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When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar's... show more

Mathelirium

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10,118 просмотров • 6 месяцев назад •via X (Twitter)

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$When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar is not a metronome. Its angle sways and wobbles, and its strength breathes over time, so the force field keeps changing. I seed about 45k test stars on near-circular orbits in a ring and evolve them by using Newton’s 2nd law step by step (velocity Verlet scheme)...the potential is time-dependent. As the bar turns, resonances trap some stars into long-lived orbital families. Others get stretched into spiral-like streams as the ring is sheared. A fraction drift into chaotic layers, where tiny timing differences blow up because the resonance boundaries are moving. What you see in the trails is phase space being rewritten...stable islands, separating boundaries, broken tori, and mixing. #Astrophysics #GalacticDynamics #BarredGalaxy #HamiltonianChaos #PhaseSpace #OrbitalResonance$

When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar is not a metronome. Its angle sways and wobbles, and its strength breathes over time, so the force field keeps changing. I seed about 45k test stars on near-circular orbits in a ring and evolve them by using Newton’s 2nd law step by step (velocity Verlet scheme)...the potential is time-dependent. As the bar turns, resonances trap some stars into long-lived orbital families. Others get stretched into spiral-like streams as the ring is sheared. A fraction drift into chaotic layers, where tiny timing differences blow up because the resonance boundaries are moving. What you see in the trails is phase space being rewritten...stable islands, separating boundaries, broken tori, and mixing. #Astrophysics #GalacticDynamics #BarredGalaxy #HamiltonianChaos #PhaseSpace #OrbitalResonance

Mathelirium

16,103 просмотров • 7 месяцев назад

Gravity makes things orbit...but what happens when the mass distribution itself is rotating and dragging phase space into resonances and chaos? In this scene we follow tens of thousands of stars in a toy barred galaxy: a flat disc with a central mass and a rotating bar of extra density that is not rigid, but wobbling, swaying, and breathing in strength over time. The stars start on almost circular orbits in a calm ring, but as the bar turns and its pattern speed and strength modulate, they feel a time-dependent gravitational shove that nudges some onto resonant tracks. 🤩😍 #Mathelirium #Astrophysics #GalacticDynamics #BarredGalaxy #PhaseSpace #HamiltonianChaos #OrbitalResonance

Gravity makes things orbit...but what happens when the mass distribution itself is rotating and dragging phase space into resonances and chaos? In this scene we follow tens of thousands of stars in a toy barred galaxy: a flat disc with a central mass and a rotating bar of extra density that is not rigid, but wobbling, swaying, and breathing in strength over time. The stars start on almost circular orbits in a calm ring, but as the bar turns and its pattern speed and strength modulate, they feel a time-dependent gravitational shove that nudges some onto resonant tracks. 🤩😍 #Mathelirium #Astrophysics #GalacticDynamics #BarredGalaxy #PhaseSpace #HamiltonianChaos #OrbitalResonance

Mathelirium

152,143 просмотров • 9 месяцев назад

The secret to antigravity is scalar longitudinal waves, as discovered by Ampere and ignored as a mathematical convenience. The A vector potential. The A field twists spacetime the same way magnetic fields twist charged particles. A conditioned EM beam can make gravitational effects by structuring a phase gradient that mimics the frame dragging of a rotating mass. An undulator or helix can twist the phase, forcing the electrons into a sinusoidal path so that it carries a coherent A vector potential.

The secret to antigravity is scalar longitudinal waves, as discovered by Ampere and ignored as a mathematical convenience. The A vector potential. The A field twists spacetime the same way magnetic fields twist charged particles. A conditioned EM beam can make gravitational effects by structuring a phase gradient that mimics the frame dragging of a rotating mass. An undulator or helix can twist the phase, forcing the electrons into a sinusoidal path so that it carries a coherent A vector potential.

Ashton Forbes

26,671 просмотров • 3 дней назад

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

54,730 просмотров • 5 месяцев назад

A sharp 90-degree turn is a brutal request for an Electromagnetic wave. Yet inside a Photonic Crystal, something remarkable can happen. The repeating dielectric structure creates ranges of frequency for which propagation through the bulk crystal is strongly suppressed. Remove a line of that crystal and suddenly the defect becomes an allowed route the EM field.

A sharp 90-degree turn is a brutal request for an Electromagnetic wave. Yet inside a Photonic Crystal, something remarkable can happen. The repeating dielectric structure creates ranges of frequency for which propagation through the bulk crystal is strongly suppressed. Remove a line of that crystal and suddenly the defect becomes an allowed route the EM field.

