big(\ln\!\big(\phi(k)\big)\Big) \Big|_{k = 0} + C^{(n)} + \equiv \frac{1}{i^n} \: \dvn{n}{}{k} \ln\!\big(\phi(k)\big) \Big|_{k = 0} \end{aligned}$$ The first two cumulants $$C^{(1)}$$ and $$C^{(2)}$$ are of particular interest, -since they turn out to be the mean and the variance respectively, -using our earlier relation: +since they turn out to be the mean and the variance respectively. +Using our earlier relation: $$\begin{aligned} C^{(1)} - &= - i \dv{}{k} \Big(\ln\!\big(\phi(k)\big)\Big) \Big|_{k = 0} + &= - i \dv{}{k} \ln\!\big(\phi(k)\big) \Big|_{k = 0} = - i \frac{\phi'(0)}{\exp(0)} = \overline{x} \\ C^{(2)} - &= - \dvn{2}{}{k} \Big(\ln\!\big(\phi(k)\big)\Big) \Big|_{k = 0} + &= - \dvn{2}{}{k} \ln\!\big(\phi(k)\big) \Big|_{k = 0} = \frac{\big(\phi'(0)\big)^2}{\exp(0)^2} - \frac{\phi''(0)}{\exp(0)} = - \overline{x}^2 + \overline{x^2} = \sigma^2 \end{aligned}$$ -Let us now define $$S$$ as the sum of $$N$$ independent variables $$x_n$$, in other words: +Now that we have introduced these tools, +we define $$t$$ as the sum +of $$N$$ independent variables $$x_n$$, in other words: $$\begin{aligned} - S = \sum_{n = 1}^N x_n = x_1 + x_2 + ... + x_N + t + \equiv \sum_{n = 1}^N x_n = x_1 + x_2 + ... + x_N \end{aligned}$$ -The probability density of $$S$$ is then as follows, where $$p(x_n)$$ are +The probability density of $$t$$ is then as follows, where $$p(x_n)$$ are the densities of all the individual variables and $$\delta$$ is the [Dirac delta function](/know/concept/dirac-delta-function/): $$\begin{aligned} - p(S) - &= \int\cdots\int_{-\infty}^\infty \Big( \prod_{n = 1}^N p(x_n) \Big) \: \delta\Big( S - \sum_{n = 1}^N x_n \Big) \dd{x_1} \cdots \dd{x_N} + p(t) + &= \int\cdots\int_{-\infty}^\infty \Big( \prod_{n = 1}^N p(x_n) \Big) \: \delta\Big( t - \sum_{n = 1}^N x_n \Big) \dd{x_1} \cdots \dd{x_N} \\ - &= \Big( p_1 * \big( p_2 * ( ... * (p_N * \delta))\big)\Big)(S) + &= \Big( p_1 * \big( p_2 * ( ... * (p_N * \delta))\big)\Big)(t) \end{aligned}$$ In other words, the integrals pick out all combinations of $$x_n$$ which -add up to the desired $$S$$-value, and multiply the probabilities +add up to the desired $$t$$-value, and multiply the probabilities $$p(x_1) p(x_2) \cdots p(x_N)$$ of each such case. This is a convolution, so the [convolution theorem](/know/concept/convolution-theorem/) states that it is a product in the Fourier domain: $$\begin{aligned} - \phi_S(k) = \prod_{n = 1}^N \phi_n(k) + \phi_t(k) + = \prod_{n = 1}^N \phi_n(k) \end{aligned}$$ By taking the logarithm of both sides, the product becomes a sum, which we further expand: $$\begin{aligned} - \ln\!\big(\phi_S(k)\big) + \ln\!\big(\phi_t(k)\big) = \sum_{n = 1}^N \ln\!\big(\phi_n(k)\big) = \sum_{n = 1}^N \sum_{m = 1}^{\infty} \frac{(ik)^m}{m!} C_n^{(m)} \end{aligned}$$ -Consequently, the cumulants $$C^{(m)}$$ stack additively for the sum $$S$$ +Consequently, the cumulants $$C^{(m)}$$ stack additively for the sum $$t$$ of independent variables $$x_m$$, and therefore the means $$C^{(1)}$$ and variances $$C^{(2)}$$ do too: $$\begin{aligned} - C_S^{(m)} = \sum_{n = 1}^N C_n^{(m)} = C_1^{(m)} + C_2^{(m)} + ... + C_N^{(m)} + C_t^{(m)} + = \sum_{n = 1}^N C_n^{(m)} + = C_1^{(m)} + C_2^{(m)} + ... + C_N^{(m)} \end{aligned}$$ We now introduce the scaled sum $$z$$ as the new combined variable: $$\begin{aligned} - z = \frac{S}{\sqrt{N}} = \frac{1}{\sqrt{N}} (x_1 + x_2 + ... + x_N) + z + \equiv \frac{t}{\sqrt{N}} + = \frac{1}{\sqrt{N}} (x_1 + x_2 + ... + x_N) \end{aligned}$$ Its characteristic function $$\phi_z(k)$$ is then as follows, @@ -176,28 +188,30 @@ For sufficiently large $$N$$, we can therefore approximate it using just the fir $$\begin{aligned} \ln\!\big( \phi_z(k) \big) &\approx i k C^{(1)} - \frac{k^2}{2} C^{(2)} - = i k \overline{z} - \frac{k^2}{2} \sigma_z^2 + = i k \mu_z - \frac{k^2}{2} \sigma_z^2 \\ + \implies \quad \phi_z(k) - &\approx \exp(i k \overline{z}) \exp(- k^2 \sigma_z^2 / 2) + &\approx \exp(i k \mu_z) \exp(- k^2 \sigma_z^2 / 2) \end{aligned}$$ We take its inverse Fourier transform to get the density $$p(z)$$, -which turns out to be a Gaussian normal distribution, -which is even already normalized: +which turns out to be a Gaussian normal distribution +and is even already normalized: $$\begin{aligned} p(z) = \hat{\mathcal{F}}^{-1} \{\phi_z(k)\} - &= \frac{1}{2 \pi} \int_{-\infty}^\infty \exp\!