## onsdag 30 november 2011

### Rapport från Möte med S om Matematik-IT

• Ge alla elever en iPad vid skolstart.
• Skapa ett nytt ämne Matematik-IT, som lär:
• IT som del av Matematik +
• Matematik med IT.
• Stöd med utbildning/fortbildning i Matematik-IT.
Återstår att se om S vill göra detta till en del av sitt utbildningsprogram. Inte omöjligt.
Forts följer...

PS. En iPad som kostar 4000 kr och används under 4 år ger en kostnad av 1000 kr/år och elev.
iPaden ersätter vanliga böcker som kostar minst dubbelt. Lärandet ökar och kostnaden minskar.

## tisdag 29 november 2011

### Sammanfattning PM Matematik-IT: Möte med S

• Vi behöver metoder som förbereder för arbete och medborgarskap i ett kunskapsintensivt och högteknologiskt samhälle.
• Sverige halkat efter när det gäller utvecklandet av lärandet med stöd av digital teknik, att utveckla lärararbetet med IKT och att utveckla skolan med samma verktyg.
• Sverige måste utarbeta en nationell strategi för digital kompetens i skolan.
• Skola fokuserad på framtidens kunskaper i stället för gårdagens typ.
JAN BJÖRKLUND:
• 1 miljard för fortbildning för att utveckla undervisningen.
• I klartext, mer katederundervisning:
• Lärare skall förklara, berätta, instruera och repetera i helklass och dessutom leda diskussion och samtal i helklass för att säkerställa att eleverna förstår det matematiska tänkandet och matematiska begreppen.
• Mindre av eget arbete mer av katederundervisning.
• Sanningen är att katederundervisning är väl beprövad och väl fungerande.
• Växande digitala klyftorna och bristen på moderna datorer i svenska skolor inget jag ligger sömnlös över.
VERKLIGHTEN:
• Om 2 år har alla elever en iPad (eller liknande)
• Läsa-Skriva-Räkna-IT: Kunskap, Samhälle, Teknologi, Arbete, Liv...
• IT = Matematik + Logik + Programmering + Data
• IT = Beräkning + Data (Google sökmotor)
• Matematik för/med IT
• Reform av Skolans Matematik
• Nytt: Mål, Innehåll, Form
• VAD, VARFÖR, HUR? Gamla svar gäller inte längre.
• Reform av Skolan
MATEMATIK I PARALYS (pga datorn):
• Hierarki: universitet-gymnasium-grundskola-förskola
• Kardinal = Professor i Matematik vid Universitet
• Splittring-Misstro: Matematiker - Didaktiker
• Alla: Skolmatematiken Funkar Inte! Katastrof!!
• 2011 = 1950 (Björklund) = 1900
• Kardinal säger inte: IT-Reform.
• Kardinal säger: Didaktiker, lärare och elever är korkade
• Kardinal säger: Ingenting kan och därför inte bör förändras
• Didaktiker kan bara gå runt i cirkel.
• PARALYS (inte bara i Sverige)
• INGEN BÄR ANSVARET: Alla lägger skulden på tidigare skola
• Vem är PÅVE?
• Nationalkommitten för Matematik (KVA)??
• Matematikersamfundet??
• Påven har abdikerat, Kyrkan står kvar, men är tom.
MISSION för S:
• Initiera reform av lärarutbildning - fortbildning: Matematik-IT
• Låt PÅVEN visa sig och bära sitt ansvar.
• Unik chans för S att Ta Initiativet
• Unik chans för S att Rädda Sveriges Konkurrenskraft
• Skola med/för IT = Skola för Alla = (S)KOLA
VÄRLDEN:
• Oslo Univ, Simula Research Lab
• Rwanda: Alla elever laptop från 9 år
• Nya Världen: Kina, Indien, Brasilien, Afrika,...
• Gamla Världen: Kris: Finans: Klimat...

