### Mathematics problems: Navier-Stokes equations

posted by @ulaulaman about #MillenniumProblems #NavierStokes #MukhtarbayOtelbayev #physics #mathematics
In fluid mechanics, the Navier-Stokes equations, developed by Claude-Louis Navier and George Gabriel Stokes, describe the motion of a fluid into the space. Given its velocity $\vec{v}$, the pressure $p$, and the kinematic viscosity $\nu$, in presence of an external force $\vec{f}$, the particles' motion in the fluid could be described by the following vector equation: $\frac{\partial \vec{v}}{\partial t} + ( \vec{v} \cdot \vec \nabla ) \vec{v} = -\vec \nabla p + \nu \Delta \vec{v} +\vec{f}(\vec{x},t)$ The trouble is that, to obtain solutions of this equation, we must introduce approximations that simplify the search of them: for example, a major difficulty is to determine the solutions in the presence of some turbulence. To this problem, that it has a physical nature, we must add another mathematical question: the difficulty in proving, given the initial conditions, the existence of smooth solutions for the equations. Given these difficulties, the Clay Mathematics Institute included it in the list of the seven Millennium Problems:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

### This is water (and snowflakes)

When water freezes, the molecules take a definite position in their relations one to another. Snow flakes are made up of molecules in similar manner, always hexagonal, but like most creation of nature, there are seldom two alike.

### The first atomic clock

The first atomic frequency standard, based on the ammonia molecule (1949).
Inventor Harold Lyons is on the right; Edward Condon, at the age the director of NBS, is on the left.
The story of the atomic clock is really interesting, because starts from a pure research and arrives to an incredible application. First of all we must start from Isidor Isaac Rabi, who started the studies about the atomic transitions, and we must arrive to Harold Lyons, who applied the devices developed during 1930s-1940s by Rabi's team(2), who awarded the Nobel Prize for these studies in 1944(1), in order to construct an atomic clock.
In particular the key paper is published in 1938, A new method of measuring nuclear magnetic moment(3)
It is the purpose of this note to describe an experiment in which nuclear magnetic moment is measured very directly. The method is capable of very high precision and extension to a large number and variety of nuclei.
A beam of particles, in the case of the first experiment they used molecules of LiCl, passed through a group of magnets, so that the nuclear spins is decoupled from each other and from the molecular rotation. At this point an additional magnetic field, this time slightly oscillating, is applied such that the spin and the nuclear magnetic moment are redirected, obtaining at the end a sort of frequency's precession(3).
At the end, Rabi and his colleagues were able to observe perfectly the separated resonance peaks of the two nuclei of lithium and chlorine and just a year later, as also promised in the conclusions of the first article, they were able to update the method using some new atoms, describing with more details the experimental apparatus used by the team:(4):

### Leonard Troland and the story of the photon's name

posted by @ulaulaman via @peppeliberti #photon #physics #LeonardTroland
Gilbert Newton Lewis was a physical chemist who used the name "photon" in order to describe the light(1, 2).
Probably unknown to Lewis and almost all contemporary physicists, the word "photon" can be found in the scientific literature as early as 1916. It was coined by the American physicist and psychologist Leonard Thompson Troland, who used it as a unit for the illumination of the retina. Although little known today, and if known at all then for his work in experimental psychology, at the time he was considered one of America's most promising scientists. When he died tragically and prematurely in 1932 by a fall from the summit of Mount Wilson in California, his death was mourned in obituaries in Science (vol. 76, pp. 26-27) and American Journal of Psychology (vol. 44, pp. 817-820).
Troland introduced the "photon" in 1916 in the article On the measurement of visual stimulation intensities
(...) as a unit for physiological stimulus intensity, defining it as follows [Troland 1917, p. 32]:
A photon is that intensity of illumination upon the retina of the eye which accompanies the direct fixation, with adequate accommodation, of a stimulus of small area, the photometric brightness of which ... is one candle per square meter, when the area of the externally effective pupil ... is one square millimeter. The physiological intensity of a visual stimulus is its intensity expressed in photons. The photon is a unit of illumination, and hence has an absolute value in meter-candles. The numerical value of the photon, in meter candles, ... will obviously be subject to some variation from individual to individual.
Troland first suggested the photon in a presentation given to the tenth annual meeting of the Illuminating Engineering Society in Philadelphia 18-20 September 1916. "I have," he said, "found it very convenient to express all intensity measures in terms of a unit retinal illumination which I have called the photon"(3). In the discussion following his talk, he mentioned as an advantage of the new unit that "the photon unit does not require so much mathematics, and I have been interested primarily in helping the psychologists, many of whom are studying vision somewhat at random."
It seems that also Joly used, in 1921, the name "photon" before Lewis, but the story of Mr. Troland it seems really interesting, and you can read his whole story on Photon: New light on an old name by Helge Kragh.
(1) Lewis G.N. (1926). The Nature of Light., Proceedings of the National Academy of Sciences of the United States of America, 12 (1) 22-29. PMID:
(2) Lewis G.N. (1926). The Conservation of Photons, Nature, 118 (2981) 874-875. DOI:
(3) Troland, L. T. (1916). Apparent brightness; its conditions and properties. Transactions of the Illuminating Engineering Society (archive.org), 11, 947-975