lördag 15 juni 2019

Demystifying the New SI Base Units.

In the previous post we observed that Planck's constant $h$ appears as a conversion factor connecting light of frequency $\nu$ with attributed energy $h\nu$ (in eV or Joule) through the photoelectric effect with the release of an electron from a surface exposed to light (of sufficient high frequency). The inner mechanics of the atoms delivering the electrons upon excitation by exposure to light does not enter into the discussion and so Planck's constant can be given a meaning in macroscopic physics, thus without quantum mechanics, as a trade between light and electron energy and then further to mechanical energy. Its role in quantum mechanics then appears as an after construction.

Let us now turn to Boltzmann's constant $k$ to see its connection to Planck's constant and macroscopic physics. Boltzmann's constant appears in Planck's universal law of blackbody radiation law of the form
  • $E(\nu ,T) = W(a)\, kT\nu^2$,
  • $W(a) = \frac{a}{\exp(a )-1}$ with $a = \frac{h\nu}{kT}$,
where $E(\nu ,T)$ is the (suitably normalised) intensity of radiation of frequency $\nu$ from a blackbody of temperature $T$ and $W(a)$ is a cut-off factor with $W(a)=1 $ for small $a$ and
$W(a)$ small for medium to large $a$, expressing Wien's displacement law stating cut-off of high frequencies. We see that Planck’s constant only appears in the cut-off factor.

Experimental observation of $E(\nu ,T)$ makes it possible to determine $W(a)$ and thus $kT$ in terms of $h\nu$, from which Boltzmann's constant $k$ can be determined with respect to a chosen scale for temperatur $T$, or the other way around as in the new SI units by specifying by definition 
  • $k=1.380650\times 10^{-23}\, J/K$,  
thus setting a new standard for Kelvin $K$ as measure of temperature. The connection between the energy measures $h\nu$ and $kT$ then shows to be

  • $h\nu_{max} \approx 2.8214391\times kT$,
where $\nu = \nu_{max}$ gives maximum of the spectrum $E(\nu ,T)$.

Again, this can be done without having to invoke quantum mechanics in its standard form with $h$ as a "smallest quantum of action" as exposed in detail on Computational Blackbody Radiationwhich presents a derivation of Planck's law using deterministic continuum physics instead of as usual statistics of discrete quanta. In particular, the new derivation captures the universality of blackbody radiation beyond specific inner atomic mechanics.

The universality of Planck's law is expressed by the fact that an ideal blackbody can take the form of a set of oscillators without very specific inner structure. In particular different blackbodies with different inner structure can share the same temperature scale.

To sum up, both Planck's constant and Boltzmann's constant are specified by definition in the new SI units, from which the new units kilogram and Kelvin can be determined by macroscopic experiments without resort to quantum mechanics in its standard form.

Hopefully this helps to demystify both Planck's and Boltzmann's constant, and the new SI units.

onsdag 12 juni 2019

New Perspective on New Unit of Mass in terms of Planck's Constant

In the 2019 redefinition of the SI base units the kilogram as unit of mass is defined in terms of Planck's constant
  • $h$ set by definition to exactly $6.62607015×10^{−34}$ Joule-second ($J\cdot s$), 
  • $Joule = Newton\times m = M\times\frac{m}{s^2}\times m=M\frac{m^2}{s^2}=Mc^2$
with $m$ meter, $s$ second, $M$ mass in kilogram and $c$ the speed of light.

This defines kilogram in terms of Planck's constant $h$, second $s$ and speed of light $c$ with meter $m$ defined in terms of $c$. The relation $E=Mc^2$, viewed as a profound discovery attributed to Einstein's relativity theory, then appears simply as a definition (of mass).

The connection to quantum mechanics comes by attributing a certain energy $h\nu$ to light of frequency $\nu$ through the law of the photoelectric effect
  • $h\nu = eV_0 + \phi = eV_0 + h\nu_0$, 
where $eV_0$ in electronVolts is the energy of a released electron with charge $e$ and $V_0$ a stopping potential in Volt, and $h\nu_0$ is the work to release an electron with $\nu_0$ a threshold frequency. This relation determines $h\approx 4.1357\times 10^{-15}$ in $eV\cdot s$, which fits with the new definition of $h$ in terms of $J\cdot s$ with the conversion $eV= 1.602176634×10^{−19} J$.

The photoelectric effect connects the macroscopic phenomena of light of different frequencies and stopping potential to the microscopic phenomenon of electron charge. In this connection there is nothing that says light of frequency $\nu$ is to be viewed as a stream of discrete photon particles of energy $h\nu$ and that Planck's constant $h$ has the physical meaning of a discrete smallest quantum of action.  Instead Planck's constant has the role of connecting light energy to electron potential energy ultimately to mechanical energy.

For a new continuum physics approach to blackbody radiation and the photoelectric effect with discrete quantum replaced by a threshold condition (as in the photoelectric effect), see Computational BlackBody Radiation.

The new definition of kilogram gives perspective on the very small size of Planck's constant $\sim 6.6\times 10^{-34}\, J\cdot s$ misleading to an idea of an absurdly small Planck length $\sim 1.6\times 10^{-35}\, m$ believed to have a physical meaning, in string theory in particular.  On the other hand, the length scale of atoms (and X-ray light of frequency about $3\times 10^{18}$) is about $10^{-10}\, m$ and that of a proton $10^{-15}\, m$, with the Planck scale 20 orders of magnitude smaller, way beyond any thinkable experimental exploration and thus meaning. Planck time  $\sim 5.3\times 10^{-44}\, s$ is even more absurd. No wonder that modern physics playing with Planck length and time is in a state of deep crisis, with the scientific madness come to full expression in the  Chronology of the Universe starting with the Planck Epoch before $10^{-43}\, s$ after Big Bang.

In Schrödingers equation $h$ multiplies the time derivative of the wave function, which means that
the atomic energy (potential + "kinetic" energy) of an eigenfunction of frequency $\nu$ is equal to $h\nu$, which comes to expression in the photoelectric effect.  Schrödinger's equation is a continuum model without any smallest quantum of action, only discrete eigenvalues representing different energies.

A reformulation of quantum mechanics in the form of a Schrödinger equation as a continuum model in real 3d space plus time without statistics, can be inspected at Real Quantum Mechanics.

To get an idea of the absurdly small Planck length scale $1.6\times 10^{-34}\, m$, one may compare with the estimated size of the observable Universe which is about $10^{27}\, m$ or $10^{33}\, \mu m$ with $\mu m$ mikrometer.

In short, Planck's constant $h$ converts light energy to mechanical energy through electron potential energy, and as such does not ask for a meaning as a "smallest quantum of action" in a mist of "quantisation" into absurdly small "quanta".

