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HYPNO FIXATIONS particle pack for VRC is OUT NOW 🌀❤️ Quest optimized! - 14 effects on ONE particle system/material - Customize color, speed, size - Seamless gesture controls Great stim toy and hypnosis focus :)

15,557 次观看 • 15 天前 •via X (Twitter)

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Final Lecture of our Statistical Mechanics Series. Lecture 2 showed how we move from the Full Phase-Space Density ρ(q₁, …, qₙ, p₁, …, pₙ, t) to smaller statistical objects by integrating out variables we do not want to keep. That gives reduced descriptions like the One-Particle Density f₁(q₁,p₁,t) and the Two-Particle Density f₂(q₁,p₁,q₂,p₂,t) This was the simplification. Now comes the catch. If the Full Density obeys Liouville’s Equation, the reduced densities do not evolve independently. The equation for one level depends on the next one and this is referred to as the BBGKY hierarchy. The One-Particle Density depends on the Two-Particle Density. The Two-Particle Density depends on the Three-Particle Density. And the chain keeps going. That happens because particles interact. Once one particle feels the rest, one-particle information is no longer enough. Correlations enter, and the lower level is fed from above. If the full Hamiltonian is H = Σᵢ pᵢ²/(2m) + Σᵢ U(qᵢ) + (1/2) Σᵢ Σⱼ≠ᵢ Φ(qᵢ − qⱼ) then reducing the full density does not make the interaction terms disappear. It leaves behind coupling to higher-order reduced densities. So, schematically, ∂f₁/∂t + transport of one particle = interaction term involving f₂ and more generally the heirarchy is such that ∂fₛ/∂t + s-particle transport = interaction term involving fₛ₊₁ Therefore, Lecture 3 is really about the price of reduction. We simplify the description, but the information we remove comes back as coupling to higher-order correlations. So, how do you actually compute anything if every level depends on the next one? This is the so-called Closure Problem. To make the hierarchy usable, you need an extra assumption that cuts the chain. You replace the exact higher-order object by an approximation in terms of lower-order ones. The most basic example is a factorized closure at the pair level, where the exact correlated Two-Particle Density is replaced schematically by a product of One-Particle Densities: f₂(q₁,p₁,q₂,p₂,t) ≈ f₁(q₁,p₁,t) f₁(q₂,p₂,t) That approximation is not exact. It throws away part of the correlation structure. But it gives you something the raw hierarchy does not... a closed equation for the lower-level description. That is why closure matters so much. Without it, the hierarchy is exact but open. With it, the theory becomes approximate but usable. Thus, the combined point of this final Statistical Mechanics post is simple. First, reduced descriptions are not closed because interactions generate correlations across levels. Second, if you want a workable Kinetic Theory, you must close the hierarchy by approximating those higher-order correlations. It is the bridge from formal many-body mechanics to equations people can actually solve. In the render, that is exactly the story you are seeing. The first part shows the hierarchy itself: one reduced level feeding the next, with lower descriptions inheriting structure from higher ones. The second part shows the closure step where the exact correlated pair level is replaced by a factorized ansatz, and that approximation gives back a closed one-particle description. That is, the animation moves from dependence to approximation, and from approximation to solvability. #StatisticalMechanics #BBGKY #ClosureProblem #KineticTheory #PhaseSpace #ReducedDistribution #HamiltonianMechanics #MathematicalPhysics #Mathematics #Physics

Mathelirium

10,970 次观看 • 3 个月前

Finally new version of text system is released on our website ! I released as update to old version but in reality it's completely new thing in material and lot's of work in verse. I'm still working on documentation or/and videos but here is a list of good stuff it does: - Material got much lighter as slots logic moved to vertex shader. - Characters can overlap and still won't be cut out compared to old version. - Character can be animated and I already implemented some (wave, scale up/down and shake). - Custom face camera logic which works with prop scale so no need to additionally specify with and height. - Rows offset automatically when scale up or down any row. - Text effects now can be applied to any character, so words or single character. - Custom glint which works perfectly with any number of rows as a single line with multiple important controls. - Higher quality outline with two parameters to control inner and outer edge. - Timer logic split into to parts, verse and material. Verse setup initial layout and reserve slots for timer digits and then each second just sends seconds value and GPU makes timer tick freeing verse TPS by huge number. - To apply effects simply need use parsing markup with custom tag where N number between 1-9. So simply can write " Hello World" and word Hello will have rainbow effect applied. Currently implemented 4 effects. - To use icons in slots from texture array is as simple as specify another tag [iconN] where N is number of icon in array. So for example if coin is icon index 0 just need to write [icon1]100K. - Progress bar is supported as well and can be snapped to row without eyeball offset. - Background image supported now too! In the video yellow one is as test and you can use any of your own. Verse side got lots of changes: - Now saves/caches layouts better. - Smart packing of data allowed drastically reduce number of parameters, which should help with network optimization! - Instead having vector parameter per each slot to just send character index and position (32x5=160 if used 5 rows) now it's only 11 vector parameters per each row so 11x5=55 which is huge save and it sends text effects index in it too! - Implemented text update queue for text which doesn't require instant update. It helps to give more breathing for other text props which needs that speed. - Materials code now use interface and wrappers for cleaner work. If you still read it then thank you! :) On our website we now run 25% sale! And if you own old version please DM me here or in Discord and I will give you promo code for 100% discount on our website! Gelos Games Fortnite #uefn #EpicPartner

AsicsoN

12,841 次观看 • 2 个月前

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Miki Djan

28,971 次观看 • 1 个月前