\big(\!-\! i k (z - \overline{z})\big) \exp(- k^2 \sigma_z^2 / 2) \dd{k} + &= \frac{1}{2 \pi} \int_{-\infty}^\infty \exp\!\big(\!-\! i k (z - \mu_z)\big) \exp(- k^2 \sigma_z^2 / 2) \dd{k} \\ - &= \frac{1}{\sqrt{2 \pi \sigma_z^2}} \exp\!\Big(\!-\! \frac{(z - \overline{z})^2}{2 \sigma_z^2} \Big) + &= \frac{1}{\sqrt{2 \pi \sigma_z^2}} \exp\!\Big(\!-\! \frac{(z - \mu_z)^2}{2 \sigma_z^2} \Big) \end{aligned}$$ Therefore, the sum of many independent variables tends to a normal distribution, regardless of the densities of the individual variables. + ## References 1. H. Gould, J. Tobochnik, *Statistical and thermal physics*, 2nd edition, diff --git a/source/know/concept/conditional-expectation/index.md b/source/know/concept/conditional-expectation/index.md index f64fa72..cd40315 100644 --- a/source/know/concept/conditional-expectation/index.md +++ b/source/know/concept/conditional-expectation/index.md @@ -41,7 +41,7 @@ Where $$Q$$ is a renormalized probability function, which assigns zero to all events incompatible with $$Y = y$$. If we allow $$\Omega$$ to be continuous, then from the definition $$\mathbf{E}[X]$$, -we know that the following Lebesgue integral can be used, +we know that the following *Lebesgue integral* can be used, which we call $$f(y)$$: $$\begin{aligned} @@ -103,6 +103,7 @@ such that $$\mathbf{E}[X | \sigma(Y)] = f(Y)$$, then $$Z = \mathbf{E}[X | \sigma(Y)]$$ is unique. + ## Properties A conditional expectation defined in this way has many useful properties, diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md index 746eb6d..9642737 100644 --- a/source/know/concept/dispersive-broadening/index.md +++ b/source/know/concept/dispersive-broadening/index.md @@ -9,10 +9,10 @@ categories: layout: "concept" --- -In optical fibers, **dispersive broadening** is a (linear) effect +In optical fibers, **dispersive broadening** is a linear effect where group velocity dispersion (GVD) "smears out" a pulse in the time domain due to the different group velocities of its frequencies, -since pulses always have a non-zero width in the $$\omega$$-domain. +since pulses always have a nonzero width in the $$\omega$$-domain. No new frequencies are created. A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation, @@ -29,7 +29,7 @@ and consider a Gaussian initial condition: $$\begin{aligned} A(0, t) - = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) + = \sqrt{P_0} \exp\!\bigg(\!-\!\frac{t^2}{2 T_0^2}\bigg) \end{aligned}$$ By [Fourier transforming](/know/concept/fourier-transform/) in $$t$$, @@ -38,7 +38,8 @@ where it can be seen that the amplitude decreases and the width increases with $$z$$: $$\begin{aligned} - A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}} + A(z,t) + = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}} \exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg) \end{aligned}$$ @@ -48,10 +49,12 @@ as the distance over which the half-width at $$1/e$$ of maximum power (initially $$T_0$$) increases by a factor of $$\sqrt{2}$$: $$\begin{aligned} - T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2} + T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} + = T_0 \sqrt{2} \qquad \implies \qquad \boxed{ - L_D = \frac{T_0^2}{|\beta_2|} + L_D + \equiv \frac{T_0^2}{|\beta_2|} } \end{aligned}$$ @@ -68,7 +71,7 @@ where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t $$\begin{aligned} \omega_{\mathrm{GVD}}(z,t) - = \pdv{}{t}\Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big) + = \pdv{}{t}\bigg( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \bigg) = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2} \end{aligned}$$ @@ -76,7 +79,7 @@ This expression is linear in time, and