The new kilogram standard is specified to high precision using a Kibble balance, where gravitational force is balanced by an electromagnetic force (between two coils), which depends on Planck's constant. Mass is then derived by measuring the local gravitational constant.  

lördag 25 maj 2019

Does Sabine Hossenfelder Exist?

Sabine Hossenfelder on BackReaction claims herself to be a modern physicist expressing truths about the state of modern physics:
  • I do not know what it means for something to be “real” or “true.” You will have to consult a philosopher on that.
  • If you want to claim that the Higgs-boson does not exist, you have to demonstrate that the theory which contains the mathematical structure called “Higgs-boson” does not fit the data. Whether or not Higgs-bosons ever arrive in a detector is totally irrelevant.
  • Here is a homework assignment: Do you think that I exist? And what do you even mean by that?
  • From November on, I will be unemployed, at least that is what it presently looks like: If you don't exist, can you be employed?
If this is the truth, no wonder that modern physics is in a state of crisis.

But Sabine's criticism of modern physics appears well founded and thus admirable. At the price of making employment difficult, which is even more admirable.  But she is not alone saying that modern  physics in a state of stalemate crisis without progress, as evidenced in the new book
The Universe Speaks in Numbers by Graham Farmelo summing up:
  • ...the slow rate of progress of the string framework may presage a more sedate pace in fundamental physics that may persist for centuries to come.
Of course the pessimism of deep crisis can be turned around into its opposite, as expressed by the leading star of modern physics Arkani-Hamed: 
  • There has never been a better, more exciting time to be a theoretical physicist.
Concerning the crisis of modern physics it is commonly accepted that one reason is that the two basic building blocks, relativity theory and quantum mechanics, are contradictory/incompatile . If you dig a little deeper you will find that the underlying reason is that both theories are unphysical as exposed in detail as Many-Minds Relativity and Real Quantum Mechanics and also here.

Two theories which are physical cannot be contradictory, because physics which exists cannot be contradictory. But unphysical theories may well be contradictory, as ghosts can have contradictory qualities.

The Special Theory of Relativity of Einstein is unphysical because the Lorentz transformation is not a transformation between physical coordinates, as strongly underlined by its inventor Lorentz, but misunderstood by the patent clerk Einstein believing that the transformed time is real and thus that time is relative. Quantum Mechanics is unphysical because its interpretation is statistical which makes it non-physical, because physics is not an insurance company. Here Einstein was right understanding that God does not play dice.

Concerning the crisis of modern physics, listen to
It is comforting to see that I am not alone in my criticism of relativity theory and quantum mechanics.

It is impossible to discuss these things with main stream physicists, since their common wisdom is that neither relativity theory nor quantum mechanics can be understood as rational science.

An example of the confusion is the hype about quantum computing with the sound criticism by Dyakonov in The Case Against Quantum Computing of course being dismissed by main stream physicist Lubos.

The confusion is exposed in an exhibition combining arts and science about quantum mechanics at Center for Contemporary Culture in Barcelona commented on at BackReaction. Here is an artistic expression of the quantum leap of an electron which infuriated Schrödinger:

In this context you are invited to a previous post on the true meaning of Planck's constant $h$ shows The text book view is that $h$ is a fundamental quantum of action connecting the energy
$E=h\nu$ to a particle/photon of light of frequency $\nu$ according the Planck-Einstein relation with light viewed as a stream of discrete particles/photons.

But it is not at all necessary to view light this way to understand the true meaning of Planck's constant, which is revealed through the way it is measured, that is through the photoelectric effect which simply connects light frequency to the energy unit of electronVolt.

Einstein gave a heuristic explanation of the photoelectric effect from an idea of light as a stream of particles. By the common Aristotle logical fallacy of confirming the assumption by observing the consequence, this has convinced modern physicists that light indeed consists of a stream of particles, which however is against all scientific rationale and a basic reason for the crisis of modern (particle) physics. Schrödinger understood that there are no particles. See posts on the photoelectric effect showing that it does not require Einstein's particle heuristics to be understood; wave mechanics serves much better!     

torsdag 16 maj 2019

Why Does a Modern Physicist Buy the CO2 Global Warming Hysteria?

Sabine Hossenfelder on BackReaction claims herself to be a physicist, yet it seems as if she without
thinking buys the CO2 Global Warming Hysteria:
  • Climate Change: There are no simple solutions.
  • The Earth is warming. Human carbon-dioxide emissions are one of the major culprits. We have known this for a long time. But in the past two decades, evidence for global warming has become more noticeable on local levels, as with seasonal shifts, extreme weather events, declines in biodiversity and, depending on where you live, droughts. And it will get worse.
She does not acknowledge that there are many knowledgeable physicists who do not buy this story, because it has no support in known laws of physics, including Dyson, Happer, Singer and many more. 

So how can it be that she jumps on the climate alarmist wagon? I have asked her, but she does want not to communicate with me, so I have to guess: Is it because of a monumental confusion concerning quantum mechanics, which makes it impossible for a modern physicist to grasp even basic physics of the climate system of the Earth, and what does that then say about modern physics?

PS To get perspective listen to Hearing of the House Committee on Natural Resources May 22.

tisdag 14 maj 2019

Yes, Quantum Mechanics Is Wrong

Sabine Hossenfelder on BackReaction has become a truth-teller about the state of modern physics, and now she tells the truth about quantum mechanics: It is wrong:
  • Yes, I am sorry. But I have a message for you from the depth of abstract math: We know that quantum mechanics is wrong. 
The same idea has been expressed by almost every notable physicist, including Roger Penrose:
  • Physics Is Wrong, From String Theory to Quantum Mechanics.
This is a funny situation, unprecedented in the history of science, and it has been like that for now
more than 100 years. 

A restart is obviously needed and my contribution is Real Quantum Mechanics. Take a look! And think! See also recent posts.

But Sabine Hossenfelder is a physicist and thus must seek to rescue the hopeless situation in one way or the other:
  • And whatever your misgivings are about quantum mechanics, there is no denying that it is useful.
So quantum mechanics is wrong, but it is nevertheless useful? That does not make sense. A theory which is wrong cannot be truly useful, because from something which is wrong cannot come something which is right. Only if the theory is right in some sense, can it be useful. But no physicist has any clue to what is right about quantum mechanics, only what is wrong!