## måndag 28 november 2011

### Renormalization in QED, Blackbody Radiation and Turbulence

There is a connection between the cut-off in Computational Blackbody Radiation and the cut-off in Renormalization of QED, namely
• avoiding an ultraviolet catastrophe of infinite energy
• by transforming high-frequency waves into low-frequency waves
• in a form of reinvestment.
In the case of blackbody radiation with an absorbing body with a cut-off proportional to temperature subject to forcing from incident waves, the reinvestment comes out as heating (increasing temperature) from incident wave frequencies above the current cut-off.

There is also a connection to turbulent dissipation with high-frequency kinetic energy being transformed into low frequency internal/heat energy. The total energy as the sum of kinetic and internal energy collects the reinvestment and is thus conserved.

Renormalization thus concerns reinvestment of what is cut off, like collecting the branches cut off from a tree and using them together with the trunk of the tree so that nothing is lost.

The basic question concerns partition of energy between the trunk and branches (on branches)
and its dependence on the size of the cut-off length.

In turbulence, the total turbulent dissipation (= drag in bluff body flow) shows to be independent of the cut-off length.

In quantum mechanics the question is if the world would look the same with a different value of Planck's constant which sets the cut-off length on atomic scales.

## fredag 25 november 2011

### Radiative vs Turbulent vs Frictional Heating

Mathematical Physics of Blackbody Radiation and Computational Turbulent Incompressible Flow (or Computational Thermodynamics) show a similarity between radiative heating and turbulent heating, which can be viewed as different forms of frictional heating.

In both cases the heating effect shows to be independent of the small coefficient of dissipation/friction connected to the heating (the viscosity in turbulent flow and coefficient of the Abraham-Lorentz recoil force in radiation).This means that macroscopic quantities such as total radiative and turbulent heating show to be insensitive to the absolute scale of the microscopic dissipation creating the heating.

This allows computational simulation of macroscopical features of radiation and turbulent flow without resolution to actual smallest physical scales. This makes in particular turbulent flow computable contrary to common perceptions of state-of-the-art.

One may compare making fire by friction with a bow and drill: What is important is the total power invested, not the specific dimensions of the bow and drill.

### Updated Computational Thermodynamics

I have put up an updated preliminary version of the upcoming book Computational Thermodynamics.

The basic idea is to find the truth by computing turbulent solutions of the Euler/Navier-Stokes equations by EG2 as stabilized General Galerkin finite element method, and understanding that EG2 can serve as a model of thermodynamics.

A basic revelation is that the concept of entropy can be replaced by turbulent dissipation to give the 2nd Law a concrete expression in terms of the physical concepts of heat energy, kinetic energy, work and turbulent dissipation, without any need of introducing the cumbersome notion of entropy with its statistical basis.

EG2 thus offers a computable deterministic model of thermodynamics opening to understanding of basic phenomena and offering the capability of real prediction, without any need of statistics.

The presentation is centered around Joule's experiment from 1845 illustrated above on the thermodynamics of an expanding gas showing fundamental aspects of heat engines, heat pumps and refrigerators.

## tisdag 22 november 2011

### Massive Swedish Delegation to CMP7 in the Sunny City of Durban

Sweden sends 45 official delegates to the upcoming COP17/CMP7 climate conference under the motto Saving Tomorrow Today:
• The 7th Session of the Conference of the Parties serving as the Meeting of the Parties (CMP7) to the Kyoto Protocol, will be held in the sunny city of Durban, South Africa.
Everybody knows that the Kyoto Protocol will not be renewed, because there is no reason to do so. One may ask why 45 official delegates from Sweden are required to not do anything? To not do anything but profit from the sunny city of Durban and get away a couple of weeks from the darkness now descending over Sweden?

### Heat Capacity of Blackbody 2?

To add to the mystery of the concept of a blackbody as a "cavity absorbing all incident radiation", we now ask about the heat capacity/specific heat of such a thing?

We know that the heat energy/internal energy per unit time (radiance) radiated from a blackbody scales with temperature T like T^4, but what is the connection between T and absorbed/radiated heat energy Q(T), that is what is the heat capacity dQ/dT of a blackbody?