Mathelirium

32,436 просмотров • 2 дней назад

Stealing Energy from a Rotating Black Hole A rotating black hole doesn’t just pull things in. It drags spacetime itself into motion. In the spinning region just outside the horizon, nothing can remain still in any normal sense. That strange rule opens a loophole. One incoming body can split, with one piece swallowed while the other escapes with more energy than it came in with. A Penrose-style heist where the difference is paid for by the black hole’s rotation. The same idea shows up in wave form. A coherent packet can skim that region and return brighter, as if the geometry itself acted like an amplifier. Split near the ergosphere and you can steal spin. Skim it as a wave and you can come back louder.

Stealing Energy from a Rotating Black Hole A rotating black hole doesn’t just pull things in. It drags spacetime itself into motion. In the spinning region just outside the horizon, nothing can remain still in any normal sense. That strange rule opens a loophole. One incoming body can split, with one piece swallowed while the other escapes with more energy than it came in with. A Penrose-style heist where the difference is paid for by the black hole’s rotation. The same idea shows up in wave form. A coherent packet can skim that region and return brighter, as if the geometry itself acted like an amplifier. Split near the ergosphere and you can steal spin. Skim it as a wave and you can come back louder.

Mathelirium

20,314 просмотров • 6 месяцев назад

A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982

A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982

Mathelirium

130,081 просмотров • 3 месяцев назад

$Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an application. We look at the theory behind electromagnetic scattering and radar clutter, which leads straight into anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum. At time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where Brownian motion shows up in the most reasonable way. Think of all the microscopic chaos: tiny motions, index fluctuations, path jitters, Doppler shifts. Over short times, all of that shows up as small random kicks to the phases θⱼ(t). If you made θⱼ(t) random in an ad-hoc way, like resampling a fresh independent angle at every instant, the field would jump around unrealistically with no physical time structure. Brownian motion is what you get when each phase takes the continuous-time limit of many tiny, independent kicks. It’s continuous in t, its variance grows the right way, and it carries just enough temporal structure to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing. It’s the clean way to compress all the small random stuff into a single process that actually matches how phases wander in time. This is called Rayleigh Scattering, but the same sum of many tiny coherent echoes shows up in lots of places...e.g. wireless multipath fading (phones/Wi-Fi), laser/optical links through atmospheric turbulence, ultrasound speckle in tissue, and sonar/underwater acoustics in rough or bubbly water. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing$

Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an application. We look at the theory behind electromagnetic scattering and radar clutter, which leads straight into anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum. At time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where Brownian motion shows up in the most reasonable way. Think of all the microscopic chaos: tiny motions, index fluctuations, path jitters, Doppler shifts. Over short times, all of that shows up as small random kicks to the phases θⱼ(t). If you made θⱼ(t) random in an ad-hoc way, like resampling a fresh independent angle at every instant, the field would jump around unrealistically with no physical time structure. Brownian motion is what you get when each phase takes the continuous-time limit of many tiny, independent kicks. It’s continuous in t, its variance grows the right way, and it carries just enough temporal structure to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing. It’s the clean way to compress all the small random stuff into a single process that actually matches how phases wander in time. This is called Rayleigh Scattering, but the same sum of many tiny coherent echoes shows up in lots of places...e.g. wireless multipath fading (phones/Wi-Fi), laser/optical links through atmospheric turbulence, ultrasound speckle in tissue, and sonar/underwater acoustics in rough or bubbly water. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing

Mathelirium

31,182 просмотров • 6 месяцев назад

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

Mathelirium

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Light Can Make Its Own Gravitational Field In gGeneral Relativity, light does not just move through Spacetime. Because it carries energy and momentum, it can also curve Spacetime.

Light Can Make Its Own Gravitational Field In gGeneral Relativity, light does not just move through Spacetime. Because it carries energy and momentum, it can also curve Spacetime.

Mathelirium

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CS Muturi: Since I am also a victim while serving in this government, I have taken this unusual step so that abductions and extra-judicial activities can be debated in the country in order to find a solution. If left unchecked, it can plunge the country into chaos and anarchy

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This is wild. Arcturus Studios is capturing real sports in full volumetric so you can explore the action from any angle. They surround the field with cameras and use Gaussian splatting to turn the game into a 3D experience. This is the future of sports.