Sabine Hossenfelder resorts to the common trick of referring to yet a more incomprehensible theory in the form of Quantum Field Theory, when basic Quantum Theory is questioned. This was used by
Einstein when he increased the bet to general relativity when his special theory of relativity was questioned on most rational grounds. This 

tisdag 7 maj 2019

Quantum Mechanics Still a Complete Mystery 2

Tim Maudlin introduces his new book Philosophy of Physics: Quantum Theory with the following credo:
  • A physical theory should clearly and forthrightly address two fundamental questions: what there is, and what it does. 
  • The answer to the first question is provided by the ontology of the theory, and the answer to the second by its dynamics. 
  • The ontology should have a sharp mathematical description, and the dynamics should be implemented by precise equations describing how the ontology will, or might, evolve.
  • The Copenhagen Interpretation, in contrast, does not. There is little agreement about just what this approach to quantum theory postulates to actually exist or how the dynamics can be unam- biguously formulated. Nowadays, the term is often used as short-hand for a general instrumentalism that treats the mathematical apparatus of the theory as merely a predictive device, uncommitted to any ontology or dynamics at all.
  •  Sometimes, accepting the Copenhagen Interpretation is understood as the decision simply to use the quantum recipe without further question: Shut up and calculate. Such an attitude rejects the aspiration to provide a physical theory, as defined above, at all. 
  • Hence it is not even in the running for a description of the physical world and what it does. More specific criticisms could be raised against this legacy of Bohr, but our time is better spent presenting what is clear than decrying what is obscure.
The Copenhagen Interpretation is the text book interpretation of quantum mechanics, which Maudlin thus refutes as meaningless. I agree totally. 

What does then Maudlin offer in the book instead of the Copenhagen Interpretation? Very little: pilot wave theory and many worlds theory, both failed attempts to give meaning to Schrödinger's wave function. The book thus gives yet another account of the mystery of quantum mechanics 100 years after its creation. 

The whole problem comes from the multi-dimensionality of Schrödinger's equation asking for a statistical unphysical interpretation. How many new books will be written on the theme that physicists do not understand the quantum mechanics they preach?

But there is light in the tunnel: Real Quantum Mechanics.

lördag 4 maj 2019

Free Will and Quantum Mechanics

Sabine Hossenfelder on BackReaction believes that quantum mechanics says that humans do not have a free will:
  • Physics deals with the most fundamental laws of nature, those from which everything else derives. These laws are, to our best current knowledge, differential equations. Given those equations and the configuration of a system at one particular time, you can calculate what happens at all other times. 
  • That is for what the universe without quantum mechanics is concerned. Add quantum mechanics, and you introduce a random element into some events. Importantly, this randomness in quantum mechanics is irreducible. It is not due to lack of information. In quantum mechanics, some things that happen are just not determined, and nothing you or I or anyone can do will determine them.
  • Taken together, this means that the part of your future which is not already determined is due to random chance. It therefore makes no sense to say that humans have free will.
This is an expression of the monumental confusion of modern physicists caused by the idea that atoms play games of roulette attributed to quantum mechanics as being based on Schrödinger's multidimensional wave equation, something which Einstein and Schrödinger never accepted.

I have followed a different line of thought viewing physics as forms of analog computation with finite precision, which can be simulated by digital computation with finite precision presented at The World as Computation including a new approach to quantum mechanics as Real Quantum Mechanics without games of roulette

In this world there is room for humans with free will, although you cannot control everything in your life, due to lack of precision. If you are not an addicted gambler this may be the world for you.

Determined or Random??

PS The idea that atoms play games of roulette has corrupted modern physics into incomprehensible voodoo-science. Macroscopics as complex systems of simple microscopics can express random behavior, but random microscopics destroys cause-effect and demands contradictory microscopics of microscopics. Contradictory science is voodoo-science, even if it is text book modern physics. 

söndag 21 april 2019

Quantum Mechanics Still a Complete Mystery 1

Lee Smolins new book Einstein’s Unfinished Revolution (lecture here) repeats yet another time what every physicist already knows:
  • I think I can safely say that nobody understands quantum mechanics. (R. Feynman)
  • Quantum mechanics is magic. (Daniel Greenberger) 
  • Everything we call real is made of things that cannot be regarded as real. (Niels Bohr) 
  • Those who are not shocked when they first come across quantum theory cannot possibly have understood it. (Niels Bohr) 
  • If you are not completely confused by quantum mechanics, you do not understand it. (John Wheeler)
  • If [quantum theory] is correct, it signifies the end of physics as a science. (Albert Einstein) 
  • I do not like [quantum mechanics], and I am sorry I ever had anything to do with it. (Erwin Schrödinger) 
  • Quantum mechanics makes absolutely no sense. (Roger Penrose)

måndag 15 april 2019

Greta Thunberg: Farlig Infantil Klimatpropaganda

Klimatalarmismen med Greta Thunberg som utvald portalfigur analyseras av av Klimatrealistene samt den franske filosofen Pascal Bruckner i en stor artikel i Le Figaro (som refereras här):
Läs och begrunda!

lördag 13 april 2019

Breakthrough: DigiMat Wins Vinnova Grant

My mathematics education reform program DigiMat together with Johan Jansson now takes a great leap forward by winning a 1.7 milj Skr grant from Vinnova:
A first version of DigiMat will be presented in the Fall 2019 in the form of a web course for Swedish mathematics teachers, and pupils. DigiMat offers a concrete realisation of the new national curriculum combining mathematics with programming. Our vision is to bring the joy and power of digital mathematics to young minds.  Stay tuned1 and tuned2 and recall this intro.

lördag 6 april 2019

Öppet Brev till Norges Statsminister om Klimatalarmismen

Nu har Klimatrealisternes Vetenskapliga Råd (där jag ingår) skickat ett brev till Norges Statsminister (se även här) om det bristande vetenskapliga underlaget för den klimatalarmism som dominerar politik och media med krav på en fossilfri värld, och som hotar med förödande konsekvenser för jordens fattiga flertal.

Ett liknande brev har tidigare skickats till Danmarks Statsminister. Dessa borde läsas även av Stefan Löven.

onsdag 20 mars 2019

Design Flaw of 737 Max vs Computational Simulation

Is the reason for the two fatal 737 Max accidents a flaw in the design of the airplane, making it prone to stall (see PS2 below), which was compensated by a possibly over-reacting control system, which the pilots could not turn off? Did FAA authorise the plane without proper safety evaluation?   Questions are piling upFBI is joining criminal investigationWikipedia,  Boeing,  New York TimesPilot training, Stability, Aviation expert, Kludge, Aviationcv, Pilots view.

Computational software used by developers of airplanes do not seem to allow simulation of the dynamics of stall and so the impact of stall on aircraft design and safety assessment must be done solely by expensive and time-consuming experiments in flight, and then also the design of the apparently needed control system. This may show to have been insufficient to make the plane safe, something which FAA did not have the capacity to check.

With our new technology of Automated Computational Mathematical Modeling with references here, the full flight of an airplane, like the 737 Max, including the full dynamics of stall, can accurately be computationally simulated as a unique capability shown in the HighLiftPW-3 Workshop, see also front page of Icarus Simulation. Such advanced technology could allow airplane makers better and faster simulations to design aircraft and assess their safety.

