From 71b9e1aa3050dd492761973ad4be73c6d65e7eb1 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 10 Apr 2021 19:58:09 +0200 Subject: Expand knowledge base --- .../know/concept/archimedes-principle/index.pdc | 94 ++++++++++++ content/know/concept/bernoullis-theorem/index.pdc | 96 ++++++++++++ .../know/concept/deutsch-jozsa-algorithm/index.pdc | 3 +- content/know/concept/gram-schmidt-method/index.pdc | 1 + content/know/concept/quantum-gate/cnot.png | Bin 0 -> 4584 bytes content/know/concept/quantum-gate/cu.png | Bin 0 -> 4625 bytes content/know/concept/quantum-gate/index.pdc | 16 +- content/know/concept/quantum-gate/swap.png | Bin 0 -> 4061 bytes content/know/concept/toffoli-gate/index.pdc | 4 +- content/know/concept/vorticity/index.pdc | 165 +++++++++++++++++++++ 10 files changed, 375 insertions(+), 4 deletions(-) create mode 100644 content/know/concept/archimedes-principle/index.pdc create mode 100644 content/know/concept/bernoullis-theorem/index.pdc create mode 100644 content/know/concept/quantum-gate/cnot.png create mode 100644 content/know/concept/quantum-gate/cu.png create mode 100644 content/know/concept/quantum-gate/swap.png create mode 100644 content/know/concept/vorticity/index.pdc (limited to 'content/know/concept') diff --git a/content/know/concept/archimedes-principle/index.pdc b/content/know/concept/archimedes-principle/index.pdc new file mode 100644 index 0000000..6335a77 --- /dev/null +++ b/content/know/concept/archimedes-principle/index.pdc @@ -0,0 +1,94 @@ +--- +title: "Archimedes' principle" +firstLetter: "A" +publishDate: 2021-04-10 +categories: +- Fluid statics +- Fluid mechanics +- Physics + +date: 2021-04-10T15:43:45+02:00 +draft: false +markup: pandoc +--- + +# Archimedes' principle + +Many objects float when placed on a liquid, +but some float higher than others, +and some do not float at all, sinking instead. +**Archimedes' principle** balances the forces, +and predicts how much of a body is submerged, +and how much is non-submerged. + +In truth, there is no real distinction between +the submerged and non-submerged parts, +since the latter is surrounded by another fluid (air), +which has a pressure and thus affects it. +The right thing to do is treat the entire body as being +submerged in a fluid with varying properties. + +Let us consider a volume $V$ completely submerged in such a fluid. +This volume will experience a downward force due to gravity, given by: + +$$\begin{aligned} + \va{F}_g + = \int_V \va{g} \rho_\mathrm{b} \dd{V} +\end{aligned}$$ + +Where $\va{g}$ is the gravitational field, +and $\rho_\mathrm{b}$ is the density of the body. +Meanwhile, the pressure $p$ of the surrounding fluid exerts a force +on the surface $S$ of $V$: + +$$\begin{aligned} + \va{F}_p + = - \oint_S p \dd{\va{S}} +\end{aligned}$$ + +We rewrite this using Gauss' theorem, +and replace $\nabla p$ by demanding +[hydrostatic equilibrium](/know/concept/hydrostatic-pressure/): + +$$\begin{aligned} + \va{F}_p + = - \int_V \nabla p \dd{V} + = - \int_V \va{g} \rho_\mathrm{f} \dd{V} +\end{aligned}$$ + +For the body to be at rest, we require $\va{F}_g + \va{F}_p = 0$. +Concretely, the equilibrium condition is: + +$$\begin{aligned} + \boxed{ + \int_V \va{g} (\rho_\mathrm{b} - \rho_\mathrm{f}) \dd{V} + = 0 + } +\end{aligned}$$ + +It is commonly assumed that $\va{g}$ has a constant direction +and magnitude $\mathrm{g}$ everywhere. +If we also assume that $\rho_\mathrm{b}$ and $\rho_\mathrm{f}$ are constant, +and only integrate over the "submerged" part, we find: + +$$\begin{aligned} + 0 + = \mathrm{g} (\rho_\mathrm{b} - \rho_\mathrm{f}) V + = \mathrm{g} (m_\mathrm{b} - m_\mathrm{f}) +\end{aligned}$$ + +In other words, the mass $m_\mathrm{b}$ of the submerged portion $V$ of the body, +is equal to the mass $m_\mathrm{f}$ of the fluid it displaces. +This is the best-known version of Archimedes' principle. + +Note