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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.

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Matheorems

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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.
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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.

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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.
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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.

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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

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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).
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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
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One of the simplest algorithms for sampling from a probability distribution is Random Walk Metropolis-Hastings. It proposes new samples by taking Gaussian-distributed steps, accepting or rejecting them to maintain the target distribution. I call this pdf the "fidget spinner".

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