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This simulation visualizes the Maxwell Boltzmann distribution by showing how gas particles move at different speeds. The emerging bell curve highlights the statistical probability of finding molecules at specific velocity states.

Matheorems

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16,018 次观看 • 18 天前 •via X (Twitter)

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Maxwell–Boltzmann distribution. The air around you is full of tiny molecules, and at room temperature most of them move faster than the speed of sound. They do not all move at the same speed. A few are slow, most cluster around a typical speed, and a rare few are extremely fast. In 1860, James Clerk Maxwell worked out how these speeds are distributed. Ludwig Boltzmann later extended the idea. Here 1,200 beads play the molecules. The faster a bead flies out, the farther it lands, so each slot collects beads of similar speed. Together, the piles trace the Maxwell–Boltzmann curve.

Maxwell–Boltzmann distribution. The air around you is full of tiny molecules, and at room temperature most of them move faster than the speed of sound. They do not all move at the same speed. A few are slow, most cluster around a typical speed, and a rare few are extremely fast. In 1860, James Clerk Maxwell worked out how these speeds are distributed. Ludwig Boltzmann later extended the idea. Here 1,200 beads play the molecules. The faster a bead flies out, the farther it lands, so each slot collects beads of similar speed. Together, the piles trace the Maxwell–Boltzmann curve.

The Math Flow

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Did you know you can create the Maxwell-Boltzmann distribution with just some balls and a motor? I built this simple device to illustrate it. This is how it works: A spinning agitator pumps energy into ~400 balls, so they bounce around and collide like gas particles in a box. They escape one at a time through a tiny hole, then fall a fixed gap into bins below. Since the hole is so small, every ball leaves with no vertical speed and takes the same time to fall. Constant fall time means how far a ball flies sideways is set purely by its speed. A faster ball goes into a further bin. So each bin is really a speed. Tally all 400 and you get the Maxwell-Boltzmann distribution.

Did you know you can create the Maxwell-Boltzmann distribution with just some balls and a motor? I built this simple device to illustrate it. This is how it works: A spinning agitator pumps energy into ~400 balls, so they bounce around and collide like gas particles in a box. They escape one at a time through a tiny hole, then fall a fixed gap into bins below. Since the hole is so small, every ball leaves with no vertical speed and takes the same time to fall. Constant fall time means how far a ball flies sideways is set purely by its speed. A faster ball goes into a further bin. So each bin is really a speed. Tally all 400 and you get the Maxwell-Boltzmann distribution.

Ethan Gao

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This animation shows how a uniform grid of points is deformed by a flow matching model. The visualization highlights how the learned flow warps the underlying space—stretching, compressing, and bending it—to transform one probability distribution into another.

This animation shows how a uniform grid of points is deformed by a flow matching model. The visualization highlights how the learned flow warps the underlying space—stretching, compressing, and bending it—to transform one probability distribution into another.

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The road to innovation is driven by simulation. Get your first look at the new docuseries exploring how simulation enables the rapid innovation transforming the way we move — on the street, off-road, or at the track. This is #DrivenBySim. Coming soon.

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How long it would take to reach the Moon at different speeds

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Most agents track price. Ours track how information propagates through markets. That difference is where the edge comes from. The visualization below is a Shannon entropy field: a model of how uncertainty collapses and spreads as new information enters a market. Each trade, liquidity shift, or derivatives flow updates a probability distribution. When information arrives, entropy collapses locally as the market incorporates it. But that update rarely happens everywhere at once. Different venues, liquidity pools, and derivative layers incorporate signals at different speeds, creating measurable information gradients across the system. One of the analytical tools in our agents’ stack models this directly: > Entropy quantifies uncertainty embedded in each probability market > KL divergence reveals conflicting distributions across venues > Information gradients show where signals have propagated—and where they haven’t yet Prediction markets and DeFi make this particularly powerful because price already represents probability. Mapping how those probability distributions update across venues turns the market into a measurable system: a live field of information flow showing where uncertainty remains—and where probability is likely to adjust next

