numerical gpu simulations of the hydrogen 1s ground state ... · het nulpuntsveld moet dan genoeg...

Numerical GPU simulations of the hydrogen 1sground state in Stochastic Electrodynamics

A fully relativistic 3D treatment

Matthew T. P. Liska

Supervised by dr. Theo M. Nieuwenhuizen

Second corrector: prof. dr. Bernard Nienhuis

30-08-2015

MSc. thesis GRAPPA physics track

Institute of Physics (IoP)

M. T. P. LiskaInstitute for Theoretical Physics P.O. Box 94485, 1098 XH Amsterdam, the Netherlands

2 MSc. thesis MTP Liska

Abstract During this research project we tested using numerical simulationswhether stochastic electrodynamics (SED) are able to explain the shape and sta-bility of the 1s hydrogen ground state.

For this we improved on previous work (Cole and Zou 2003) by a full 3Dtreatment of the zero-point field, longer run times and the inclusion of relativisticcorrections plus spin-orbit coupling. On top of this, we compared our results toa conjecture for the angular momentum and energy distribution of an electron inthe 1s ground state.

This was made possible by mathematical simplications and using a self-developedhigh performance OpenCL code utilising the enormous power of graphics process-ing units (GPUs). The results are in all cases with or without the inclusion ofposition dependence and/or relativistic effects similar. However, the answer to thestability question is negative.

While we can achieve close correlation to a hydrogen ground state on inter-mediate time scales using controlled conditions, we also see serious deficiencies athigh eccentricities and inconsistent results for different cutoffs of the zero-pointfield. Another result is that our system ionises on timescales similar to 106 elec-tron orbits. This is in contrast to previous simulations (Cole and Zou 2003), whodidn’t encounter ionisation on shorter timescales.

We also propose a remedy for this ionisation to be investigated in future byimproving on the used point charge approximation for the electron.

On the hydrogen ground state in Stochastic Electrodynamics 3

Contents

1 Dutch summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 The zero-point field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 The zero-point field: Physical representation . . . . . . . . . . . . . . . . . . . . 93.2 The zero-point field: Numerical representation . . . . . . . . . . . . . . . . . . . 103.3 The zero-point field: Fixed vs moving cuttoffs . . . . . . . . . . . . . . . . . . . 11

4 Deriving the equations of SED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1 Deriving the equations of SED: The coulomb force . . . . . . . . . . . . . . . . 124.2 Deriving the equations of SED: The Abraham-Lorenz force . . . . . . . . . . . 124.3 Deriving the equations of SED: Final equation of motion . . . . . . . . . . . . . 134.4 Deriving the equations of SED: Switching to Bohr units . . . . . . . . . . . . . 144.5 Deriving the equations of SED: Canonical equations of motion . . . . . . . . . 14

5 Relativistic corrections in the hydrogen problem . . . . . . . . . . . . . . . . . . . . 166 Conjecture for the ground state phase space density . . . . . . . . . . . . . . . . . . 187 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.1 Numerics: Limitations of previous simulations . . . . . . . . . . . . . . . . . . . 207.2 Numerics: Our code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.1 Computation: CPUs not becoming faster . . . . . . . . . . . . . . . . . . . . . . 228.2 Computation: The tremendous speed of GPUs . . . . . . . . . . . . . . . . . . 228.3 Computation: Implementing our algorithm on a GPU . . . . . . . . . . . . . . 248.4 Computation: Used system and performance . . . . . . . . . . . . . . . . . . . . 24

9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.1 Results: Moving cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.2 Results: Fixed cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Other work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.1 Other work: Theoretical prediction about ionisation . . . . . . . . . . . . . . . 2910.2 Other work: A protocol for the electron’s conserved quantities . . . . . . . . . . 29