1. If the cavity is filled with a perfect gas, then Q scales like T.

2. If the cavity is filled with a solid, then Q should scale like T^2 according to Mathematical Physics of Blackbody Radiation.

3. If the cavity filled with a photon gas, then Q scales like T^4.

We thus get a heat capacity dQ/dT scaling with temperature in a wide range from T^0 = constant to T^3.

This is yet another indication that the concept of a blackbody is confusing since it is not at all clear what physics it models, and thus is unfortunate as a scientific concept.

## söndag 20 november 2011

### Blackbody as Bottle with Peephole?

A blackbody is a theoretical idealized object described as something "absorbing all incident radiation" commonly pictured as a cavity or empty bottle/box in which waves/photons are bouncing back and forth between walls at a certain temperature defining the temperature of the cavity. The bottle has a little peephole through which radiation is escaping to be observed, as indicated in the above common illustration of a blackbody.

A blackbody is supposed to capture an essential aspect of the radiation from a real body like the visible glow from a lump of iron at 1000 C, the Sun at 6000 C or the invisible infrared faint glow of a human body at 37 C.

But why is a lump of iron, the Sun or a human body thought of as an empty bottle with a peephole?

Yes, you are right: It is because Planck used this image in his proof of Planck's Law of blackbody radiation based on statistics of energy quanta/photons in a box. Planck's mathematical proof required a certain set up and that set up came to define the idealized concept of a blackbody as an empty bottle with peephole. But to actually construct anything near such a blackbody is impossible.

It is natural to ask if with another proof of Planck's Law, the concept of blackbody would be different, possibly closer to reality?

In Mathematical Physics of Blackbody Radiation I give a different proof of Planck's Law with a different concept of blackbody as a lattice of vibrating atoms absorbing and emitting radiation as electromagnetic waves, which models a real body like a lump of iron, and not a fictional empty bottle with a peephole.

This shows the role of mathematics in the formation of concepts of the World:
• With a strange mathematical proof the World may appear strange and incomprehensible.
• With a natural mathematical proof the World my become comprehensible.

## lördag 19 november 2011

### Interaction Light-Matter or Soul-Body

One of the big mysteries of physics is the interaction between immaterial light and material matter, or in a wider context the interaction between immaterial soul/mind and material body.

Descartes believed that the interaction soul-body took place in the little pineal gland in the center of the brain. Modern neurobiology does not give much support to Descartes' idea but has not really any better theory and so the mystery of how soul and body interact remains to be resolved.

What does then modern physics say about the interaction of light and matter? There are two competing theories depending on the nature of light as
1. Deterministic electromagnetic waves described by Maxwell's equations.
2. Statistics of massless particles.
2. connects to Newton's old corpuscular theory of light, which was revived by Einstein in 1905 after it had been declared dead and had been replaced by Maxwell's wave theory in the late 19th century.

2. became popular because it offered a resolution to the light-matter interaction problem by simply side-stepping the whole question by claiming that everything is (statistics of) particles: If light is not an immaterial electromagnetic wave phenomenon, but simply some sort of material particles (albeit without mass, but never mind) then there is no wave-matter problem to resolve!

Clever, but maybe too clever since after all light is an electromagnetic wave phenomenon. This brings us back to 1. and the real question of how an immaterial wave can interact with a material body?

In Mathematical Physics of Blackbody Radiation I suggest a resolution with immaterial waves interacting with matter by wave resonance and statistics replaced by finite precision computation. This is a resolution in terms of waves with electromagnetic wave motion interacting with wave motion in matter ultimately also consisting of electromagnetic waves.

The wave-matter interaction problem is thus in this case resolved by understanding that everything is (finite precision) wave and wave resonance, both light and matter. In the wider context: everything is soul and soul resonance.

We have thus two possible solutions of the light-matter interaction problem:
1. Everything is (finite precision) deterministic wave and wave resonance.
2. Everything is statistics of particles and collision of particles.
Maxwell and Schrödinger said 2. What do you say?

## torsdag 17 november 2011

How quickly will a hot (black) body cool off if all exterior heat sources are shut off and the
heat energy stored in the body radiates from the body? How quickly does the Earth surface cool off during night?

