Complexity
The Princeton Institute for Advanced Study was as calm as ever.
Shen Qi visited his physics mentor, Witten, and shared his thoughts. "Navier and Stokes actually used calculus to explain Fluid Motion. Although the Navier-Stokes equations have not yet found a mathematical solution, Navier's name is engraved on the Eiffel Tower, and Stokes is Professor Lucas of Cambridge University."
Witten said, "In comparison, Stokes was a stronger mathematical strategist. Most of the mathematical processing in the Navier-Stokes equations came from Stokes, and he has received a greater commemoration, with craters named after him on both the Moon and Mars."
"I'm good at differentials, integrals, and partial differentials, but I'm a complete novice when it comes to Fluid Motion. I've been studying Fluid Motion recently," Shen Qi reported on his recent progress in physics.
Witten asked, "So you want to find the mathematical solution to the Navier-Stokes equations?"
"It's difficult, but I'm trying my best. It takes a lot of time and a little bit of just the right inspiration. So, Edward, let's continue talking about the Yang- Mills equations. Compared to the Navier-Stokes equations, the Yang- Mills equations involve a much more complex field."
"Classical mechanics, Quantum Mechanics, Quantum Field Theory, energy chain waves and cosmology, and your String Theory —QFT alone is giving me a headache, and GUT seems like a legend." Shen Qi's distress. QTF is an abbreviation for Quantum Field Theory, while GUT is the Grand Unification theory that all physicists dream of.
"So you want to solve both the Navier-Stokes equations and the Yang- Mills equations at the same time?" Edward Witten asked in surprise.
Shen Qi shook his head and said, "This is too difficult. I'd like to hear your opinion."
Witten: "I can't offer any opinions right now. Let's talk about it after you get your PhD in physics."
"Oh, right, Edward, I came with a question. We all know that Condensed Matter Physics is more like a bridge, connecting Quantum Mechanics and classical physics. We can find research reports related to the Yang- Mills equation in Condensed Matter Physics, and we can also find marginal intersections of the Navier-Stokes equation. So what are your thoughts on the complexity mentioned in Condensed Matter Physics?" Shen Qi asked.
The concept of complexity was first proposed by PW Anderson, and Witten explained it in his own words, "In the Physics Community, complexity is still an unresolved issue. From the image of spin lattices to the well-known Turing machine, from the Second Law of Thermodynamics to the Devil's Staircase, from the Miedui experiment to Chaos Theory, from turbulence to earthquakes, seemingly unrelated things are connected and follow certain laws, and that is complexity."
"However, most of the laws of complexity currently rely mainly on the intuition of physicists, such as PW Anderson, and we lack the necessary quantitative explanations. Hey, kid, let me ask you, what can you associate with the Graphitic Carbon Nitride that you study every day, based on the principles of complexity?" Witten took the opportunity to elaborate and chatted happily.
Shen Qi immediately replied, "Friction."
Witten was intrigued. "Friction? That's a very novel answer. Could you elaborate on your concept of friction?"
Shen Qi rubbed his hands together. "Matter has defects. I define material defects using topological methods based on Homotopy Group. Professor Hackman, based on my paper published in PRL, used physicochemical methods such as doping and trimming to improve and reorganize the molecular structure of Graphitic Carbon Nitride. This is friction at the microscopic level."
Shen Qi kept rubbing his hands together, faster and faster, until his palms were burning hot. "Friction, friction, the devil's pace. If you expand the carbon atoms in Graphitic Carbon Nitride by about 10 to the power of 17 times, it becomes the size of the Earth. Friction, friction, can the Earth's flaws be optimized through friction? This is the complexity I can think of. Unfortunately, I can't provide quantitative support; it's just an intuition without any evidence."
"Then let's find the evidence." Witten concluded the conversation for the day.
Shen Qi rubbed his hands together all the way back to his office, unable to stop.
Complexity theory in Condensed Matter Physics is a very mysterious thing. Once you start studying it, Shen Qi becomes obsessed.
"Friction, friction."
Shen Qi rubbed the words for the entire afternoon, repeatedly chanting them.
Seemingly without any particular purpose, Shen Qi casually opened the ADF software on his computer and found the simulated molecular structure diagram of GaN crystal. It looked like a wallpaper, strictly following the rules of symmetry, with white spheres representing gallium atoms and black spheres representing nitrogen atoms, crisscrossing and extending infinitely.
Shen Qi manipulated the mouse, rubbing it against the mouse pad.
By performing TDDFT calculations on GaN Quantum Dots, the ultraviolet-visible Absorption Spectrum of GaN Quantum Dots can be obtained.