PS1 The Jas 39 Gripen is a fighter airplane designed and manufactured by the Swedish aerospace company Sabb, which is inherently unstable (relaxed stability) to allow quick turns and thus needs to be stabilized by automatic active control of small canards (wings) ahead of the main wings. The plane crashed twice from pilot-induced oscillations with a the pilot seeking to compensate an over-reacting control system, in 1989 during landing and in 1993 over central Stockholm.  A commercial aircraft is not designed to be unstable, but to be positively stable.

PS2 Stall means that when the inclination of a wing vs forward motion (angle of attack) is too large (around 20 degrees for a common wing), then the lift to drag ratio suddenly drops from 15 before stall to 2 after stall and the wing drastically loses functionality. The Maneuvering Characteristics Augmentation System (MCAS) was developed for the 737 MAX to prevent stalls in flaps-retracted, low-speed, nose-up flight.

PS3 Boeing is working on an update of MCAS to get 737 Max approved by FAA to fly again, but the problem may be bigger than just a software update asking for a redesign of the airplane, see here.

torsdag 14 mars 2019

Icarus Digital Math

Icarus Digital Math is getting attention at KTH and the world:

Stay tuned! Several exciting projects in pipe line!

onsdag 13 mars 2019

Brev till Statsministern om Klimatalarmismens Bristande Vetenskapliga Grund

Ett brev har skickats till Danmarks Statsminister Lars Løkke Rasmussen av Johannes Krüger, Professor emeritus, dr.scient, Institut for Geovidenskab og Naturforvaltning Københavns Universitet Geocenter Danmark, med följande inledning:
  • Spørgsmålet om årsagen til nutidens klimaforandringer og de mange følgevirkninger er meget mere kontroversielt, end det fremstilles af IPCC og nyhedsmedierne, for videnskabelige målinger og observationer viser noget andet end de utilstrækkelige computerproducerede klimamodeller.
Brevet avser att göra Danmarks Statsminister medveten om klimatalarmismens bristande vetenskapliga grund.

Ett liknande brev kommer att skickas till Norges Statsminister och borde skickas även till Sveriges Statsminister.

Till detta kan man lägga Høringssvar til NOU 2018:17 Klimarisiko og norsk økonomi samt Roy Spencer i Washington Times samt Richard Lindzen osv...

The Illusion of a Wing with Laminar Boundary Layer

Aerodynamicists have long and still carry a dream of a laminar wing as a wing of special shape like the P-51 Mustang with special flow characteristics, including less drag as compared to that of a traditional wing as pictured below:

Our New Theory of Flight revealing the true Secret of Flight shows that this dream is built on misconceptions going back to the boundary layer theory of Prandtl/Schlichting based on the idea of a no-slip boundary condition with fluid particles sticking to the surface of the wing with zero velocity thus creating a thin boundary layer, which can be laminar or turbulent, connecting to the free stream flow.

The dream of a laminar wing would then be a wing with a laminar boundary layer on the upper surface of the wing with small skin friction, which hopefully would be the case for a wing with special shape, like that of P-51. The New Theory shows that this can only be a dream or illusion, because flow with a laminar boundary layer separates on the crest of a wing and generates little lift.

The New Theory of Flight builds on the observation that for the high Reynolds number relevant for a wing, the boundary layer is always turbulent with an effect of small skin friction on the main flow, which can be modeled by Navier-Stokes equations with a slip (small friction) boundary condition instead of the dictate of Prandtl to use a no-slip condition.

The New Theory shows that it is the slip boundary condition modeling the action of a turbulent boundary layer, which prevents the flow from separating on the crest and delays separation until the trailing edge (before stall). You can see this in the video below showing computational solution of Navier-Stokes with a slip boundary condition which agrees very well with experiments, as documented in the references in Solution of the Clay Navier-Stokes Problem.

  1. With a slip boundary condition there is no boundary layer and the flow can stay non-turbulent attached until turbulent separation at the trailing edge and thus give large lift. 
  2. With a no-slip boundary condition and laminar boundary layer the flow separates on the crest and gives little lift. 
  3. With a no-slip boundary condition the boundary layer is in fact always turbulent with an action on the main flow like a slip boundary condition. 
  4. The dream of drag reduction of a laminar wing is already cashed in by a slip/small friction boundary condition. 
  5. It is a turbulent boundary layer which can be modeled with slip, and not a laminar boundary layer even though its skin friction is small. 
A laminar wing in the sense of having laminar boundary layer, does not work.

A laminar wing in the sense of a flow with a slip boundary condition without layer and turbulence until the trailing edge, works fine. In this sense a standard wing thus already has made the dream come true, but that is not what aerodynamicists call a laminar wing.

måndag 4 mars 2019

PON om Klimatförnekare i Samhällets Utkant

Pressens opinionsnämnd PON har kommit med sitt beslut avseende min anmälan av artikeln i Svenska Dagbladet om Climate Sense 2018 med följande motivering:
  1. Du upplevde att artikeln gav en kränkande bild av mötets arrangörer, föreläsare och deltagare i allmänhet, och speciellt din person i (stor)bild.
  2. (Du angav att) Reportern hade inte rapporterat sakligt från det vetenskapliga mötet. Artikeln var istället inriktad på misskreditering av mötets deltagare med tillmälen som "klimatförnekare av manligt kön", "i samhällets utkant", "personer med auktoritära attityder", "låg empati och hög dominans", "inte längre i yrkesaktiv ålder" kopplade till "fossilindustri, auktoritära tider, nationalism och populism", som "gräver sig allt djupare ned i förnekelse ju mer klimatpolitiken skärps med brett politiskt stöd", och "byggt hela sina liv på att vara med i verksamhet som släpper ut koldioxid."
PONs bedömning: (se länken ovan)
  • För pressetiskt klander krävs att anmälaren är utpekad och att han eller hon omnämns på kränkande eller nedsättande sätt.
  • Du är utpekad eftersom Du förekommer i (stor)bild.
  • Du associeras i viss mån med uppgifterna i artikeln om de andra deltagarna och andra klimatförnekare. Det är dock i huvudsak allmänna påståenden om klimatförnekare som grupp och du är inte direkt kopplad till någon av uppgifterna.
  • Den publicitetsskada du förorsakats får därför anses vara begränsad.
  • Mot bakgrund av detta finns det sammantaget inte skäl att rikta pressetisk kritik mot tidningen. 
Vad kan man då säga om detta? Man får väl ändå säga att tillmälena enligt 2. är mycket nedsättande (vad skulle vara värre?) och att PONs påstående att jag inte kan kopplas till någon av dessa (genom mitt uppträdande i storbild) inte kan vara riktigt. Men PON säger inte att skadan är ringa, bara att den är "begränsad" dvs inte oändligt stor. Med en sådan måttstock undrar man om PON överhuvudtaget kan fälla någon artikel. Kanske det ändå fanns skäl att rikta lite pressetisk kritik i alla fall? Eller är PON lika tandlös som FI?