1. Mathematical Physics of Blackbody Radiation gives the following cooling curve for the temperature T(t) as function of time t starting at t=0:
• T(t) = T(0)/(1 + C t)^0.5
as a consequence of the fact that the stored heat energy E scales like T^2 while the heat loss scales like T^4 by Stefan-Boltzmann's Law which also defines the constant C.

2. The standard answer is different:
• T(t) = T(0)/(1 + C t)^0.33
with E scaling instead like T.

Which answer is best? Experiments appear to favor 1 with its quicker decay, also shown in the above picture from Comlab.

The difference between 1 and 2 may be the difference between a blackbody in the form of a solid or a gas, where the blackbody nature/spectrum of a gas may be questioned.

## onsdag 16 november 2011

### Josef Stefan: Backradiation and DLR Pure Fiction!!

Josef Stefan 1879: Backradiation and DLR is pure fiction!!

Let us quote from Josef Stefan's On the Relation between Radiation and Temperature from 1879 (my translation of page 411):
• The absolute value the heat energy emission from a radiating body cannot be determined by experiment. An experiment can only determine the surplus of emission over absorption, with the absorption determined by the emission from the environment of the body.
• However, if one has a formula for the emission as a function of temperature (like Stefan-Bolzmann's Law), then the absolute value of the emission can be determined, but such a formula has only a hypothetical meaning.

We see that Stefan understood very well that only the net/surplus has a physical meaning.

Stefan would thus say that the mantra of CO2 alarmism of backradiation or Downwelling Longwave Radiation DLR from the atmosphere to the warmer Earth surface, is pure fiction.

Are CO2 alarmists relying on Stefan-Boltzmann's Law willing to listen to what Stefan has to say? Or can the work of a dead scientist be misused at will without any consequence?

## tisdag 15 november 2011

### Möte med S: Matematikutbildning för IT-Samhället

PM inför Möte med Socialdemokraterna i Riksdagens Utbildningsutskott 30/11

S har inbjudit mig till möte om matematikutbildning med följande utgångspunkt:
• Socialdemokraterna stödjer en investering i matematik men vi vet ännu inte hur regeringen vill lägga upp programmet och kommer självklart ha synpunkter på hur så bör ske.
• Vad gäller en modernisering av skolan för att passa it-samhället så har vi har under flera års tid kritiserat regeringen för att inte ta informationsteknologin på allvar, vi vet att detta skulle kunna ha stor betydelse för t ex undervisningen i matematik.
• Vi skulle gärna ta del mer av hur du tycker att matematikundervisningen och it i skolan bör utvecklas.
Inför mötet hoppas jag att Mikael Damberg/S vill ta del av:
mina nya debattinlägg på Newsmill och Teknikdebatt (med äldre här):
rapport från mitt möte med Utbildningsdepartmentet Nov 09, inlägg på min blogg om Matematik-IT och Mathematics Education, artiklar på Google Knol om Mathematics/Science Education, samt följande debattartikel som kommer att publiceras i Ny Teknik:

Om ett par år kommer alla elever att ha en egen laptop eller iPad från skolstart, som ett naturligt uttryck för att skolan är en del av IT-samhället med alla dess möjligheter till aktivt lärande i nya former.

Detta ställer nya krav på skolan och lärarutbildningen måste nu reformeras så att alla elever kan ges en god förberedelse för liv och arbete i IT-samhället. Detta gäller alla ämnen men särskilt i IT-intensiva tekniska ämnen av avgörande betydelse för Sveriges konkurrenskraft.

Detta har oppositionen med sin utbildningspolitiske talesman Mikael Damberg (S) förstått, som framgår av hans artikel på SKLs Blogg om eSamhället. Men det har inte utbildningsminister Jan Björklund, som i sitt nya initiativ Mattelyftet beordrat Skolverket att utarbeta en plan för att omskola alla mattelärare till mer av "traditionell katederundervisning", till en kostnad av 2.6 miljarder.

