The new proof broadly consists of three steps: derive the macroscopic theory from the mesoscopic one; derive the mesoscopic theory from the microscopic one; and then stitch them together in a single derivation of the macroscopic laws all the way from the microscopic ones.
The first step was previously understood, and even Hilbert himself contributed to it. Deriving the mesoscopic from the microscopic, on the other hand, has been much more mathematically challenging. Remember, the mesoscopic setting is about the collective behavior of vast numbers of particles. So Deng, Hani and Ma looked at what happens to Newton’s equations as the number of individual particles colliding and ricocheting grows to infinity and their size shrinks to zero. They proved that when you stretch Newton’s equations to these extremes, the statistical behavior of the system—or the likely behavior of a “typical” particle in the fluid—converges to the solution of the Boltzmann equation.
The purpose of this work is twofold. First we extend the derivation of Boltzmann’s equation in [26] to the periodic setting Td (d = 2,3). Second, we connect this kinetic limit to the hydrodynamic limit in the abovecited works, to obtain a full derivation of the fluid equations starting from Newton’s laws on the particle system, thereby completing Hilbert’s original program. We summarize the main theorems as follows: • Theorem 1: Derivation of the Boltzmann equation on Td (d = 2,3). Starting from a Newtonian hard-sphere particle system on the torus Td (d = 2,3) formed of N particles of diameter ε undergoing elastic collisions, and in the Boltzmann-Grad limit Nεd−1 = α, we derive the Boltzmann equation (1.1) as the effective equation for the one-particle density function of the particle system. • Theorem 2: Derivation of the incompressible Navier-Stokes-Fourier system from Newton’s laws. Starting from the same Newtonian hard-sphere particle system on the torus Td (d = 2,3) close to global equilibrium, and in an iterated limit where first N → ∞, ε → 0 with α = Nεd−1 f ixed and then α → ∞ separately (there are also other variants, see Theorem 2), we derive the incompressible Navier-Stokes-Fourier system as the effective equation for the macroscopic density and velocity of the particle system. • Theorem 3: Derivation of the compressible Euler equation from Newton’s laws. Starting from the same Newtonian hard-sphere particle system on the torus Td (d = 2,3), and in an iterated limit where first N → ∞, ε → 0 with α = Nεd−1 fixed and then α → ∞ separately (there are also other variants, see Theorem 3), we derive the compressible Euler equation as the effective equation for the macroscopic density, velocity, and temperature of the particle system.
My BIL was stuck in that region for a day before he and his family could return to Yangon. Pretty scary stuff.
Massive steel beams shaking like cooked spaghetti. You're not wrong.Seeing a window turn into a wall of liquid and then solid again is not something I will forget.
A good quake is as perspective-altering as a good trip.
Yikes. That's new for me.Massive steel beams shaking like cooked spaghetti. You're not wrong.
Not a particularly pleasant experience.Yikes. That's new for me.
No she was in Anchorage and Anchorage didn’t get hit by the tsunami.Was she affected by the tsunami?
Of course, when we are looking at which language is closest to Latin, we have to take into consideration what we mean by Latin. For example, there is Classical Latin and Ecclesiastical Latin. Also, Vulgar Latin has to be taken into consideration. Ecclesiastical Latin is based on Late Vulgar Latin with Italian pronunciation. Sardinian would certainly not be closest language to this version of Latin but to an earlier version of Vulgar Latin or Classical Latin. What is interesting is that there are actually many words that exist only in Sardinian and Romanian that descend from Latin, which lead us to interesting conclusions. It could be argued that they were both the most isolated versions of Latin and were in certain ways, more conservative.
Studies of rock and dust from asteroid Bennu delivered to Earth by NASA’s OSIRIS-REx (Origins, Spectral Interpretation, Resource Identification and Security–Regolith Explorer) spacecraft have revealed molecules that, on our planet, are key to life, as well as a history of saltwater that could have served as the “broth” for these compounds to interact and combine.
The findings do not show evidence for life itself, but they do suggest the conditions necessary for the emergence of life were widespread across the early solar system, increasing the odds life could have formed on other planets and moons.
“NASA’s OSIRIS-REx mission already is rewriting the textbook on what we understand about the beginnings of our solar system,” said Nicky Fox, associate administrator, Science Mission Directorate at NASA Headquarters in Washington. “Asteroids provide a time capsule into our home planet’s history, and Bennu’s samples are pivotal in our understanding of what ingredients in our solar system existed before life started on Earth.”
Detailed in the Nature Astronomy paper, among the most compelling detections were amino acids – 14 of the 20 that life on Earth uses to make proteins – and all five nucleobases that life on Earth uses to store and transmit genetic instructions in more complex terrestrial biomolecules, such as DNA and RNA, including how to arrange amino acids into proteins.
Scientists also described exceptionally high abundances of ammonia in the Bennu samples. Ammonia is important to biology because it can react with formaldehyde, which also was detected in the samples, to form complex molecules, such as amino acids – given the right conditions. When amino acids link up into long chains, they make proteins, which go on to power nearly every biological function.
These building blocks for life detected in the Bennu samples have been found before in extraterrestrial rocks. However, identifying them in a pristine sample collected in space supports the idea that objects that formed far from the Sun could have been an important source of the raw precursor ingredients for life throughout the solar system.