Most agents track price. Ours track how information propagates through markets. That difference is where the edge comes from. The visualization below is a Shannon entropy field: a model of how uncertainty collapses and spreads as new information enters a market. Each trade, liquidity shift, or derivatives flow updates a probability distribution. When information arrives, entropy collapses locally as the market incorporates it. But that update rarely happens everywhere at once. Different venues, liquidity pools, and derivative layers incorporate signals at different speeds, creating measurable information gradients across the system. One of the analytical tools in our agents’ stack models this directly: > Entropy quantifies uncertainty embedded in each probability market > KL divergence reveals conflicting distributions across venues > Information gradients show where signals have propagated—and where they haven’t yet Prediction markets and DeFi make this particularly powerful because price already represents probability. Mapping how those probability distributions update across venues turns the market into a measurable system: a live field of information flow showing where uncertainty remains—and where probability is likely to adjust next

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A visualisation of different velocity factors in a PCB. Both simulations use the same excitation and port parameters. Only the permittivity is different, showing how the signal travels much faster in low Dk dielectrics. This simulation shows an embedded trace. A signal on an outer layer also travels faster because part of the EM field propagates through air, which has a Dk very close to 1

A visualisation of different velocity factors in a PCB. Both simulations use the same excitation and port parameters. Only the permittivity is different, showing how the signal travels much faster in low Dk dielectrics. This simulation shows an embedded trace. A signal on an outer layer also travels faster because part of the EM field propagates through air, which has a Dk very close to 1

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I started this GPU implementation of interacting particles and it's already blowing my mind! Have a look at the stuff I found! (Thread) #madewithunity #indiedev #Simulation

I started this GPU implementation of interacting particles and it's already blowing my mind! Have a look at the stuff I found! (Thread) #madewithunity #indiedev #Simulation

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This is one of my most favorite tools: TPS → the heartbeat of the market That small line at the top? That’s TPS - ticks per second. • Low TPS = quiet accumulation or decay. • Spikes in TPS = sudden aggression, algo dumps, or a liquidity hunt. Why it matters: scalps live & die on velocity. A TPS spike + price movement = can signal real aggression. Rule: when TPS spikes across multiple exchanges, treat the move as higher-probability and monitor trades amount. If only one exchange shows velocity = be cautious (venue specific).

This is one of my most favorite tools: TPS → the heartbeat of the market That small line at the top? That’s TPS - ticks per second. • Low TPS = quiet accumulation or decay. • Spikes in TPS = sudden aggression, algo dumps, or a liquidity hunt. Why it matters: scalps live & die on velocity. A TPS spike + price movement = can signal real aggression. Rule: when TPS spikes across multiple exchanges, treat the move as higher-probability and monitor trades amount. If only one exchange shows velocity = be cautious (venue specific).

When Sailing

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The Gaussian Wave Packet ✍️ It is a way to describe a quantum particle, like an electron, that is located in a certain area of space instead of being spread out everywhere. It forms a bell-shaped curve, where the peak of the curve indicates the most likely place to find the particle. This shape is special because it offers the best way to pack a particle; it meets the physical limit set by the Heisenberg Uncertainty Principle, striking the best balance between knowing a particle's position and its momentum. However, because this packet consists of many different overlapping waves, it is naturally unstable for a free particle. Over time, these internal waves travel at slightly different speeds, which causes the bell curve to gradually flatten and widen. This process is known as dispersion. In simpler terms, the longer a quantum particle travels freely, the more unclear its exact location becomes. Video : Jeedecode

The Gaussian Wave Packet ✍️ It is a way to describe a quantum particle, like an electron, that is located in a certain area of space instead of being spread out everywhere. It forms a bell-shaped curve, where the peak of the curve indicates the most likely place to find the particle. This shape is special because it offers the best way to pack a particle; it meets the physical limit set by the Heisenberg Uncertainty Principle, striking the best balance between knowing a particle's position and its momentum. However, because this packet consists of many different overlapping waves, it is naturally unstable for a free particle. Over time, these internal waves travel at slightly different speeds, which causes the bell curve to gradually flatten and widen. This process is known as dispersion. In simpler terms, the longer a quantum particle travels freely, the more unclear its exact location becomes. Video : Jeedecode

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