bobopopo

Futures

[Commentary Donald J. Trump Truth Social Posts On X @TrumpTruthOnX]

Iran’s Military is a complete and total mess. Much of it, like their Navy and Air Force, doesn’t even exist anymore - They have been completely defeated. Iran is all talk and no action. The Bully of the Middle East is DEAD!!! They’ve taken too long to negotiate a deal that would have been great for them, now they will have to pay the price!!! President DONALD J. TRUMP

( TruthSocial: Jun 10 2026, 7:03 AM ET )​​​‍​​‌‍​​‌‍​​​​​​​‌‍​​​​​​​​‌‍​​​‌‍​​​​​​‌‍​​​​​‌‍​​​​​​​​‌‍​​​​​​​‌‍​​​‌‍​​​‌‍​​​​​​​​​​‌‍​​​‌‍​​​​​​‌‍​​​​​​​​‌‍​​​​​‌‍​​​​​​​​​​‌‍

[bobopopo @bobopopo9]

雨是对风的描述w1

[Mathelirium @mathelirium]

Statistical Mechanics Lecture 2.

Lecture 1 made the first big move in Statistical Mechanics. Instead of tracking one exact microstate (q,p), we moved to a Phase-Space Density

ρ(q,p,t)

and the key result was that this density is transported by the microscopic dynamics through Liouville’s Equation:

∂ρ/∂t + Σᵢ q̇ᵢ ∂ρ/∂qᵢ + Σᵢ ṗᵢ ∂ρ/∂pᵢ = 0

Lecture 1 taught us two things. First, the exact microstate is too complicated to follow directly. Second, uncertainty itself moves through Phase Space in a precise mathematical way.

However, do we really need the full density ρ(q,p,t), over all coordinates and all momenta?

In principle, yes, because it is the complete statistical description of the system. But in many-particle systems, we usually do not know that full density explicitly, and even if we did, working with it directly would be hopeless.

The same issue shows up all over modern research in High-Dimensional Probability, Machine Learning, Diffusion Models, Bayesian Inference, and related areas. So in practice, we use approximations, symmetry, model assumptions, and Numerical Methods to get the smaller statistical objects we actually care about.

This is often called the idea of a Reduced Description.

Suppose the full microscopic state is

(q,p) = (q₁, …, qₙ, p₁, …, pₙ)

with full density

ρ(q₁, …, qₙ, p₁, …, pₙ, t)

If we only care about one particle, we integrate out the variables we are not keeping. That gives the One-Particle Reduced Density

f₁(q₁,p₁,t) = ∫ ρ(q₁, …, qₙ, p₁, …, pₙ, t) dq₂ … dqₙ dp₂ … dpₙ

If we want two particles, we keep two and integrate out the rest

f₂(q₁,p₁,q₂,p₂,t) = ∫ ρ(q₁, …, qₙ, p₁, …, pₙ, t) dq₃ … dqₙ dp₃ … dpₙ

So, the rule is simple. A Reduced Density is obtained by integrating the full density over the variables you do not want to keep.

Why is this useful?

Because most physical questions do not ask for everything. They ask for Particle Density, Velocity Distribution, Energy Distribution, Pressure, Temperature, or the behavior of one small part of the system.

This is the second important simplification of Statistical Mechanics.

First, we stopped tracking one exact Phase-Space point and moved to a density. Now, we stop insisting on the full density and move to Reduced Densities.

In the animation, that is exactly the logic you are seeing. I used a coupled two-degree-of-freedom Hamiltonian

H(q₁,q₂,p₁,p₂) = (p₁² + p₂²)/2 + U(q₁) + (w₂²/2)q₂² + εq₁q₂

with

U(q₁) = a q₁⁴/4 − b q₁²/2

and showed the full density as a moving cloud in the variables (q₁,p₁,q₂), while the hidden variable p₂ is encoded by color.

Beneath that cloud sits a dynamic reduced plane, and that plane cycles through all six pairwise two-variable reduced densities that can be formed from

ρ(q₁,q₂,p₁,p₂,t).

These are

f_q₁q₂(q₁,q₂,t)

f_q₁p₁(q₁,p₁,t)

f_q₁p₂(q₁,p₂,t)

f_q₂p₁(q₂,p₁,t)

f_q₂p₂(q₂,p₂,t)

f_p₁p₂(p₁,p₂,t)

As the plane moves from one pair of variables to another, its shape changes because we are changing which microscopic coordinates are being kept and which are being integrated out. The thin luminous filaments from the cloud down to the plane make that reduction feel like a live projection rather and not a disconnected second plot.

In conclusion, the real point of Lecture 2 is that we are not abandoning the microscopic world. We are compressing it into forms we can actually use.

#StatisticalMechanics #PhaseSpace #ReducedDistribution #ProbabilityDensity #HamiltonianMechanics #MathematicalPhysics #Mathematics #Physics

[bobopopo @bobopopo9]

taco，taco，taco again

[Shanaka Anslem Perera ⚡ @shanaka86]

BREAKING: Donald Trump just said the Strait of Hormuz will be “jointly controlled, and maybe controlled by me and the Ayatollah.”

The same man who 36 hours ago threatened to “obliterate” Iranian power plants now proposes co-managing the world’s most important oil chokepoint with the leader of the country he is bombing. In the same breath, Trump said: “It is not the Supreme Leader; nobody has heard from him, and we don’t know if he is living.”