that if $\rho_\mathrm{b} > \rho_\mathrm{f}$, then, +even if the entire body is submerged, +the displaced mass $m_\mathrm{f} < m_\mathrm{b}$, +and the object will continue to sink. + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/content/know/concept/bernoullis-theorem/index.pdc b/content/know/concept/bernoullis-theorem/index.pdc new file mode 100644 index 0000000..5ff5225 --- /dev/null +++ b/content/know/concept/bernoullis-theorem/index.pdc @@ -0,0 +1,96 @@ +--- +title: "Bernoulli's theorem" +firstLetter: "B" +publishDate: 2021-04-02 +categories: +- Physics +- Fluid mechanics +- Fluid dynamics + +date: 2021-04-02T15:05:08+02:00 +draft: false +markup: pandoc +--- + +# Bernoulli's theorem + +For inviscid fluids, **Bernuilli's theorem** states +that an increase in flow velocity $\va{v}$ is paired +with a decrease in pressure $p$ and/or potential energy. +For a qualitative argument, look no further than +one of the [Euler equations](/know/concept/euler-equations/), +with a [material derivative](/know/concept/material-derivative/): + +$$\begin{aligned} + \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} + = \va{g} - \frac{\nabla p}{\rho} +\end{aligned}$$ + +Assuming that $\va{v}$ and $\va{g}$ are constant in $t$, +it becomes clear that a higher $\va{v}$ requires a lower $p$: + +$$\begin{aligned} + \frac{1}{2} \nabla \va{v}^2 + = \va{g} - \frac{\nabla p}{\rho} +\end{aligned}$$ + + +## Simple form + +For an incompressible fluid +with a time-independent velocity field $\va{v}$ (i.e. **steady flow**), +Bernoulli's theorem formally states that the +**Bernoulli head** $H$ is constant along a streamline: + +$$\begin{aligned} + \boxed{ + H + = \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} + } +\end{aligned}$$ + +Where $\Phi$ is the gravitational potential, such that $\va{g} = - \nabla \Phi$. +To prove this theorem, we take the material derivative of $H$: + +$$\begin{aligned} + \frac{\mathrm{D} H}{\mathrm{D} t} + &= \va{v} \cdot \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + + \frac{\mathrm{D} \Phi}{\mathrm{D} t} + + \frac{1}{\rho} \frac{\mathrm{D} p}{\mathrm{D} t} +\end{aligned}$$ + +In the first term we insert the Euler equation, +and in the other two we expand the derivatives: + +$$\begin{aligned} + \frac{\mathrm{D} H}{\mathrm{D} t} + &= \va{v} \cdot \Big( \va{g} - \frac{\nabla p}{\rho} \Big) + + \Big( \pdv{\Phi}{t} + (\va{v} \cdot \nabla) \Phi \Big) + + \frac{1}{\rho} \Big( \pdv{p}{t} + (\va{v} \cdot \nabla) p \Big) + \\ + &= \pdv{\Phi}{t} + \frac{1}{\rho} \pdv{p}{t} + + \va{v} \cdot \big( \va{g} + \nabla \Phi \big) + \va{v} \cdot \Big( \frac{\nabla p}{\rho} - \frac{\nabla p}{\rho} \Big) +\end{aligned}$$ + +Using the fact that $\va{g} = - \nabla \Phi$, +we are left with the following equation: + +$$\begin{aligned} + \frac{\mathrm{D} H}{\mathrm{D} t} + &= \pdv{\Phi}{t} + \frac{1}{\rho} \pdv{p}{t} +\end{aligned}$$ + +Assuming that the flow is steady, both derivatives vanish, +leading us to the conclusion that $H$ is conserved along the streamline. + +In fact, there exists **Bernoulli's stronger theorem**, +which states that $H$ is constant *everywhere* in regions with +zero [vorticity](/know/concept/vorticity/) $\va{\omega} = 0$. +For a proof, see the derivation of $\va{\omega}$'s equation of motion. + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/content/know/concept/deutsch-jozsa-algorithm/index.pdc b/content/know/concept/deutsch-jozsa-algorithm/index.pdc index b2b3d98..a891dd1 100644 --- a/content/know/concept/deutsch-jozsa-algorithm/index.pdc +++ b/content/know/concept/deutsch-jozsa-algorithm/index.pdc @@ -4,6 +4,7 @@ firstLetter: "D" publishDate: 2021-04-08 categories: - Quantum information +- Algorithms date: 2021-04-08T10:31:45+02:00 draft: false @@ -59,7 +60,7 @@ $$\begin{aligned} \end{aligned}$$ Starting on the left from two qubits $\ket{0}$ and $\ket{1}$, -we apply the Hadamard gate $H$ to both: +we apply the [Hadamard gate](/know/concept/quantum-gate/) $H$ to both: $$\begin{aligned} \ket{0} \ket{1} diff --git a/content/know/concept/gram-schmidt-method/index.pdc b/content/know/concept/gram-schmidt-method/index.pdc index 88488dd..0b02eee 100644 --- a/content/know/concept/gram-schmidt-method/index.pdc +++ b/content/know/concept/gram-schmidt-method/index.pdc @@ -4,6 +4,7 @@ firstLetter: "G" publishDate: 2021-02-22 categories: - Mathematics +- Algorithms date: 2021-02-22T21:36:08+01:00 draft: false diff --git a/content/know/concept/quantum-gate/cnot.png b/content/know/concept/quantum-gate/cnot.png new file mode 100644 index 0000000..9fae3ce Binary files /dev/null and b/content/know/concept/quantum-gate/cnot.png differ diff --git a/content/know/concept/quantum-gate/cu.png b/content/know/concept/quantum-gate/cu.png new file mode 100644 index 0000000..465f09c Binary files /dev/null and b/content/know/concept/quantum-gate/cu.png differ diff --git a/content/know/concept/quantum-gate/index.pdc b/content/know/concept/quantum-gate/index.pdc index cd09094..189145f 100644 --- a/content/know/concept/quantum-gate/index.pdc +++ b/content/know/concept/quantum-gate/index.pdc @@ -208,6 +208,10 @@ but not always in the basis of $\ket{0}_1$, $\ket{1}_1$, $\ket{0}_2$ and $\ket{1 With that said, the first two-qubit gate is $\mathrm{SWAP}$, which simply swaps $\ket{\psi_1}$ and $\ket{\psi_2}$: + + + + $$\begin{aligned} \boxed{ \mathrm{SWAP} = @@ -233,6 +237,10 @@ $$\begin{aligned} Next, there is the **controlled NOT gate** $\mathrm{CNOT}$, which "flips" (applies $X$ to) $\ket{\psi_2}$ if $\ket{\psi_1}$ is true: + + + + $$\begin{aligned} \boxed{ \mathrm{CNOT} = @@ -252,7 +260,13 @@ $$\begin{aligned} &= c_{00} \ket{00} + c_{01} \ket{01} + c_{11} \ket{10} + c_{10} \ket{11} \end{aligned}$$ -More generally, each one-qubit gate $U$ can be turned into a **controlled** $U$ **gate**: +More generally, from every one-qubit gate $U$, +we can define a two-qubit **controlled U gate** $\mathrm{CU}$, +which applies $U$ to $\ket{\psi_2}$ if $\ket{\psi_1}$ is true: + + + + $$\begin{aligned} \boxed{ diff --git a/content/know/concept/quantum-gate/swap.png b/content/know/concept/quantum-gate/swap.png new file mode 100644 index 0000000..fa20442 Binary files /dev/null and b/content/know/concept/quantum-gate/swap.png differ diff --git a/content/know/concept/toffoli-gate/index.pdc b/content/know/concept/toffoli-gate/index.pdc index 84e7656..da5f2a5 100644 --- a/content/know/concept/toffoli-gate/index.pdc +++ b/content/know/concept/toffoli-gate/index.pdc @@ -60,8 +60,8 @@ i.e. OR is NAND of NOT $A$ and NOT $B$: Thanks to its reversibility and universality, -the Toffoli gate is interesting in quantum computing, -where it is often referred to as the CCNOT gate. +the Toffoli gate is interesting for quantum computing. +Its [quantum gate](/know/concept/quantum-gate/) form is often called **CCNOT**. In the basis $\ket{A} \ket{B} \ket{C}$, its matrix is: $$\begin{aligned} diff --git a/content/know/concept/vorticity/index.pdc b/content/know/concept/vorticity/index.pdc new file mode 100644 index 0000000..5aec049 --- /dev/null +++ b/content/know/concept/vorticity/index.pdc @@ -0,0 +1,165 @@ +--- +title: "Vorticity" +firstLetter: "V" +publishDate: 2021-04-03 +categories: +- Physics +- Fluid mechanics +- Fluid dynamics + +date: 2021-04-03T09:24:42+02:00 +draft: false +markup: pandoc +--- + +# Vorticity + +In fluid mechanics, the **vorticity** $\va{\omega}$ +is a measure of the local circulation in a fluid. +It is defined as the curl of the flow velocity field $\va{v}$: + +$$\begin{aligned} + \boxed{ + \va{\omega} + \equiv \nabla \cross \va{v} + } +\end{aligned}$$ + +Just as curves tangent to $\va{v}$ are called *streamlines*, +curves tangent to $\va{\omega}$ are **vortex lines**, +which are to be interpreted