När så småningom klimatalarmismens svenska historia kommer att skrivas, så kanske detta PONs bidrag kan vara en del.

PS1 Till skaran av klimatförnekare i samhällets utkant kan vi lägga prof Lennart Bengtsson, Sveriges mest meriterade klimatforskare, som i dagens Expressen säger att det finns ingen anledning till oro. Jag har bett Anna Hedenmo, enligt föregående post, säga hur hon hanterar en röst som denna samt frågat om PK har någon synpunkt på PONs beslut (eller är PK underställd PON?)

PS2 Titta på SVTs Ekdahl och Ekdahl som 12/2 behandlade Klimatet med följande inledning
  • Klimatfrågan är vår stora ödesfråga. 
  • Vi skall säga att vi följer FNs klimatpanels rekommendationer.
  • Förutsättningarn för kvällens diskussion är att det finns ett klimathot.
  • Det finns dom som inte tror på det, men det skall vi inte diskutera.  
Det kan man kalla åsiktskorridor, eller rättare taget censur, utövad av public service.
Titta sedan på den kritiska granskningen av programmet av Lars Bern. 

lördag 2 mars 2019

Publicistklubben och klimatfrågan

Tege Tornvall föreslog Publicistklubbens ordförande Anna Hedenmo att ordna en debatt om klimathotets realitet och ev. bristande information om observerade klimatfakta. På detta svarade Hedenmo så här:

Hej Tege,
Jag har förstått att du vill ha ett mer utförligt svar på ditt mail om klimatfrågan. Det ska du få.

Eftersom en absolut majoritet av ledande klimatforskare i dag är eniga om att klimatförändringarna är en följd av mänsklig påverkan, anser jag att en PK-debatt på det temat varken är seriös eller meningsfull. Samma inställning finner du i de flesta större medier, sen ett antal år.

I andra frågor, som flyktingfrågan eller metoo, har jag som du kanske känner till tidigare efterlyst en mer tillåtande debatt, utan åsiktskorridor. Men när det gäller klimatfrågan ser jag ingen anledning att debattera det som i dag måste betraktas som ett faktum.

Vi i Publicistklubben får ständigt förslag på debattämnen. En del tar vi fasta på, andra väljer vi bort. Du är med andra ord inte den enda medlemmen som inte har fått gehör för sina förslag.

Jag beklagar om du inte upplever att du får något för medlemsavgiften. Men kanske kan du söka något av våra stipendier?

Anna Hedenmo

Med anledning av detta beklämmande och nonchalanta bemötande av ett seriöst förslag har jag skickat följande brev till Hedenmo med begäran om svar.

Till Anna Hedenmo
ordf i Publicistklubben

Tege Tornvall meddelar att Du avfärdat hans förslag till debatt i PK om klimatfrågan, med motivering att: 
  • "jag ser ingen anledning att debattera det som i dag måste betraktas som ett faktum".
Jag ställer nu följande frågor till Dig, som jag önskar får besvarade för publicering på min välbesökta blogg http://claesjohnson.blogspot.com :
  1. Du välkomnar på Publicistklubbens hemsida med "vår uppgift är att slå vakt om det fria ordet, som en livsviktig del av demokratin". Om Du nu värnar om det fria ordet, hur kan Du censurera debatt om mänsklighetens ödesfråga nummer 1: klimatfrågan? 
  2. Vad är det som "i dag måste betraktas som ett faktum"?
  3. Vad är innebörden av “måste” i Ditt uttalande? Vem uttalar detta “måste"?
Claes Johnson
prof em i tillämpad matematik KTH

Hedenmos svar kommer att publiceras så snart det inkommit. 

Nedan är Hedenmos icke-svar på mina frågor i form av osorterade vaga påståenden, som Hedenmo samlat ihop utifrån sina kunskaper om klimatet och som hon tydligen "måste" komma dragande med, följt av mitt avslutande brev. 

Hej Claes och Tege,

Nedan ser ni fakta i frågan. 
Nu sätter jag punkt för denna mailkonversation.

Beträffande Publicistklubben, så är det helt frivilligt att vara medlem. 

Anna Hedenmo 

Jordens klimat håller på att värmas upp och det är en konsekvens av människans utsläpp av koldioxid och andra växthusgaser. Koldioxiden i atmosfären absorberar den utgående värmestrålningen. Ökande halter innebär därför en ökad absorption och planeten värms upp. Det påverkar jordens vädersystem, leder till smältande glaciärer, havsnivåhöjningar (både till följd av termisk expansion och smältande glaciärer), samt en ökande avsmältning av havsisen i Arktis och shelf-isarna i Antarktis. Det ökande energiinnehållet i atmosfären leder till fler och mer extrema väderhändelser. Regionalt kan klimatet bli såväl blötare som torrare och perioder med extremvärme blir allt vanligare. På längre sikt kan avsmältningen av de stora isarna på Grönland och Antarktis kraftigt komma att påverka havsnivåerna. 

Den globala CO2-utsläppskurvan planade ut mellan 2014—2016 men har sedan 2017 börjat öka igen. En annan viktig indikator är koncentrationen av CO2 i atmosfären. Den mäts direkt i atmosfären sedan 1957 och med hjälp av borrkärnor i is (för att få längre mätserier). Före industrialiseringen låg den på 280 ppm. I dag ligger den på 405 ppm. Det kommer att ta lång tid för minskande utsläpp att ge ett utslag i sjunkande koncentrationer av CO2. Utsläpp av koldioxid har funnits i miljontals år. Vid till exempel en skogsbrand släpps stora mängder CO2 ut i atmosfären. När sedan växter och träd tar upp kolet igen skapas ett naturligt kretslopp. Men när man eldar kol/olja/gas frigörs kol som under miljontals år funnits i jordskorpan och som inte är en del av det naturliga kretsloppet. 

Sedan 1900 har jordens genomsnittliga temperatur ökat med 1,0 grad men med variation mellan enskilda år. I vissa regioner, som Arktis går uppvärmningen mycket snabbare. Sverige är delvis ett arktiskt land och i norra Sverige går klimatförändringarna just nu snabbare än det globala genomsnittet. Uppvärmningen går snabbare över land än över hav. Den globala uppvärmningen sker snabbare över norra halvklotet än södra halvklotet.

Som jämförelse kan nämnas att skillnaden mellan dagens globala medeltemperatur och den globala medeltemperaturen under de kallaste perioderna under den senaste istiden är 5-7 grader. Detta ger ett perspektiv på det faktum att en uppvärmning över 1,5 eller 2 grader kan få mycket stora konsekvenser. 

