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ℏistas
เข้าร่วมเมื่อ 16 ส.ค. 2022
Two baristas serving up special topics in mathematical physics.
Random Matrices in Unexpected Places: Atomic Nuclei, Chaotic Billiards, Riemann Zeta #SoME2
Chapters:
0:00 Intro
2:21 What is RMT
7:12 Ensemble Averaging/Quantities of Interest
13:30 Gaussian Ensemble
18:03 Eigenvalues Repel
28:08 Recap
29:08 Three Surprising Coincidences
32:44 Billiards/Quantum Systems
36:00 Reimann Zeta
~~~~~~~~~~~~~~~~~~~~~~~~~ Errata + Clarifications ~~~~~~~~~~~~~~~~~~~~~~~~
05:01 The covariance matrix should be denoted by C instead of X.
06:07 To be clear: H_ij gives what physicists call the transition amplitude to go from state j to i. To get the actual transition probability, one takes the magnitude squared, P(i to j) = |H_ij|^2. Because H_ij and H_ji are complex conjugates of each other, both entries encode the same information and yield the same probabilities, so the probability is symmetric P(i to j ) = P(j to i).
06:17 When we say the eigenvalues are interpreted as energies, the list on the RHS really should read (E_1, E_2, … ).
06:18 The equation in the bottom right should be H_ij = H_ji*.
07:00 To be clear, we demand the covariance matrix C to be symmetric such that C = C^T and the Hamiltonian matrix H to be Hermitian such that H = H^\dagger (physicists’ notation) or H = H* (mathematicians’ notation).
16:48 In the expression for Z, the whole argument of the exponential should be multiplied by a factor of N, and the off-diagonal entries should have a factor of 2.
17:20 See previous remark about H_ij encoding transition probabilities.
33:05 For the Sinai billiard: To be clear, the potential is infinity outside the walls of the square arena and in the circular barrier in the center, but is zero on the interior of the arena.
~~~~~~~~~~~~~~~~~~~~~~~ References [in construction] ~~~~~~~~~~~~~~~~~~~~~
Analytic continuation in the Riemann zeta function: 1) video by 3b1b 2) notes from Math 259 at Harvard.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Credits ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Animation team: Artin Kim, Peter Gao, Adin Ackerman, Abrun Nereim.
Original music by: Abrun Nereim, Adin Ackerman.
Image for title sequence: Cafe Venetia.
~~~~~~~~~~~~~~~~~~~~~~~~~ Acknowledgements ~~~~~~~~~~~~~~~~~~~~~~~~~
This video wouldn't have been possible if it wasn't for the animation team!
CJ would also like to thank the following:
- D.Z.S. for introducing and enthusiasting me to/about quantum chaos and RMT.
- S.S. for beautiful class on RMT.
-- Livan, Novaes, and Vivo for these beautiful and accessible introductory notes: arxiv.org/pdf/1712.07903.pdf.
0:00 Intro
2:21 What is RMT
7:12 Ensemble Averaging/Quantities of Interest
13:30 Gaussian Ensemble
18:03 Eigenvalues Repel
28:08 Recap
29:08 Three Surprising Coincidences
32:44 Billiards/Quantum Systems
36:00 Reimann Zeta
~~~~~~~~~~~~~~~~~~~~~~~~~ Errata + Clarifications ~~~~~~~~~~~~~~~~~~~~~~~~
05:01 The covariance matrix should be denoted by C instead of X.
06:07 To be clear: H_ij gives what physicists call the transition amplitude to go from state j to i. To get the actual transition probability, one takes the magnitude squared, P(i to j) = |H_ij|^2. Because H_ij and H_ji are complex conjugates of each other, both entries encode the same information and yield the same probabilities, so the probability is symmetric P(i to j ) = P(j to i).
06:17 When we say the eigenvalues are interpreted as energies, the list on the RHS really should read (E_1, E_2, … ).
06:18 The equation in the bottom right should be H_ij = H_ji*.
07:00 To be clear, we demand the covariance matrix C to be symmetric such that C = C^T and the Hamiltonian matrix H to be Hermitian such that H = H^\dagger (physicists’ notation) or H = H* (mathematicians’ notation).
16:48 In the expression for Z, the whole argument of the exponential should be multiplied by a factor of N, and the off-diagonal entries should have a factor of 2.