Björklund motiverar sitt direkta statliga ingripande med en minskning enligt PISA från 510 år 2000 till 494 år 2009, dvs en minskning med 3%. Björklund beskriver minskningen om 3% som så alarmerande att staten "inte stillatigande kan titta på" utan måste diktera undervisningen, som i ett totalitärt samhälle.

Skolverket tar i sin plan dock avstånd från "en dominerande traditionell skol- och undervisningskultur", dvs Björklunds "katederundervisning". Resultatet är en statlig förvaltning i kramp, en lärarutbildning i paralys och en skola utan framtidshopp. Björklunds miljarder skulle kunna användes för att leda skolan in i IT-samhället, inte minst genom en reformerad skolmatematik eftersom datorn är en matematikmaskin, men stora resurser kommer nu istället att förödas på meningslös revision utan IT.

Björklund kör över hela svenska folket, inklusive elever, föräldrar, lärare, lärareutbildare, fack och statlig förvaltning, med ett meningslöst kostsamt statligt diktat, men kritiken av Mattelyftet har hittills varit svag och splittrad. Skolverket försöker lamt att obstruera men alla lärarutbildare med ansvar för skolans undervisning säger ingeting. Det är inte lätt att säga nej till 2.6 miljarder, även om ändamålet är tveksamt och PISA-resultaten bara kommer att fortsätta att sjunka.

Men skattebetalarna vill inte att miljarder förslösas på ett meningslöst illa genomtänkt statligt diktat. Detta ger oppositionen en lysande möjlighet att ta initiativet i skolpolitiken med en offensiv framtidsinriktad skolpolitik där resurser istället används för leda skolan in i IT-samhället.

Mikael Damberg förstår vad Björklund inte förstår, och det skall bli mycket intressant att se vad ett kommande möte med Damberg kan leda till. Jag har förgäves sökt få tala med Björklund men hans dörr är stängd. På en direkt fråga om de växande digitala klyftorna och bristen på moderna datorer i svenska skolor från Damberg svarar Björklund att "det är inget som han ligger sömnlös över". Nej, Björklund sover tungt och drömlöst och det är inte bra för svensk skola.

Min blogg ger mer information om matematik och matematikundervisning i IT-samhället. När alla barn om ett par år kommer att ha en laptop eller iPad i skolväskan kommer skolans arbete att förändras. Vi är snart där och måste förbereda oss.

Sammanfattning:
• Skolan måste reformeras för att passa det nya IT-samhället:
• IT = ord, bild, ljud, siffror, diagram... i digital form: ettor och nollor
• IT bygger på matematik: logik, geometri, algebra, Calculus, beräkning
• Matematikämnet ger grunden för IT i skolan
• Matematikämnet förändras med IT: Från formler till beräkningsalgoritmer
• IT-Matematik: logik, programmering, beräkning, simulering
• Genomgripande reform av mål, innehåll och form från åk 1
• Nya läroplaner, nya läromedel
• Ny lärarutbildning i matematik med IT
• Prototyp: Body and Soul, Mathematical Simulation Technology.
Datorspel som ny pedagogisk metod:

Body and Soul använder datorspel på två sätt:
1. Använda färdiga datorspel för att lära sig läsa-skriva-räkna
2. Göra egna datorspel som tillämpning av matematik-programmering.
Datorspel ger feed-back, kan stimulera flera sinnen, fånga intresse och skapa motivation.

Formell kunskap vs reell kunskap:

Den gamla real/gymnasieskolan (Björklunds) gav formell kunskap för få i hierarki:
• latin, matematik, grammatik, svåra ord, skriva på tyska
Dagens skola skall ge reell kunskap för många i jämlikhet:
• tala och skriva: svenska, engelska,...
• använda dator för kalkyl, ekonomi, simulering,...
• lära sig hur man kan lära sig olika saker
• lära sig att man kan förstå och skapa mening.
Mål för S-Matematik:
• ge alla redskap för att klara sig i IT-samhället: ord, bild, matematik, simulering
• kombination av manuell/praktisk och språklig/teoretisk kunskap
• matematik grunden för IT
• logik, orsak-verkan, korrekthet
• ABF tradition i modern tappning: ge arbetaren IT-redskap
• att kunna styra sitt liv
• att kunna uttrycka sig
• att kunna förstå text, bild, diagram, formler
• att förstå logik
• att kunna styra datorn: tex bildbehandling i Photoshop
• textbehandling
• att kunna göra film, presentation, analys, datorspel i vid mening.
Viktiga S-aspekter:
• interaktivt lärande: feed-back
• elevers särskilda behov kan tillfredställas
• fler elever kan engageras
• utbredd matematikskräck kan motverkas
• matematik en ges annan mening än prestige och gradering.
Möjlighet för S att vinna nästa val:

Björklund tittar defensivt bakåt mot 1950-talets realskola för de få och ger därmed S en enastående möjlighet att ta initiativet med en offensiv skolpolitik för de många för dagens och morgondagens samhälle. S söker ett vinnande valprogram. (S)kolfrågan kan vara svaret.

Media:
Juholt SVD 26/11:
• Vi vill bryta den ökade segregationen i skolan och bygga Sverige starkt med en kunskapsbaserad ekonomi.
• Att skapa ett utbildningssystem där samtliga elever når målen i grundskolan och alla har en fullständig gymnasieexamen före 25 års ålder.
• 5 miljarder extra till stadsbudgeten fokuserat på skola och utbildning.
Om CJ:
• prof i tillämpad matematik Chalmers 1981-2007, KTH 2007-
• 100 artiklar, 10 böcker, 15 studenter
• ICI Highly Cited lista av världens mest citerade forskare
• Humboldt Research Prize
• reform av ingenjörsmatematik: Body&Soul.

## måndag 14 november 2011

### Heat Capacity of Blackbody

In Mathematical Physics of Blackbody Radiation I derive an expression for the heat capacity C(T) of a blackbody in the form of a forced damped wave equation as function of temperature T, of the form
• C(T)= c T
with c a constant, thus with a linear dependence on T. This fits pretty well with observations for several solid metals for temperatures T smaller than 200 K, and for diamond up to 500 K, as shown above.

Recall that the theory of specific heats was studied intensely in the beginning of the 20th century: Einstein and Debye offered improvements for low temperatures of the classical Dulong-Petit Law C(T) = constant, using arguments from statistical mechanics.

I do not use statistical mechanics but instead a concept of finite precision computation.

### The Secret of Flow Separation Uncovered

Real flow as potential flow modified by vortical slip separation with point stagnation.

The flow around a body moving through a slightly viscous incompressible fluid like water or air at subsonic speeds, attaches at the front as fluid particles approach the body and separates in the back as fluid particles part from the body.

In slightly viscous flow fluid particles glide along the surface with small skin friction which can be approximated as a slip boundary condition.

Observation shows laminar attachment and turbulent separation, with the flow being close to potential flow (inviscid irrotational flow with slip) prior to separation. In potential flow the separation is simply reverse attachment and thus real flow differs from potential flow at separation.

Ever since the basic mathematics of fluid mechanics was formulated by Euler and d'Alembert in the mid 18th century, a prime goal has been to describe slightly viscous flow around a body as potential flow subject to some modification at separation:
• Prandtl as the father of modern fluid mechanics described the modification as a boundary layer effect from a no-slip boundary condition.
• Kutta and Zhukovsky as fathers of modern flight mechanics described the modification as large scale circulation around a wing section.
In the new article Analysis of Separation in Turbulent Incompressible Flow together with Johan Hoffman (submitted), we show that the answer is neither boundary layer nor circulation, but instead
• vortical slip separation with point stagnation.
We give evidence in the form of
• mathematical analysis of basic instability of potential flow at separation
• computational solution of Navier-Stokes equations
• experimental observation.
We thus present strong evidence that the dream of Euler and d'Alembert can be fulfilled by describing slightly viscous flow around a body as
• potential flow modified by vortical slip separation with point stagnation.
The article gives the details.