The spectral curves are very clear. GaN has five absorption peaks with different absorption intensities between 300-1500 nm. The strongest absorption peak, centered at 401 nm, mainly comes from the electronic transition from HOMO-11 to LUMO (95%).
"Friction, friction, expansion and collapse."
A schematic diagram of the planar molecular structure of GaN crystal was cut back by Shen Qi. The simulation experimental calculation results verified the mixed dislocations in GaN.
Dislocation slips along the plane formed by the dislocation line and the Burgers vector, while dislocation climb is achieved by the expansion or contraction of the edge half-atom plane of the dislocation.
Following this line of thought, based on the dynamics of the two-dimensional lattice OFC model in the case of 0≤α≤1/4, Shen Qi instinctively made a mathematical calculation.
The calculations are based on the steady Navier-Stokes equations with Dirichlet boundary conditions.
Currently, we can only assume that the Navier-Stokes equations hold true. For over a century, scientists have assumed that the Navier-Stokes equations hold true, resulting in a wealth of theoretical research findings that have been applied in practice.
If the Navier-Stokes equations did not hold, how could airplanes fly?
The Navier-Stokes equations are constantly being performed in nature, but people cannot find their mathematical solutions.
Shen Qi's thoughts kept jumping from Gallium Nitride, the core material of blue light-emitting diodes in semiconductor lighting, to the Navier-Stokes equation.
Two seemingly unrelated entities, interwoven with the principles of complexity in Condensed Matter Physics, are driven by friction, friction.
Because they are completely unrelated, the computational workload is extremely large.
Shen Qi calculated until late at night, forgetting to eat and forgetting to go home.
Ou Ye, who was alone in her room, received a phone call, "What time are you coming home?"
"Oh, time to go home, it's 12 o'clock." Shen Qi carried a pile of draft papers back to the faculty dormitory.
"Are you very busy?" Ou Ye heated up the chicken soup she had prepared at 7 o'clock and placed it on the dining table.
"I'll have some soup first. You help me calculate a set of data. You don't need to know the background. Just solve this equation using the low-order finite element method, and that's it." Shen Qi assigned Ou Ye a calculation task. He then drank the chicken soup, which tasted good, but there was too much salt in it.
"This is... Dirichlet boundary conditions, Navier-Stokes equations?" When Ou Ye was an undergraduate, she studied Mathematical Physics Methods and got 99 points. The reason she was denied the remaining point was because the female teacher of Mathematical Physics Methods said that Ou Ye was too beautiful.
"Have you forgotten all the Mathematical Physics Methods?" Shen Qi asked with a smile.
"Underestimating me?" Ou Ye rolled her eyes at Shen Qi, her Calculation Ji heart burning brightly in her youth.
In the time it takes to eat a bowl of chicken soup, without knowing the ins and outs, Ou Ye performed the most efficient calculation using purely mathematical methods.
"Wow, Ye, you've got amazing skills! You can even cut down NS with hot spring water!" Shen Qi checked the calculation results again.
This calculation result left both Shen Qi and Ou Ye somewhat puzzled.
υ1(x1, x2)=(x1δ(i+j)+x2γ(ij-1))(x1δ(i+j+1)^2+x2γ(i+j-1)^2)...
υ2(x1, x2)=(x1δ(i+k)+x2γ(i-k+1))(x1δ(i+k-1)^2+x2γ(i-k+1)^2)...
"I don't know what these calculation results represent or what their specific meaning is. From the perspective of mathematical calculation principles alone, they are correct results." Shen Qi could only explain it this way. He shook his head. "I went to Witten this morning and got some inspiration. Then I racked my brains all day, thinking about crystal defects, dynamics, Navier-Stokes equations, my mind was jumping around like crazy, I was so excited, and that's how I came up with this meaningless mathematical result."
"Not every mathematical calculation result has practical significance," Ou Ye said.
"Hey, you're getting more and more enlightened, Little Yezi." Shen Qi turned on the TV, intending to watch some TV before going to bed.
CNN is broadcasting a news report, "A 6.5-magnitude earthquake struck the Pacific Ocean southwest of San Diego yesterday, triggering a tsunami that hit San Diego, but fortunately caused no casualties."
Shen Qi jumped up with a start, rubbing, rubbing!
Earthquakes are caused by friction within the Earth.
Television footage showed that the epicenter was located on the seabed of the Pacific Ocean, at coordinates of 32 degrees 30 minutes north latitude and 117 degrees 19 minutes west longitude.