11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


1 Dutch summary

Quantum mechanica was en is zeer succesvol geweest in het maken van voor-spellingen in de atomaire fysica. Zo kunnen de fijn- en hyperfijnstructuur vanonder andere het waterstofatoom verklaard worden, wat erg belangrijk was in decontext van het bestuderen van sterspectra begin vorige eeuw. Ook werd aange-toond dat deeltjes net als golven interferentiepatronen kunnen vertonen, wat eropwijst dat deeltjes en golven eigenschappen van elkaar kunnen bezitten. Nog steedswordt in de wetenschap veelvuldig gebruik gemaakt van voorspelling uit de quan-tum mechanica, zoals in de chipindustrie. Met andere woorden, het heeft zich inontwikkeld tot een zeer goed functionerende effectieve theorie.

Quantum mechanica werd op fundamenteel niveau echter niet altijd goed ont-vangen, omdat zijn puur probabilistisch karakter onverenigbaar is met onze in-tuitie. Dit probabilistisch karakter houdt in dat de uitkomst van een quantummechanisch experiment vooraf niet te voorspellen is. Albert Einstein schijnt ooitte hebben gezegd: ’God does not play dice’. In de quantummechanica bepaalt degolffunctie, welke uit toestanden met verschillende kansen bestaat, de uitkomst vaneen experiment pas na een meting. Dan stort de golffunctie pas in en neemt eenzekere toestand in. Een beroemd voorbeeld is Schrodinger’s kat, welke opgeslotenin een doos is en een superpositie van een levende en dode kat is. Stel we openende doos en vinden een dode kat, kunnen we dan vaststellen dat de kat dood was?Volgens de quantummechanica niet. Sterker nog, deze zegt dat als we misschieneven gewacht hadden met de ’meting’ we een levende kat vonden.

Deze en soortgelijke contra-intuitieve resultaten uit de quantum mechanicawapperen nog steeds de discussie aan of er een sub-quantum mechanische theoriebestaat, die het bijvoorbeeld wel mogelijk zou maken te voorspellen of de kat levendis. Een veelbelovende theorie is de Stochastische electrodynamica (SED). Deze steltdat er een zogenaamd nulpuntsveld rond ons heen aanwezig is, de grondtoestandvan het elektromagnetische veld, welke alle fotonen ’draagt’. Klassiek gezien wordtdit nulpuntsveld door elektromagnetische golven gerepresenteerd, die natuurlijkeen kracht op materie om ons heen kunnen uitoefenen. De stochastische electro-dynamica probeert met behulp van dit nulpuntsveld de resultaten uit de quantummechanica na te bootsen, maar dan op een betere en meer fundamentele manier.

De Stochastische elektrodynamica is echter eind vorige eeuw in verval is ger-aakt, omdat het te weinig vooruitgang had geboekt in het verklaren van niet-lineare problemen zoals de grondtoestand van het waterstofatoom. Daar gaat menuit van een klassiek elektron dat in een Kepler baan rond het waterstofatoom be-weegt. Uit de relativistische elektrodynamica zou volgen dat het elektron al zijnenergie zou uitstralen en op de kern van het atoom belanden, wat natuurlijk on-juist is. Het nulpuntsveld moet dan genoeg energie terugggeven der middel vaneen elektromagnetische kracht op het elektron om het stabiel te houden en dequantum-mechanische vorm van de grondtoestand te verklaren.

Begin vorig decennium hebben niet-relativistische computerresultaten gesug-gereerd dat de stochastische elektrodynamica de grondtoestand van het water-stofatoom inderdaad kon verklaren. Deze simulaties waren echter gelimiteerd dooreen tekort aan computationele kracht indertijd. Zo konden deze alleen in 2D enmet een beperkte selectie van resonante modes uit het nulpuntsveld worden uit-gevoerd. Dit was bovenop een zeer korte draaitijd, die de volledige dynamica nietin kaart kon brengen.


De afgelopen jaren is de computationele fysica echter heel snel vooruitgegaan.Buiten de wet van Moore werd het mogelijk om met behulp van videokaarten,ontworpen voor het renderen van graphics in computerspellen, vele codes metmeer dan een factor 100 te versnellen.