As an application we uncover The Secret of Flight (upcoming book) and The Mathematical Secret of Flight (article and talk).

## söndag 13 november 2011

### The Poynting Vector

The Poynting vector $E\times H$ representing the energy flux of an electromagnetic field.

Mathematical Physics of Blackbody Radiation gives a new wave based proof of Planck's radiation law in the form
• $Radiation + Heating = Forcing$,
where
• $Forcing = \int\int f^2(x,t)\, dxdt$
with $f(x,t)$ the amplitude of the incoming wave.

The measure of $Forcing$ as the (integral of the) square $f^2$ of the amplitude $f$ connects to the Poynting vector $E\times H$ representing the energy flux of an electromagnetic field with $E$ the electric field and $H$ the magnetic field having the same amplitude.

## tisdag 8 november 2011

### Is the Economical Crisis a Climate Crisis?

Two facts fly in our faces today:
1. EU is in a deep economical crisis.
2. EU has the most ambitious plan for reduction of CO2 emissions.
Is there a connection? Many questions present themselves:
• Has the financial market already adjusted to the new reality of a carbon-free society projected by the Royal Swedish Academy of Sciences to 2050?
• Would the economical crisis evaporate if EU leaders gave up their plans for a Great Transformation?
• Was the climate crisis invented to give clever EU political leaders a mission to lead the World into a new future?
• Is the economicial crisis triggered by the climate crisis now driving the same clever EU politicians out of business ?

## måndag 7 november 2011

### The Essence of Planck's Radiation Law and Its Proof

Planck trying to convince a disbelieving Bohr that his proof of Planck's Law is correct. On the blackboard: Maxwell's equations.

The derivation of Planck's Law (of blackbody radiation) given by Planck is based on particle statistics borrowed from thermodynamics with little connection to the real physics of radiation as interaction of matter and electromagnetic wave motion.

Planck was forced into particle statistics because a radiation law based on wave motion seemed to lead a Rayleigh-Jeans law with an "ultraviolet catastrophe" with energy exploding to infinity like $\nu^2$ with frequency $\nu$ without upper bound. By a statistical assumption that highly energetic waves are rare Planck gave physics a way to avoid the catastrophe which earned almost infinite fame and a Nobel Prize.

But Planck left a question without answer: Is it possible to derive a correct radiation law without resort to statistics, using instead classical electrodynamics described by Maxwell's wave equations?

In Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation I show that this is indeed possible, by replacing statistics by a much more basic physical assumption of finite precision computation. Let me here presents the essence of my argument:

I consider mathematical wave model of the form:
• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f$
where the subindices indicate differentiation with respect to space $x$ and time $t$, and
1. $U_{tt} - U_{xx}$: material force from vibrating string with U displacement
2. $- \gamma U_{ttt}$ is Abraham-Lorentz (radiation reaction) force
3. $- \delta^2U_{xxt}$ is a viscous force
4. $f$ is exterior forcing,
and $\gamma$ and $\delta^2$ are (small) positive coefficients subject to a frequency dependent switch from outgoing radiation with to absorption/internal heating defined below.

The finite precision computation is represented by the $\delta^2$ viscosity which sets a smallest scale in space of size $\delta$ by damping frequencies higher than $\frac{1}{\delta}$.

The coefficients $\gamma$ and $\delta$ are chosen so that only one is positive for a certain frequency $\nu$ in a spectral decomposition, with a switch from $\gamma >0$ to $\delta >0$ at $\nu = \frac{T}{h}$, where $T$ is temperature and $h$ is a fixed precision parameter representing atomic dimension in the string of atoms modeled by the wave equation.

The switch thus defines a temperature dependent cut-off frequency $\frac{T}{h}$ with the effect that only frequencies below the cut-off are emitted as radiation while frequencies above cut-off are stored as internal heat energy of the string.