Den globala havsnivåhöjningen, havet har höjts med 20 cm senaste 100 åren. Haven absorberar värme vilket också gör att havet expanderar. Dessutom bidrar inlandsisars och glaciärers avsmältning. Haven blir också allt mer försurade i takt med ökande koncentrationer av koldioxid. Detta tillsammans med stigande havsvattentemperaturer bidrar till blekning av koraller och kan få andra mycket stora konsekvenser för ekosystemen i haven. 

Forskarna har inte uppmätt fler extrema väderhändelser, men de är intensivare och kraftigare än förr. 2018 var ett extremt väderår. Det går till historien som det fjärde varmaste sedan mätningarna inleddes. 20 av de 22 senaste åren har varit de varmaste sedan mätningarna inleddes. Varje fraktion av en grad leder till extremare väder händelser som mer intensiva orkaner, torka och konsekvenser som skogsbränder, översvämningar, döende korallrev, ökning av smittsamma sjukdomar. https://public.wmo.int/en/media/press-release/wmo-climate-statement-past-4-years-warmest-record

I oktober i år publicerade FN:s klimatpanel IPCC sin 1,5 c-rapport beställd av världens regeringar 2015. I den beskrivs konsekvenserna av 1,5 c graders uppvärmning av jordens medeltemperatur, som enligt IPCC med stor sannolikhet kommer inträffa 2030—2052. 

Forskarna uppmanar nu världens länder att tredubbla sina ansträngningar att fasa ut förbränningen av olja, naturgas och kol. Tidsfönstret krymper snabbt, och en minskning på 70 procent bör ske till 2030, enligt forskarnas rekommendationer.

Här är mitt svar till Anna:

Tack Anna för Ditt svar på mitt brev, som jag lagt upp min blogg.

Dock svarar Du inte på mina frågor utan kommer istället med en samling mer eller mindre lösa påståenden (s k fakta) som Du på något sätt samlat ihop efter de kunskaper i ämnet Du besitter. 

Vad man kan säga om detta är att varje påstående skulle kunna tas till utgångspunkt för en diskussion med syfte att klargöra dess signifikans och avgöra dess sanningshalt. 

Men Du censurerar varje sådan diskussion hos PK, och Du avslutar vår mailkonversation.

För att reda ut det hela föreslår jag att Du bjuder in mig eller Tege till Min Sanning (om klimatet)!

Vänliga hälsningar

PS Publicistklubben inbjuder till debatt den 11 mars över ämnet:
  • Vad kan politiken göra för att värna journalistiken i tider av filterbubblor och misstro mot etablerade medier?
Frågan kan också formuleras: Hur kan censuren göras mer effektiv? 

torsdag 28 februari 2019

Planck's Desperate Mad Ugly Ad Hoc Trick: The Quantum

Planck's reasoning was mad, but his madness has that divine quality that only the greatest transitional figures can bring to science. (Abraham Pais in The Science and Life of Albert Einstein)

...the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be... (Planck on the statistical mechanics basis of his radiation law)

Sabine Hossenfelder on Backreaction gives praise to the new book Breakfast with Einstein by Chad Orzel:
  • Physics is everywhere, that is the message of Chad Orzel’s new book “Breakfast with Einstein,” and he delivers his message masterfully.
  • In contrast to many recent books about physics, Orzel stays away from speculation, and focuses instead on the many remarkable achievements that last century’s led to.
Planck was not happy with his desperate mad ugly ad hoc trick of the quantum

Chapter 2 of the book has the title The Heating Element: Planck's Desperate Trick with the objective of describing the birth of quantum mechanics attributed to Planck's (ugly ad hoc) trick of avoiding the apparent ultraviolet catastrophe of classical wave mechanics by introducing the concept of a smallest package of energy named quantum: 
  • This “quantum hypothesis” does the necessary trick of cutting off the amount of light at high frequencies—exactly where the ultraviolet catastrophe happens. 
  • Planck initially introduced the quantum hypothesis thinking it was a “desperate mathematical trick.” 
  • Despite the many successes of his formula and the personal fame it brought him, Max Planck himself was never particularly satisfied with his quantum theory.
  • He regarded the quantum hypothesis as an ugly ad hoc trick, and he hoped that someone would find a way to get from basic physical principles to his formula for the spectrum without resort- ing to that quantum business. 
  • Once the idea was out there, though, other physicists picked it up and ran with it, most notably a certain patent clerk in Switzerland—leading to a complete and radical transformation of all of physics.
I have presented an alternative theory based on finite precision computation, which meets Planck's wish of explaining the black body spectrum from basic classical wave mechanics physics, which is presented on Computational BlackBody Radiation. Why not take a look, and see if you get enlightened? By a physical theory of blackbody radiation.

The idea of finite precision computation is the same as that used in a new explanation of the the 2nd law of thermodynamics discussed in the previous post on Boltzmann and his explanation based on (ugly ad hoc) statistics.

The master of ugly ad hoc tricks is Roger Stone as documented in his new book Stone's Rules. Such tricks can take you to the top of both science and politics! They can give you fame, but evidently not happiness. Another master of this game was the patent clerk in Switzerland, who also was unhappy with his theories, in particular the theory of the quantum he picked up from Planck, which gave him such immense fame:
  • If I would be a young man again and had to decide how to make my living, I would not try to become a scientist or scholar or teacher. I would rather choose to be a plumber or a peddler in the hope to find that modest degree of independence still available under present circumstances.
  • All these fifty years of conscious brooding have brought me no nearer to the answer to the question, “What are light quanta?”. Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.
  • For the most part I do the thing which my own nature drives me to do. It is embarrasing to earn so much respect and love for it. 
  • Why is it that nobody understands me, and everybody likes me? (Einstein in New York Times, March 12, 1944) 
PS1 Often a truth about science, or rather a truth about a shortcoming of some scientific theory, is more honestly expressed in popular science, as the truth of the ugly ad hoc science of the quantum in Orzel's book (because the audience is supposed to be ignorant), than in some professional scientific context hiding the shortcoming in some cover-up (because the audience is supposed to be knowledgable and critical). Therefore it is interesting to read popular science also for a scientist.

PS2 Planck's desperate ugly ad hoc trick (which originates from Boltzmann) has caused a lot of confusion among physicists. For example, quantum mechanics, which is not understood by any serious honest physicist,  is supposed to have some mysterious connection to the quantum of energy of Planck, but the fact is that quantum mechanics is based on Schrödinger's equation, which is a continuum mechanical model and not a discrete model build from small packets of energy. The confusion is exhibited in Real Quantum Mechanics offering a new form of and new view on Schrödinger's equation with the common confusion eliminated.  But it is not easy to get a discussion going on the fundamentals of quantum mechanics, since the confusion is so monumental resulting from a  desperate mad ugly ad hoc trick, supposed to be the foundation of modern physics. No wonder that physics is in crisis. See also Dr Faustus of Modern Physics.