as the "axes" that $\va{v}$ is circulating around. + +The vorticity is a local quantity, +and the corresponding global quantity is the **circulation** $\Gamma$, +which is defined as the projection of $\va{v}$ onto a close curve $C$. +Then, by Stokes' theorem: + +$$\begin{aligned} + \boxed{ + \Gamma(C, t) + \equiv \oint_C \va{v} \cdot \dd{\va{l}} + = \int_S \va{\omega} \cdot \dd{\va{S}} + } +\end{aligned}$$ + + +## Ideal fluids + +For an inviscid, incompressible fluid, +consider the *Bernoulli field* $H$, which is defined as: + +$$\begin{aligned} + H + \equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} +\end{aligned}$$ + +Where $\Phi$ is the gravitational potential, +$p$ is the pressure, and $\rho$ is the (constant) density. +We then take the gradient of this scalar field: + +$$\begin{aligned} + \nabla H + &= \frac{1}{2} \nabla \va{v}^2 + \nabla \Phi + \frac{\nabla p}{\rho} + \\ + &= \va{v} \cdot (\nabla \va{v}) - \Big( \!-\! \nabla \Phi - \frac{\nabla p}{\rho} \Big) +\end{aligned}$$ + +Since $-\nabla \Phi = \va{g}$, +the rightmost term is the right-hand side of +the [Euler equation](/know/concept/euler-equations/). +We substitute the other side of said equation, yielding: + +$$\begin{aligned} + \nabla H + &= \va{v} \cdot (\nabla \va{v}) - \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + = \va{v} \cdot (\nabla \va{v}) - \pdv{\va{v}}{t} - (\va{v} \cdot \nabla) \va{v} +\end{aligned}$$ + +We isolate this equation for $\pdv*{\va{v}}{t}$, +and apply a vector identity to reduce it to the following: + +$$\begin{aligned} + \pdv{\va{v}}{t} + = \va{v} \cdot (\nabla \va{v}) - (\va{v} \cdot \nabla) \va{v} - \nabla H + = \va{v} \cross (\nabla \cross \va{v}) - \nabla H +\end{aligned}$$ + +Here, the definition of the vorticity $\va{\omega}$ is clear to see, +leading us to an equation of motion for $\va{v}$: + +$$\begin{aligned} + \boxed{ + \pdv{\va{v}}{t} + = \va{v} \cross \va{\omega} - \nabla H + } +\end{aligned}$$ + +More about this later. +Now, we take the curl of both sides of this equation, giving us: + +$$\begin{aligned} + \nabla \cross \pdv{\va{v}}{t} + = \nabla \cross (\va{v} \cross \va{\omega}) - \nabla \cross (\nabla H) +\end{aligned}$$ + +On the left, we swap $\nabla$ with $\pdv*{t}$, +and on the right, the curl of a gradient is always zero. +We are thus left with the equation of motion of the vorticity $\va{\omega}$: + +$$\begin{aligned} + \boxed{ + \pdv{\va{\omega}}{t} + = \nabla \cross (\va{v} \cross \va{\omega}) + } +\end{aligned}$$ + +Let us now return to the equation of motion for $\va{v}$. +For *steady* flows where $\pdv*{\va{v}}{t} = 0$, in which case +[Bernoulli's theorem](/know/concept/bernoullis-theorem/) applies, +it reduces to: + +$$\begin{aligned} + \nabla H + = \va{v} \cross \va{\omega} +\end{aligned}$$ + +If a fluid has $\va{\omega} = 0$ in some regions, it is known as **irrotational**. +From this equation, we see that, in that case, $\nabla H = 0$, +meaning that $H$ is a constant in those regions, +a fact sometimes referred to as **Bernoulli's stronger theorem**. + +Furthermore, irrotationality $\va{\omega} = 0$ +implies that $\va{v}$ is the gradient of a potential $\Psi$: + +$$\begin{aligned} + \va{v} + = \nabla \Psi +\end{aligned}$$ + +This fact allows us to rewrite the Euler equations in a particularly simple way. +Firstly, the condition of incompressibility becomes the well-known Laplace equation: + +$$\begin{aligned} + 0 + = \nabla \cdot \va{v} + = \nabla^2 \Psi +\end{aligned}$$ + +And second, the main equation of motion for $\va{v}$ states +that the quantity $H + \pdv*{\Psi}{t}$ is spatially constant +in the irrotational region: + +$$\begin{aligned} + \pdv{\va{v}}{t} + = \nabla \pdv{\Psi}{t} + = - \nabla H + \quad \implies \quad + \nabla \Big( H + \pdv{\Psi}{t} \Big) + = 0 +\end{aligned}$$ + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. -- cgit v1.2.3