17:20 See previous remark about H_ij encoding transition probabilities.
33:05 For the Sinai billiard: To be clear, the potential is infinity outside the walls of the square arena and in the circular barrier in the center, but is zero on the interior of the arena.
~~~~~~~~~~~~~~~~~~~~~~~ References [in construction] ~~~~~~~~~~~~~~~~~~~~~
Analytic continuation in the Riemann zeta function: 1) video by 3b1b 2) notes from Math 259 at Harvard.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Credits ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Animation team: Artin Kim, Peter Gao, Adin Ackerman, Abrun Nereim.
Original music by: Abrun Nereim, Adin Ackerman.
Image for title sequence: Cafe Venetia.
~~~~~~~~~~~~~~~~~~~~~~~~~ Acknowledgements ~~~~~~~~~~~~~~~~~~~~~~~~~
This video wouldn't have been possible if it wasn't for the animation team!
CJ would also like to thank the following:
- D.Z.S. for introducing and enthusiasting me to/about quantum chaos and RMT.
- S.S. for beautiful class on RMT.
-- Livan, Novaes, and Vivo for these beautiful and accessible introductory notes: arxiv.org/pdf/1712.07903.pdf.
มุมมอง: 104 482
My pleasure of stumbling upon this video can be only matched by the sorrow of realizing it's the only one in the channel😢
I can imagine how much time you spent to do this video. But don't give up! Visualization of advanced math like this is very rare and important for me.😊
PLS MAKE MORE VIDEOS
Great video!
This was great! Love that it’s actually written in the language of theoretical physics instead of the language of indecipherable mathematics notation
...have they connected this to moonshine theory yet?
You are a legend
Great video! However, the audio volume is quite low.
6:40 ji
Ok i think i Skipped the definition of Eigenvalues Spacings 18:51
Why you stopped after just one video 🤨🤨🤨 You really great, plz continue we don't have a lot of content related to random matrix or random NLA
Consider changing the channel name a bit. It's pretty hard to find your channel from TH-cam search even if you remember the name because of the hbar
Thank you for this --- what is the source of the background music?
Recommending playback speed 0.75
all hail the youtube algorithm I just got done coding a matrix multipler and a determinant calculator in python, which also necessitated coding a random matrix generator to generate some samples matrices to feed into the calculators. I even used a -10 - 10 range, but with a small bias towards 0s (test the efficiency of the determinant calculator, it should be able to start from the row or column with the most 0s to avoid extra work.) and a switch to allow complex numbers to occasionally pop in. My obsession with coding matrices has been vindicated. I need to tell my friends.... oh wait they've gone.
OH, it's pronounced "h-baristas." very clever lmao
Amazing
Why do you call it the wigner-dyson distribution? I've always seen this denoted as the Rayleigh distribution
amazing video, needs more love
This is so cool! We only covered random vectors in probability but our professor mentioned the concept of random matrices and more abstract random variables. Really cool to finally see them in action ❤
❤❤
13:57 We don't care about rotations? They are not real symmetric, sort of the exact opposite. We don't care about an object that has been randomly rotated?
4:00
What a great video. Sadly this is the only one. Can I ask what happened that you stopped making this amazing content?
Nice use of manim
Thankyou for helping me write my Random Matrix Theory section for my Masters thesis on Quantum Chaos!
Truly enjoyable, hope to see more on this! Maybe demonstration of applicability of RMT to more problems, such as finance?
I’am waiting for a follow up video
Will you continue this series?
Awesome video. Top quaility content. Started self studying on random matrix theory.