PS3 Recall that it was Einstein who introduced the idea that light is made of discrete chunks of energy $h\nu$ as photons with $h$ Planck's constant in Joulesecond and $\nu$ frequency, in his heuristic Law of the photoelectric effect $h\nu + W = eU$ with $W$ work to release an electron
and $eU$ in electron volt eV with $U$ the stopping potential in volt and e the charge of an electron. I argue in Mathematical Physics of BlackBody Radiation and related blogg posts that the Law is to viewed as a frequency threshold condition, which has no relation to any idea of light as consisting of discrete photons or light quanta, which according to the above quote was also the view of the late Einstein.

The Law shows that Planck's constant $h$ appears as a conversion between energy related to light frequency $\nu$ (in Joule) and electron energy (in eV), for which Einstein received the Nobel Prize in Physics in 1921 with explicit mention that he did not get the Prize for his theories of relativity). 

PS4 Schrödinger's equation connects energy related to light frequency and electron energy and it is thus no wonder that the Planck constant appearing in Schrödinger's equation is the same as that in the Law of the photoelectric effect. Mathematical Physics of BlackBody Radiation also gives evidence that the Law of the photoelectric effect is a consequence of Schrödinger's equation, within a continuum model without photon particles and reference to Einstein's heuristic argument that a photon of sufficient energy can kick out an electron. 


fredag 22 februari 2019

Boltzmann 175 vs 2nd Law by Finite Precision Computation

Ludwig Boltzmann 1844-1906

Lubos on the Reference Frame recalls the 175th birthday of Ludwig Boltzmann:
  • Yesterday, Ludwig Eduard Boltzmann would have had a chance to celebrate his 175th birthday if he hadn't killed that chance by hanging himself at age of 62...
  • Boltzmann's reason powering the suicide were intellectually driven frustrations.
  • If he were resurrected and if he were around, he would probably ask me whether there's a reasonable chance that the people will get more reasonable when it comes to the ideas required for his new statistical picture of thermodynamics and physics in general. I would probably answer "No" and he would hang himself again. 
Lubos then enters into a defence of Boltzmann's 2nd law based on statistics and the related
Copenhagen interpretation of quantum mechanics with electrons randomly jumping around atom kernels, something which Einstein and Schrödinger never accepted. 

The problem Boltzmann tried to solve, with its tragic ending, is how formally reversible systems can show to have irreversible solutions. Boltzmann showed that you can hang yourself, but he could not un-hang himself, and he sought the explanation in statistics. Unsuccessfully according to Lubos, because still today people cannot understand what he was saying, about entropy and a 2nd law based on nonsensical statistics saying that something with a higher probability is more likely to happen.  I think this is not because people/scientists are stupid, which Lubos claims, but because what Boltzmann says makes sense only to Lubos.

I have presented a different explanation based on finite precision computation. This says that the reason that you cannot un-do things is lack of precision, and that all physics as well as digital computation is realised in finite precision. This means that you can enter a labyrint (the woods/world) with finite precision, like taking a step forward in time, but you cannot find your way out of the labyrint or retrace your path through the woods, go back in in time, because your are limited by finite precision. The arrow of time is an expression of finite precision computational physics. This is meaningful physics and different from Boltzmann's empty idea that the world moves form less probable to more probable states or from more ordered to less ordered states (with a start/bigbang as most ordered state inexplicable). 

I thus offer an explanation of the 2nd law of thermodynamics presented in many earlier blog posts 
and the book Computational Thermodynamics explaining that finite precision solutions of formally reversible system like the Euler equations of fluid mechanics can show to be irreversible, e.g. by the emergence of turbulence. This directly connects to a resolution of the Clay Navier-Stokes Problem reported in previous posts

The catch is that formally reversible systems can have irreversible solutions if precision is finite, and of course precision cannot be infinite, not in digital computation and neither in the physical world.

torsdag 21 februari 2019

Hamming and Tartar on Clay Navier Stokes Problem

Richard Hamming (1915-98)

The mathematician Richard Hamming said:
  • Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
An example is given in the official formal formulation of the Clay Navier-Stokes Problem by Fefferman, which does not mention the world turbulencewhich in the informal presentation is central:
  • Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
Everybody, except Fefferman, understands that turbulence is the secret hidden in Navier-Stokes and that the Clay problem, to be more than an intellectual sport standing in the way for sensible information, should ask about a mathematical theory unlocking the secret of turbulence. 

The informal presentation is sensible, while the formal presentation is nothing but an intellectual sport, which neither has practitioners since no progress towards a resolution has been made since 2000, or rather since 1932 when Leray proved existence of weak solutions with uniqueness or wellposedness left completely open.  

We have presented a resolution to a reformulated Clay Problem offering sensible information about the physical process of turbulence, by computation. We hope there are some sensible people that can show a reaction to our resolution. We show by computation that weak solutions exist, are non-smooth/turbulent and have wellposed mean-values such as drag and lift. 

Hamming also said:
  • The purpose of computing is insight, not numbers. … 
  • [But] sometimes … the purpose of computing numbers is not yet in sight.
Yes, we find that being able to compute (turbulent) solutions to Navier-Stokes equations opens to gain insight into the nature and manifestation of turbulence.  DFS is in sight and gives insight! 

So, what insight has DFS brought? Here is one major revelation:
  • bluff body flow = potential flow + turbulent 3d rotational slip separation.
Bluff body flow is thus computable by DFS, which offers a revolutionary new capacity to CFD with a vast field of applications for all sorts of vehicles or life moving through air and water, and the fluid mechanics is understandable!

Also note what the mathematician Luc Tartar says in the presentation of his book on Navier-Stokes:
  • To an uninformed observer, it may seem that there is more interest in the Navier-Stokes equation nowadays, but many who claim to be interested show such a lack of knowledge about continuum mechanics that one may wonder about such a superficial attraction. 
  • Could one of the Clay Millennium Prizes be the reason behind this renewed  interest?
  • Reading the text of the conjectures to be solved for winning that particular prize leaves the impression that the subject was not chosen by people interested in continuum mechanics, as the selected questions have almost no physical content.
  • The problems seem to have been chosen in the hope that they will be solved by specialists of harmonic analysis...
  • I  hope that this particular set of lecture notes...may help the readers understand a little more about the physical content of the equation, and also its limitations, which many do not seem to be aware of.
And as before: the pure mathematicians Fefferman, Constantin and Tao in charge of the problem formulation refuse to participate in any form of discussion.  Why? Lack of knowledge about continuum mechanics, with focus instead on harmonic analysis?