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Justin Ryan

61,942 просмотров • 6 месяцев назад

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics

Mathelirium

35,799 просмотров • 2 месяцев назад

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

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

Mathelirium

26,893 просмотров • 2 месяцев назад

Black Hole Can Ring Like a Bell A Black Hole can vibrate after it is disturbed. Instead of settling instantly, its horizon releases gravitational waves in fading tones called quasinormal modes. This is commonly known as Ringdown. Each tone carries information about the black hole’s final mass and spin, which is why physicists call this Black-Hole Spectroscopy. In a way, when Spacetime strikes a Black Hole, the Black Hole answers with its own gravitational sound. #BlackHole #GeneralRelativity #GravitationalWaves #Spacetime #Einstein #Astrophysics #Physics #Mathematics #Cosmology

Black Hole Can Ring Like a Bell A Black Hole can vibrate after it is disturbed. Instead of settling instantly, its horizon releases gravitational waves in fading tones called quasinormal modes. This is commonly known as Ringdown. Each tone carries information about the black hole’s final mass and spin, which is why physicists call this Black-Hole Spectroscopy. In a way, when Spacetime strikes a Black Hole, the Black Hole answers with its own gravitational sound. #BlackHole #GeneralRelativity #GravitationalWaves #Spacetime #Einstein #Astrophysics #Physics #Mathematics #Cosmology

Mathelirium

30,802 просмотров • 1 месяц назад

🧱 #Live3D #Live2D #HandTracking 🧱 I spent some time refining the way I use Warp Deformers to create lateral physics-based swinging motion in Live2D. With a new production concept, I tried to achieve the highest possible level of mathematical precision. At the core are two fundamental swinging structures, which I can mix in different proportions to create different qualities of motion. The first is parallel swinging. Both sides of the Warp Deformer remain perfectly parallel throughout the motion, making it look somewhat like a sheet of paper or a tassel. The second is bending swinging. The angle between the two sides of the Warp Deformer changes as it moves, making it look more like a flexible tube or a tail. （0:12）Interestingly, when I layer the first and second structures together, the bottom forms an X-shaped pattern as it swings. It looks a little like a spine or some kind of mechanical structure. （0:23）The third structure is an equal blend of the first and second. Its characteristics are much more neutral, combining the qualities of both. （0:35）The fourth and fifth structures are created by smoothly blending the first structure into the center of the second, and the second into the center of the first, respectively. This makes them appear softer and more flexible, somewhat like a piece of fabric. （0:28）Interestingly, when I blend the fourth and fifth structures equally, the result is exactly identical to the third structure. This also demonstrates that, despite their apparent complexity, the two are actually perfect mirror images of each other. Every physics-based motion shown here uses a two-segment pendulum structure, with a swinging range of ±90°. In theory, when curled to its limit, it can bend upward into a perfect semicircle. To most people, this may look as boring as a brick. But to me, this is exactly where the romance of Live2D physics lies. Does anyone else feel the same way?

🧱 #Live3D #Live2D #HandTracking 🧱 I spent some time refining the way I use Warp Deformers to create lateral physics-based swinging motion in Live2D. With a new production concept, I tried to achieve the highest possible level of mathematical precision. At the core are two fundamental swinging structures, which I can mix in different proportions to create different qualities of motion. The first is parallel swinging. Both sides of the Warp Deformer remain perfectly parallel throughout the motion, making it look somewhat like a sheet of paper or a tassel. The second is bending swinging. The angle between the two sides of the Warp Deformer changes as it moves, making it look more like a flexible tube or a tail. （0:12）Interestingly, when I layer the first and second structures together, the bottom forms an X-shaped pattern as it swings. It looks a little like a spine or some kind of mechanical structure. （0:23）The third structure is an equal blend of the first and second. Its characteristics are much more neutral, combining the qualities of both. （0:35）The fourth and fifth structures are created by smoothly blending the first structure into the center of the second, and the second into the center of the first, respectively. This makes them appear softer and more flexible, somewhat like a piece of fabric. （0:28）Interestingly, when I blend the fourth and fifth structures equally, the result is exactly identical to the third structure. This also demonstrates that, despite their apparent complexity, the two are actually perfect mirror images of each other. Every physics-based motion shown here uses a two-segment pendulum structure, with a swinging range of ±90°. In theory, when curled to its limit, it can bend upward into a perfect semicircle. To most people, this may look as boring as a brick. But to me, this is exactly where the romance of Live2D physics lies. Does anyone else feel the same way?