You know it's serious when it switches to a second narrator! And I'm gonna be watching for that distribution; it immediately reminded me of potential energy in chemical bonds! 🤯
This is a huge amount of information thrown at me very very fast. It felt like you guys were trying to speak as fast as possible and the animations kept changing before i could understand what they were trying to convey. It's an extremely accurate and complete (i assume) introduction to random matrix theory but most of the people i see in the comments who get it are people who already work with rmt. For someone who has all the linear algebra background but has never seen this subject before i'd need to watch this at half speed at most and rewatch the animations to try to suss out what was being communicated. I think you guys did a great job for an early go at this but damnnn. Just go a little slower. You don't need to fit the entire thing into a single video. This video could be twice as long and still difficult. It's really cool stuff, you're doing great
I love your video about Random Matrix Theory specially the spectrum distribution of eigenvalues in correlation matrices. I am a scientist, Electrical Engineer and it is great to see how physicists apply RMT to handle for example the complexity of the Hamiltonian of the Uranium nucleus. I will be alert to your next great video. Thank You
<3
There is a z missing when you write the euler product for the zeta
Gaussian and exponential distributions are fundamentally different in their tail behaviour. I don’t believe the transition from correlated to independent can lead to such drastic change in the distribution of spacing. Can you provide a link to a scientific paper that highlights this result? Or is it something that you proved yourself?
Great content - I’d definitely recommend going slower, breaking things up into separate videos and emphasizing/repeating key points. You have such a wealth of useful information- give it to the people in digestible chunks. But clearly very thoughtful & informed presentation.
Awesome Video!
🎯 Key Takeaways for quick navigation: 00:00 📚 Atomic nuclei, chaotic billiards, and the Riemann Hypothesis are seemingly unrelated, but they're all connected through random matrix theory (RMT). 03:08 🧮 Random Matrix Theory (RMT) is a model for matrices with random entries, used to describe various systems such as atomic nuclei and chaotic billiards. 08:22 🔍 RMT focuses on computing averages of unitarily invariant quantities, often related to the eigenvalues of matrices. 14:03 📜 The Gaussian Ensemble is a fundamental random matrix model, providing simplicity for calculations and capturing physical behavior. 17:48 🧪 Studying eigenvalue spacings reveals that correlated eigenvalues differ from uncorrelated ones, shedding light on randomness and structure in matrices. 21:58 📊 The probability distribution of eigenvalue spacings in correlated events or eigenvalues follows a Gaussian distribution multiplied by a linear term in the spacing. 24:17 📐 The distribution of spacings suggests that for small values of the spacing, it's proportional to the spacing itself, resembling a distance from the origin in polar coordinates. 26:23 🧮 The derived distribution for correlated eigenvalues matches the Wigner surmise and is proportional to `s * e^(-s^2)`, showing that eigenvalues repel each other as they get closer. 27:29 💥 The key difference between Poissonian and Wigner-Dyson distributions lies in the limit as the spacing goes to zero; the latter's limit is zero, representing repulsion, while the former's limit is one. 35:46 🎭 Complex functions like the Riemann Zeta function's zeros display the same spacing distribution as eigenvalues of random matrices, leading to insights about prime number distribution and quantum chaos
very good work, said very well. I would suggest that with NN's one might be able to determine a halucination from a real thought or to determin if NNs are human aligned in the NN by looking at the distributution and see how closely it matches the original data. it wont be exact but it would be countable , I have done something similar with my bag of marbles sort , on sets that are in the trillions you can select 10 percent of the binary values randomly and get back the original, with a quadrillion even better ratios. you need a perfect random generator though at some point to insure that random selection is truely made. I do not know if cryptographic random is enough though. Any thoughts?
This is an amazing video. This is one of the only videos I have seen that explains the topic without requiring one to watch several hours of a lecture series to even get a basic understanding. Thank you. Additionally, your channel description is intriguing. Two baristas who make mathematical physics videos .... there is an interesting story for sure.
Oh god I'm excited to watch this. My bachelor's scientific initiation (and also the course finish work) was about Random Matrices. Absolutely no one in my university (UNICAMP) was working with them, including my mentors, so I had to study them on my own, which was hard but exciting. I miss research.
Can't wait for more!
i need to brush up on a lot of math before i rewatch this
Unfortunately, video format didn't help this lecture at all. It would be easier just to read a script
This is really well explained and interesting.
I am so glad to have stumbled to this video. Amazing content expressed in a beginner friendly manner. Thanks a lot.
mathematics is my God and my staff I pray like this and break minds in half
Can someone explain what is s, P(s) and q(s) at 19:12? q(s) was defined via P(s), but P(s) and s are not clear to me. Thanks in advance
I have developed a new theory, I have called Partitions Trigonometric and I have discovered something amazing. I can do X Rays with these equations applied to Z Riemann Equation.
th-cam.com/video/BBzZt-NWdqA/w-d-xo.html