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This scene asks a stranger question : What happens if the material has negative refractive index? As the electron moves through the metamaterial slab, the emitted radiation no longer follows the familiar Cherenkov geometry (see below). The energy flow forms a backward-directed cone, extending opposite to the electron's motion.... show more

Mathelirium

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13,390 次观看 • 24 天前 •via X (Twitter)

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This is a simulation of a Hydrogen electron described by a superposition of several highly excited Rydberg states. The superposition forms a localized quantum wavepacket that moves around the atom, spreads, splits, and later reforms as the different energy components return to phase.

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Riddle: Where does the energy flow in this steady state DC circuit with a copper block? And where do the Poynting vectors point? As a bonus, extend the size of the copper block to the size of the earth.

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A rocket cuts through a solved Navier-Stokes flow, forcing the fluid to separate around its body and roll into a long turbulent wake. Using the D2Q9 lattice-Boltzmann method, the scene reveals vorticity, strain, Q-criterion vortex cores and wall shear.

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An intense pulse of laser light enters a plasma and what follows is almost unbelievable. The laser pushes plasma electron aside, carving out a charged cavity. As the plasma rushes back, it creates an electric wake that can trap and accelerate another bunch of electrons. This is known as Laser Wakefield Acceleration (LWFA) and is being explored for future accelerator Physics. For this animation, we used a reduced particle-in-cell (PIC) method, coupling particle motion to the self-consistent plasma field on a computational mesh.

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A Luneburg lens bends an Electromagnetic wave without a curved boundary. Instead its dielectric constant changes continuously from the centre to the edge. This forces different parts of the wavelength to travel at different speeds. Here, we solve the full 2D TM EM wave equation on an unstructured triangular mesh using the Finite Element Method (FEM).

A Luneburg lens bends an Electromagnetic wave without a curved boundary. Instead its dielectric constant changes continuously from the centre to the edge. This forces different parts of the wavelength to travel at different speeds. Here, we solve the full 2D TM EM wave equation on an unstructured triangular mesh using the Finite Element Method (FEM).

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The positron is the antimatter partner of the electron and together they form positronium... a tiny quantum system with no nucleus. We used the time-dependent Schrödinger equation to evolve the electron and positron clouds. Their motion comes from their quantum states changing over time and the color from phase.

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What Shape Does Chaos Take Before It Looks Random? Take one particle moving in the Hénon-Heiles Hamiltonian H = ½(pₓ² + pᵧ²) + ½(x² + y²) + λ(x²y - ⅓y³) Its motion is forced by Hamilton’s equations ẋ = pₓ ẏ = pᵧ ṗₓ = -x - 2λxy ṗᵧ = -y - λ(x² - y²) The saddle energy is Eₛ = 1/(6λ²) Below that energy, the particle remains trapped. But a tiny packet of nearby initial conditions stretches into filaments, folds through the potential, and begins to look almost random. However, it is not random. Hamiltonian flow preserves the hidden phase-space structure. What looks like chaos in ordinary space is still a deterministic loom of energy conserved, equations obeyed, geometry stretched without being erased. #HamiltonianMechanics #ChaosTheory #PhaseSpace #DynamicalSystems #MathematicalPhysics #NonlinearDynamics #HenonHeiles #SymplecticGeometry #ScienceAnimation

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Now, we allow the electromagnetic energy to propagate cavity by cavity through a longer optical pathway.

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It is often said that the lift on a wing is generated because the flow moving over the top surface has a longer distance to travel and therefore needs to go faster. This common explanation is actually wrong.

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Mathelirium

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In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

45,802 次观看 • 6 个月前

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

Mathelirium

10,806 次观看 • 4 个月前

An Electron flies above a metal grating without touching it. As the electron passes, each groove scatters its electromagnetic near field at a slightly different time. The emissions interfere and produce what's called Smith-Purcell radiation. This effect is being explored for tunable Terahertz radiation, biomedical spectroscopy, and high-speed communications.

An Electron flies above a metal grating without touching it. As the electron passes, each groove scatters its electromagnetic near field at a slightly different time. The emissions interfere and produce what's called Smith-Purcell radiation. This effect is being explored for tunable Terahertz radiation, biomedical spectroscopy, and high-speed communications.

Mathelirium

21,005 次观看 • 29 天前

This is a visualization of a Rydberg wavepacket. In simple terms it is a way of showing an electron in a very highly excited quantum state. The Math comes from the Schrödinger equation for hydrogen where the electron is described by a wavefunction rather than as a tiny ball moving on a fixed path. The yellow sphere at the centre is the proton, not the electron. The colored 3D surface around it shows places where the electron is more likely to be, i.e. a probability cloud.

This is a visualization of a Rydberg wavepacket. In simple terms it is a way of showing an electron in a very highly excited quantum state. The Math comes from the Schrödinger equation for hydrogen where the electron is described by a wavefunction rather than as a tiny ball moving on a fixed path. The yellow sphere at the centre is the proton, not the electron. The colored 3D surface around it shows places where the electron is more likely to be, i.e. a probability cloud.

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17,580 次观看 • 29 天前

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27,496 次观看 • 1 年前