And remember:
  • What is computable is understandable. (Pythagoras)
Luc Tartar


onsdag 20 februari 2019

From Equation to Solution

This is a continuation of the previous post on the role of functional analysis, more precisely the role of the finite element method as a form of computational functional analysis.

We start with the basic partial differential equation of physics and mechanics, Poisson's equation:
  • $-\Delta u(x) = f(x)$ for $x\in\Omega$,
  • $u(x)=0$ for  $x\in\Gamma$, 
where $\Omega$ is a domain in space with boundary $\Gamma$, $f(x)$ is a given function defined on $\Omega$ and $u(x)$ is the solution to the equation defined on $\Omega$ and $\Gamma$. The game is: Given $f(x)$ find $u(x)$ satisfying Poisson's equation.

We can think of the differential equation $-\Delta u(x)=f(x)$ as expressing force balance at the point $x$ with $u(x)$ the deflection of an elastic membrane under a transversal force or load $f(x)$, in case $\Omega$ is two-dimensional.  There are endless other interpretations.

So far so good, the partial differential equation $-\Delta u=f$ captures complex physics in very compact beautiful mathematical form, and so is marvellous, but there is one caveat: The formulation of the equation gives no clue to how to determine the solution $u(x)$. The equation is like a rebus without any hint of resolution.

It is here that functional analysis enters by offering a reformulation of the differential equation $-\Delta u =f$ into variational form: Find $u\in V$ such that
  • $\int_\Omega \nabla u\cdot\nabla v\, dx = \int_\Omega fv\, dx$ for all $v\in V$,       (1) 
where $V$ is a collection (function space) of possible solutions, from which a best possible solution $u(x)$ is determined by the relation (1). Formally (1) is obtained by multiplying the differential equation $-\Delta u=f$ on both sides with an arbitrary function $v\in V$ and integrating over $\Omega$ using integration by parts to see that (using that $v=0$ on $\Omega$)  
  • $-\int\Delta uv\, dx =\int_\Omega\nabla u\cdot\nabla v\, dx$. 
In the finite element method the space $V$ consists of piecewise polynomial functions over a triangulation of $\Omega$ and (1) is a linear system of algebraic equations, which can be solved by Jacobi iteration or Gaussian elimination. 

The differential equation as unsolvable rebus has thus been reformulated into variational form which allows a best possible solution to be computed by standard linear algebra software.  Here functional analysis enters in the variational formulation and the construction of the finite element space $V$.

The great thing is now that the same method works for virtually any (partial) differential equation, in particular the differential equations of science and technology: Reformulating the differential equation into variational form allows computation of best possible (approximate) solution. 

This is realised in the FEniCS Project which is software automating the whole process consisting of 
  • reformulation into variational form, 
  • construction of finite element space $V$,
  • computation of solution by linear algebra. 
The crown jewel is automated computation of best possible solution of Navier-Stokes equations which we claim resolves the Clay Navier-Stokes Problem and makes turbulent flow computable and thus understandable, for the first time. And this is only the beginning of a FEniCS revolution.

We understand that the differential equation $-\Delta u(x)=f(x)$ expresses local force balance (at the point x), while the solution $u(x)$ comes out as a global effect depending on $f(y)$ for all $y$ and not just $f(x)$. This means that to determine $u(x)$ requires computation collecting many local inputs to one global output.

The mathematics of Jacobi iteration then corresponds to the physics of relaxation where the system reacts to reduce force imbalance. Gaussian elimination (or even better multi-grid) is more efficient than Jacobi iteration, which allows mathematics to take a short-cut to solution compared to physical relaxation.

PS The Navier-Stokes-Euler equations for incompressible flow contains the equation
  • $\nabla\cdot u=0$ 
expressing the incompressibility, together with an equation expressing force balance according to Newton's 2nd law. The equation $\nabla\cdot u=0$ does not express force balance and appears more like a regulation stipulating a certain property of the solution (incompressibility) than a true law of physics like Newton's 2nd law. In DFS (near) incompressibility is instead expressed as a pressure law of basic form
  • $\Delta p=\frac{\nabla\cdot u}{\delta}$  
where $\delta > 0$ is a small parameter, with the effect of forcing $\nabla\cdot u$ to be small by pressure as an expression of some physics. The lesson is that a differential equation without solution procedure is only half of the story.  Stating laws without means of enforcing the laws may be empty.

tisdag 19 februari 2019

Banach and DFS and Clay Navier-Stokes Problem

This is an exercise in preparation for participation in a film about the Polish mathematician Stefan Banach who advanced functional analysis as mathematics describing relations between functions or analogies between analogies. My punch line is that the finite element method, as the subject of my work, is (nothing but) computational functional analysis following the spirit of Banach.

The crown of my work, together with Johan Hoffman and Johan Jansson, is Direct Finite Element Simulation DFS as solution of the Navier-Stokes-Euler equations without turbulence model or complicated wall model from a principle of best possible solution, in a situation where there is no exact solution. DFS brings revolutionary new capacity to Computational Fluid Dynamics CFD, which we (as a show case) claim resolves the Clay Navier-Stokes Problem by computation.

Functional analysis was formed by the mathematician Hilbert at the switch to modernity around 1900, with contributions from the Swedish mathematician Fredholm, and was further developed by Banach starting in 1920.  A prime objective was to justify mathematical models in the form of partial differential equations of solid and fluid mechanics and electromagnetics formulated during the 19th century by Laplace, Fourier, Navier, Stokes and Maxwell, by answering basic questions concerning existence and uniqueness of solutions, as well a construction of solutions by computation.

The basic element of functional analysis is a collection of functions named Hilbert space or Banach space equipped with a structure or geometry generalising that of ordinary three dimensional space. The solution of a given partial differential equation is then an element of a suitably chosen Hilbert or Banach space in basic cases determined by a principle of energy minimisation. The differential equation, which is impossible to solve directly by symbolic computation with pen and paper,  is thus reformulated into a minimisation problem over a function space, which allows construction of solutions as a limits of functions with decreasing energy computed according to the Banach Contraction Mapping Theorem.

Starting in the 1950s this form of computational functional analysis has been developed under the name of the finite element method into a universal method for computing solutions of the differential equations of science and engineering bringing revolutionary new capacities.  This success story was darkened only by Navier-Stokes-Euler equations of fluid mechanics, which were believed to demand computational power beyond anything which could be envisioned, the reason being the phenomena of turbulence and thin boundary layers involving small scales too costly to resolve computationally, the impossibilities presented in NASA CFD Vision 2030.

We show that with DFS the NASA CFD Vision 2030 is realised already today. By computational functional analysis in the spirit of Banach.

DFS and functional analysis gives a new perspective on differential equations representing ideal physics, however with uncomputable or non-existing exact solutions as in the case of Navier-Stokes-Euler,  and reformulations in terms of functional analysis with computable approximate solutions representing real physics.