depending on the sign of $$\beta_2$$, frequencies on one side of the pulse arrive first, and those on the other side arrive last. The effect is stronger for smaller $$T_0$$: -this makes sense, since short pulses are spectrally wider. +this makes sense, since shorter pulses are spectrally wider. The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/) leads to many interesting effects, diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md index cf252c0..976758b 100644 --- a/source/know/concept/holomorphic-function/index.md +++ b/source/know/concept/holomorphic-function/index.md @@ -9,13 +9,13 @@ layout: "concept" --- In complex analysis, a complex function $$f(z)$$ of a complex variable $$z$$ -is called **holomorphic** or **analytic** if it is complex differentiable in the -neighbourhood of every point of its domain. +is called **holomorphic** or **analytic** if it is **complex differentiable** +in the vicinity of every point of its domain. This is a very strong condition. As a result, holomorphic functions are infinitely differentiable and equal their Taylor expansion at every point. In physicists' terms, -they are extremely "well-behaved" throughout their domain. +they are very "well-behaved" throughout their domain. More formally, a given function $$f(z)$$ is holomorphic in a certain region if the following limit exists for all $$z$$ in that region, @@ -23,14 +23,17 @@ and for all directions of $$\Delta z$$: $$\begin{aligned} \boxed{ - f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} + f'(z) + = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} } \end{aligned}$$ We decompose $$f$$ into the real functions $$u$$ and $$v$$ of real variables $$x$$ and $$y$$: $$\begin{aligned} - f(z) = f(x + i y) = u(x, y) + i v(x, y) + f(z) + = f(x + i y) + = u(x, y) + i v(x, y) \end{aligned}$$ Since we are free to choose the direction of $$\Delta z$$, we choose $$\Delta x$$ and $$\Delta y$$: @@ -56,9 +59,9 @@ $$\begin{aligned} } \end{aligned}$$ -Therefore, a given function $$f(z)$$ is holomorphic if and only if its real -and imaginary parts satisfy these equations. This gives an idea of how -strict the criteria are to qualify as holomorphic. +Therefore, a given function $$f(z)$$ is holomorphic if and only if +its real and imaginary parts satisfy these equations. +This gives an idea of how strict the criteria are to qualify as holomorphic. @@ -70,7 +73,8 @@ provided that $$f(z)$$ is holomorphic for all $$z$$ in the area enclosed by $$C$ $$\begin{aligned} \boxed{ - \oint_C f(z) \dd{z} = 0 + \oint_C f(z) \dd{z} + = 0 } \end{aligned}$$ @@ -86,34 +90,36 @@ $$\begin{aligned} &= \oint_C u \dd{x} - v \dd{y} + i \oint_C v \dd{x} + u \dd{y} \end{aligned}$$ -Using Green's theorem, we integrate over the area $$A$$ enclosed by $$C$$: +Using *Green's theorem*, we integrate over the area $$A$$ enclosed by $$C$$: $$\begin{aligned} \oint_C f(z) \dd{z} &= - \iint_A \pdv{v}{x} + \pdv{u}{y} \dd{x} \dd{y} + i \iint_A \pdv{u}{x} - \pdv{v}{y} \dd{x} \dd{y} \end{aligned}$$ -Since $$f(z)$$ is holomorphic, $$u$$ and $$v$$ satisfy the Cauchy-Riemann -equations, such that the integrands disappear and the final result is zero. +Since $$f(z)$$ is holomorphic, $$u$$ and $$v$$ satisfy the Cauchy-Riemann equations, +such that the integrands disappear and the final result is zero. {% include proof/end.html id="proof-int-theorem" %} -An interesting consequence is **Cauchy's integral formula**, which -states that the value of $$f(z)$$ at an arbitrary point $$z_0$$ is -determined by its values on an arbitrary contour $$C$$ around $$z_0$$: +An interesting consequence is **Cauchy's integral formula**, +which states that the value of $$f(z)$$ at an arbitrary point $$z_0$$ +is determined by its values on an arbitrary contour $$C$$ around $$z_0$$: $$\begin{aligned} \boxed{ - f(z_0) = \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z - z_0} \dd{z} + f(z_0) + = \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z - z_0} \dd{z} } \end{aligned}$$ {% include proof/start.html