📐Hephaestus📏Live2D匠人魂

19,362 просмотров • 21 дней назад

$Today we introduce Stochastic Differential Equations (SDEs). I find that the best way to introduce these complex concepts is to look at an application. This is part I of the lecture🙂 We look at the theory behind electromagnetic scattering/radar clutter which leads to anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum...at time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where the magic of Brownian motion appears naturally and in the most reasonable way. Think of all the microscopic chaos...tiny motions, index fluctuations, path jitters, Doppler shifts that shows up as small random kicks to the phases θⱼ(t) over very short times. If you just made θⱼ(t) random in an ad-hoc way (say, resampling independent angles at each time), the field would jump around unrealistically with no temporal structure. Brownian motion is what you get when you let each phase take the continuous-time limit of many tiny, independent kicks...it’s continuous in t, it has the right cumulative variance growth, and it remembers just enough of its past to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing...it’s the clean way to compress all the small random stuff into a single process that actually matches how the phases wander in time. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing$

Today we introduce Stochastic Differential Equations (SDEs). I find that the best way to introduce these complex concepts is to look at an application. This is part I of the lecture🙂 We look at the theory behind electromagnetic scattering/radar clutter which leads to anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum...at time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where the magic of Brownian motion appears naturally and in the most reasonable way. Think of all the microscopic chaos...tiny motions, index fluctuations, path jitters, Doppler shifts that shows up as small random kicks to the phases θⱼ(t) over very short times. If you just made θⱼ(t) random in an ad-hoc way (say, resampling independent angles at each time), the field would jump around unrealistically with no temporal structure. Brownian motion is what you get when you let each phase take the continuous-time limit of many tiny, independent kicks...it’s continuous in t, it has the right cumulative variance growth, and it remembers just enough of its past to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing...it’s the clean way to compress all the small random stuff into a single process that actually matches how the phases wander in time. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing

Mathelirium

55,319 просмотров • 8 месяцев назад

Once you swallow the fact that Brownian motion is continuous everywhere and differentiable nowhere, Itô’s quadratic variation is the next quiet shock waiting for you: Ordinary smooth curves have zero quadratic variation...if you chop time into tiny pieces and add up (increment)² over the interval, that sum goes to zero as your partition gets finer. But for Brownian motion that same sum doesn’t vanish, it locks onto the length of the time interval.😲 So, you have this path with no velocity anywhere, but its squared wiggles accumulate in a completely rigid, deterministic way. Over time T, the total quadratic variation is exactly T. 🫡 That’s the trade the universe makes here...you lose the classical derivative, but you gain a new calculus where the “dB times dB” term behaves like dt, and suddenly integrals against Brownian motion make sense and you can do stochastic calculus. 📝 Brownian motion refuses to be smooth, but its roughness is so structured that when you look at it through the lens of quadratic variation, the randomness averages out and you’re left with a clean clock ticking underneath the noise. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #Ito #QuadraticVariation

Once you swallow the fact that Brownian motion is continuous everywhere and differentiable nowhere, Itô’s quadratic variation is the next quiet shock waiting for you: Ordinary smooth curves have zero quadratic variation...if you chop time into tiny pieces and add up (increment)² over the interval, that sum goes to zero as your partition gets finer. But for Brownian motion that same sum doesn’t vanish, it locks onto the length of the time interval.😲 So, you have this path with no velocity anywhere, but its squared wiggles accumulate in a completely rigid, deterministic way. Over time T, the total quadratic variation is exactly T. 🫡 That’s the trade the universe makes here...you lose the classical derivative, but you gain a new calculus where the “dB times dB” term behaves like dt, and suddenly integrals against Brownian motion make sense and you can do stochastic calculus. 📝 Brownian motion refuses to be smooth, but its roughness is so structured that when you look at it through the lens of quadratic variation, the randomness averages out and you’re left with a clean clock ticking underneath the noise. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #Ito #QuadraticVariation

Mathelirium

100,544 просмотров • 8 месяцев назад

A Quantum wave takes two paths around an enclosed magnetic flux and reunites with a phase offset that no local magnetic field measurement can explain. Threads represent the probability current.

A Quantum wave takes two paths around an enclosed magnetic flux and reunites with a phase offset that no local magnetic field measurement can explain. Threads represent the probability current.

Mathelirium

30,287 просмотров • 15 дней назад

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

Mathelirium

47,651 просмотров • 9 месяцев назад