He is proposing to jointly control the strait with a man he simultaneously says may be dead.

While Trump was saying this, the IDF confirmed it had just completed a “wide-scale wave of strikes” on Tehran using over 100 munitions. Targets included a Quds Force command post, an IRGC aerial defence headquarters, an IRGC Ground Forces headquarters in “a large military compound in the heart of Tehran,” a Quds Force intelligence headquarters, a Ministry of Defence naval cruise missile manufacturing site, and multiple ballistic missile, warhead, and electronics research facilities. The IDF reports it has destroyed approximately 330 of Iran’s estimated 470 ballistic missile launchers. Iranian fire has collapsed from 90 missiles per day on Day 1 to approximately 10. The IAF is now hunting the remaining 140.

While Trump was saying this, Iran’s Foreign Ministry spokesperson stated for the third time today that there have been “no direct or indirect talks with Washington” and that Iran’s position on the Strait of Hormuz “has not changed in any way.”

While Trump was saying this, Iran delivered the sentence that should be read by every head of state on Earth: “You struck our hospitals, we did not do the same. You struck our emergency centres, we did not do the same. You struck our schools, we did not do the same. But if you strike electricity, we will strike electricity.”

That is not a threat. That is a ledger. Iran is publicly recording what it has NOT done to establish the legal and moral basis for what it WILL do. The ledger says: we absorbed hospitals. We absorbed schools. We absorbed emergency centres. Electricity is the line. Cross it and the regional grid becomes a battlefield. Every desalination plant. Every hospital ventilator. Every water treatment system in the Gulf that runs on power.

Trump then claimed Iran agreed to hand over “nuclear dust.” He said “we’ve agreed to that.” Iran has agreed to nothing publicly. Trump is negotiating in public with a counterparty that says it is not at the table, proposing to share a strait with a leader whose existence he questions, claiming a nuclear deal that the other side has not acknowledged, while his air force drops 100 munitions on the capital of the country he says he is dealing with.

Rosatom is evacuating Bushehr nuclear plant to a skeleton crew. The 5-day clock is ticking. The 140 remaining launchers are being hunted. The strait is closed. And the man who controls the world’s largest military just proposed co-managing the world’s most important waterway with a man who may not be alive to sign the agreement.

The molecules do not negotiate with ghosts. The molecules wait.

Full analysis: https://t.co/iFmUcarGdV

[にこそく @nicosokufx]

欧州天然ガス「一時35%急騰」🚨

イランの攻撃で
カタール 世界最大 LNGプラントに被害

・複合施設に「甚大な被害」
・欧州やアジアでガス価格の高止まりが長期化する恐れ

[J_Wise_geology @J_Wise_geology]

Gold price- introducing the "Dead Elephant Bounce"

[Narendra Modi @narendramodi]

Tomorrow, 3rd January, is a very special day for those passionate about history, culture and the ideals of Bhagwan Buddha.

At 11 AM, the Grand International Exposition of Sacred Piprahwa Relics related to Bhagwan Buddha, ‘The Light & the Lotus: Relics of the Awakened One’, will be inaugurated at Rai Pithora Cultural Complex in Delhi.

This Exposition brings together:

The Piprahwa relics repatriated after more than a century.

Authentic relics and archaeological materials from Piprahwa that are preserved in the collections of the National Museum, New Delhi and the Indian Museum, Kolkata.

[Vieira de Leiria. 🇵🇹 @Francis00191138]

Conhecer os DIAS DE PRECEITO é parte de viver a fé com responsabilidade e amor.

[Massimo @Rainmaker1973]

The structure of various chemical compounds under the microscope.

[🔬 helnium]

[bobopopo @bobopopo9]

@cissan_9984 肥尾

[bobopopo @bobopopo9]

@tj_research 守株待兔

[bobopopo @bobopopo9]

@tj_research 逻辑有些简单了

[bobopopo @bobopopo9]

@iamericlintw 花莲吴三桂

[*Walter Bloomberg @DeItaone]

U.S. 10-YEAR TREASURY YIELDS FALL AROUND 10 BPS TO 3.904%

[bobopopo @bobopopo9]

cool

[The Kobeissi Letter @KobeissiLetter]

BREAKING: President Trump says he is not making a deal with China unless it solves the trade deficit.

Trump was just asked about the stock market decline and responded, "some times you have to take medicine."

Trump continues to double down on his trade war.

[Nossa Senhora do Rosário @NSenhoraRosario]

São José cuida de nossa família como cuidaste da Sagrada Família!
Protege a nossa família dos perigos como protegeste tua esposa, Maria. 🙏🙌❤️🕊️

[bobopopo @bobopopo9]

@Prince_Fynnd Avoid eating seafood and irritating foods. Drink as much water as possible to speed up metabolism and get more rest.

[bobopopo @bobopopo9]

@Prince_Fynnd This is urticaria. I have had it twice since I was a child. It may be caused by allergies or an overactive immune system. There is no specific medicine. Avoid scratching or taking a hot bath. Even if it itches, you must be patient.

[bobopopo @bobopopo9]

Should be prepared for the stock to fall, he can't win

[Elon Musk @elonmusk]

Every independent voter I know is voting for @realDonaldTrump

[Prof Simon @skwirrellmate]

link

https://t.co/L88IUMVlWx