The essence of the proof is the following energy balance established by a spectral analysis assuming periodicity in space and time:
• $\int \gamma U_{tt}^2dxdt + \int \delta^2U_{xt}^2dxdt = \epsilon \int f^2 dxdt$
with $\epsilon\approx 1$ is a coefficient of emissivity (= absorptivity), which we write in condensed form as
• $R + A = F$, or in a spectral decomposition $R_\nu + A_\nu = F_\nu$, or in words
• Radiation + Absorption = Forcing,
where $R = \int \gamma U_{tt}^2dxdt$, $A=\int \delta^2U_{xt}^2dxdt$ and $F=\epsilon \int f^2 dxdt$.

As just said, the switch effectively means that
• $R_\nu = F_\nu$ for $\nu <\frac{T}{h}$
• $A_\nu = F_\nu$ for $\nu >\frac{T}{h}$,
which expresses that the forcing $F_\nu$ is remitted as outgoing radiation for $\nu<\frac{T}{h}$, and is absorbed and stored as internal heat energy for $\nu > \frac{T}{h}$.

The coefficient $\epsilon$ represents emissivity = absorptivity, in accordance with Kirchhoff's Law. An essential aspect is that $\epsilon$ is independent of $\gamma$ and $\delta$, which expresses that radiation (to up to the constant $\epsilon$) is independent of the nature of the radiating/absorbing body.

Planck's law in the form $R+A = F$ with only one of $R$ and $A$ non-zero for each frequency
is a variant of Planck's classical law with a sharp switch instead of a continuous transition from radiation to absorption/heating.

Planck's law in the form $R + A = F$ thus contains the classical law for radiation as $R=F$ but also the further information that absorption into internal heat energy $A=R$ only occurs for frequencies above cut-off.

The proof that $R + A = F$ is non-trival and expresses a fundamental aspect of near-resonance
in systems of oscillators with small damping. The proof shows Planck's Law $R_\nu\sim \gamma T\nu^2$ for $\nu <\frac{T}{h}$ and Kirchhoff's Law, without resort to statistics.

The effects of Planck's statistics in modern physics are subject to an investigation in Dr Faustus of Modern Physics: Is it true that Planck's technique for avoiding the ultraviolet catastrophe led to a much bigger catastrophe of abandoning rationality in physics?

## onsdag 2 november 2011

### Generic Model of Resonant Interaction Matter-Wave

• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f$
where the subindices indicate differentiation with respect to space $x$ and time $t$, and
1. $U_{tt} - U_{xx}$: material force from vibrating string with U displacement
2. $- \gamma U_{ttt}$: radiation pressure from outgoing radiation = emission
3. $- \delta^2U_{xxt}$: viscous force from internal heating = absorption
4. $f$ is exterior forcing,
and $\gamma$ and $\delta^2$ are (small) positive constants connected to dissipative losses as outgoing radiation = emission and internal heating = absorption. Let me here clarify the physics behind the dissipative terms:
The corresponding dissipation energies are obtained by multiplication of the forces with $U_t$ followed by integration (by parts) in space-time over a period, assuming periodicity, to give
Recall that damping in the wave equation $U_{tt} - U_{xx} = f$ can take the from
$\nu U_{t}$ with $\nu$ a viscosity coefficient, or as above $-\gamma U_{ttt}$ and $- \delta^2U_{xxt}$, thus with the following derivatives:
• $U_{t}$
• $U_{ttt}$
• $U_{xxt}$
where $U_t$ appears in acoustic viscous damping, and can be seen as a simplified version of both $U_{ttt}$ and $U_{xxt}$.

The above model thus can be viewed as the generic model of wave motion with damping describing of interaction matter (string) and electromagnetic waves (radiation term and forcing) with a very rich area of application including:
The mathematical analysis exhibits a fundamental phenomenon of near resonance in wave motion with small damping, as explained in How to Push a Swing: In near resonance the forcing is in phase with the displacement, while in perfect resonance (with large damping) the forcing is in phase with the velocity. The effect is that in near resonance the exterior forcing is balanced by both the string force and the Abraham-Lorentz radiation force, while in perfect resonance only by the radiation force.

Near resonance appears when the radiative damping is small ($\gamma$ is small), which is the real case of radiation with full interaction between matter (string) and radiation.