id="proof-int-formula" -%} -Thanks to the integral theorem, we know that the shape and size -of $$C$$ is irrelevant. Therefore we choose it to be a circle with radius $$r$$, -such that the integration variable becomes $$z = z_0 + r e^{i \theta}$$. Then -we integrate by substitution: +Thanks to the integral theorem, we know that +the shape and size of $$C$$ are irrelevant. +Therefore we choose it to be a circle with radius $$r$$, +such that the integration variable becomes $$z = z_0 + r e^{i \theta}$$. +Then we integrate by substitution: $$\begin{aligned} \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z - z_0} \dd{z} diff --git a/source/know/concept/ion-sound-wave/index.md b/source/know/concept/ion-sound-wave/index.md index 8749f1a..6a9dcff 100644 --- a/source/know/concept/ion-sound-wave/index.md +++ b/source/know/concept/ion-sound-wave/index.md @@ -49,7 +49,7 @@ $$\begin{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: +are assumed to satisfy: $$\begin{aligned} \pdv{n_{i0}}{t} = 0 @@ -63,11 +63,7 @@ $$\begin{aligned} \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$$ is linear: +Inserting this decomposition into the momentum equations yields new equations: $$\begin{aligned} m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t} @@ -77,17 +73,19 @@ $$\begin{aligned} &= - q_e (n_{e0} \!+\! n_{e1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_e k_B T_e \nabla (n_{e0} \!+\! n_{e1}) \end{aligned}$$ -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: +Using the assumed properties of $$n_{i0}$$, $$n_{e0}$$, $$\vb{u}_{i0}$$ and $$\phi_0$$, +and discarding products of perturbations for being too small, +we arrive at the below equations. +Our choice $$\vb{u}_{i0} = 0$$ lets us linearize +the [material derivative](/know/concept/material-derivative/) +$$\mathrm{D}/\mathrm{D} t = \ipdv{}{t}$$ for the ions: $$\begin{aligned} m_i n_{i0} \pdv{\vb{u}_{i1}}{t} - &= - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1} + &\approx - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1} \\ 0 - &= - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1} + &\approx - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1} \end{aligned}$$ Because we are interested in linear waves, @@ -123,7 +121,7 @@ to get a relation between $$n_{e1}$$ and $$n_{e0}$$: $$\begin{aligned} i \vb{k} \gamma_e k_B T_e n_{e1} = - i \vb{k} q_e n_{e0} \phi_1 - \quad \implies \quad + \qquad \implies \qquad n_{e1} = - \frac{q_e \phi_1}{\gamma_e k_B T_e} n_{e0} \end{aligned}$$ @@ -159,13 +157,13 @@ $$\begin{aligned} \approx \pdv{n_{i1}}{t} + n_{i0} \nabla \cdot \vb{u}_{i1} \end{aligned}$$ -Then we insert our plane-wave ansatz, +Into which we insert our plane-wave ansatz, and substitute $$n_{i0} = n_0$$ as before, yielding: $$\begin{aligned} 0 = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1} - \quad \implies \quad + \qquad \implies \qquad \vb{k} \cdot \vb{u}_{i1} = \omega \frac{n_{i1}}{n_{i0}} = \omega \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}} @@ -187,9 +185,9 @@ $$\begin{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, +Nevertheless, some authors tend to ignore this fact, thereby making the so-called **plasma approximation**. -We will not, and therefore turn to Gauss' law: +We will not, and thus turn to Gauss' law: $$\begin{aligned} \varepsilon_0 \nabla \cdot \vb{E} @@ -244,7 +242,7 @@ $$\begin{aligned} } \end{aligned}$$ -Curiously, unlike a neutral gas, +Curiously, unlike in a neutral gas, this velocity is nonzero even if $$T_i = 0$$, meaning that the waves still exist then. In fact, usually the electron temperature $$T_e$$ dominates $$T_e \gg T_i$$, diff --git a/source/know/concept/lagrange-multiplier/index.md b/source/know/concept/lagrange-multiplier/index.md index a0b22aa..ce5418f 100644 --- a/source/know/concept/lagrange-multiplier/index.md +++ b/source/know/concept/lagrange-multiplier/index.md @@ -127,8 +127,22 @@ about the interdependence of a system of equations then $$\lambda$$ is not even given an expression! Hence it |