We hebben in dit onderzoek gebruik gemaakt van deze nieuwe technieken omnog een keer de stochastische electrodynamica te testen. Hiervoor hebben we eeneigen 3D relativistische code ontwikkeld, die de grondtoestand van het waterstofa-toom probeerde te berekenen onder aanname van een nulpuntsveld en klassiekemechanica.

Wij vonden in tegenstelling tot eerdere resultaten een iets minder mooie corres-pondentie met de grondtoestand van het waterstofatoom (figuur 1). Het elektronbleek ook te ioniseren, omdat het bij banen met hoge eccentriciteit te veel energieopneemt. Dus we kunnen concluderen dat de huidige implementatie van de theorieminder goed werkt dan de eerdere minder nauwkeurige simulaties suggereerden.

Desalnietemin is er toch een duidelijk verband tussen onze gesimuleerde verdel-ing en de quantummechanische grondtoestand van het waterstofatoom zichtbaar.Dit suggereert dat er misschien ergens in de gebruikte theorie iets ontbreekt of eenfoute aanname verstopt ligt, wat de theorie zou kunnen redden.

Zo is het mogelijk om nog in de toekomst andere vormen van het nulpuntsvelduit te proberen of het electron een realistische ladingsverdeling toe te kennen. Erzijn namelijk voor beide theoretische argumenten te bedenken die het probleemvan de ionisatie zouden kunnen oplossen.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 ÈEÈ

0.5

1.0

1.5

2.0

2.5

P

1 2 3 4 5 r

0.1

0.2

0.3

0.4

0.5

P

Fig. 1 a): Histogram voor de energie en straalverdeling van het elektron als gemeten in desimulaties met in rood de voorspelling uit de quantum-mechanica


2 Introduction

The statistical interpretation of quantum mechanics has been a long standingdispute. SED (stochastic electrodynamics) for example claims that everywhere inour universe there is a so called zero-point electromagnetic field (ZPF), a groundstate remnant from almost excellently working quantum field theory (QFT). ThisZPF would explain all quantum mechanical behaviour of particles and waves, fromthe double slit experiment to the hydrogen atom all the way to the probabilisticSchrodinger equation with wavefunctions as solutions. Thus it would introduce aso called local hidden variable, based on the local value of the ZPF and be incontradiction with mainstream thinking where quantum mechanics constitute themost fundamental theory.

This debate eventually turned into a merely phylosophical one since quantummechanics works excellent as an effective theory at least, with SED being deemedirrelevant and/or ruled out based on controversial experiments. Nevertheless itis interesting to give SED a rethought, because we can test SED for non-linearproblems such as the 1s state of hydrogen atom using cutting-edge simulationsimpossible 10 years ago.

The biggest argument in favour of a purely statistical interpretation are theBell inequalities, which claim to rule out the existance a local hidden variable.However previous experiments, which measured the quantities needed to plug intoBell’s formula, may be misinterpreted since differences in the experimental setupbetween measurements have not been taken into account. Each component of suchan experimental setup possibly has a different hidden variable distribution due todifferences in the experimental setup/context in which the quantaties are mea-sured. The violation of Bell’s inequalities thus merely demonstrates that differentcontexts can’t be combined [3]. Thus it may still be worthwile to keep an openmind and test if SED is able to reproduce it’s quantum mechanical equivalent forvarioust problems.

The derivation of the Schrodinger equation from first principle SED has beenreported [4,5]. The authors show that the Liouville equation describing spatialand temporal evolution of the probability distributions of the position and theangular momentum for a free particle reduces under action of a zero-point field toa Schrodinger like equation under assumption of energy balance, which assumesthat the energy going into the system through a zero-point field is equal to the totalenergy radiated by the Abraham-Lorentz force. The authors also show that thecommutation relation [x,p] reduces to its quantum mechanical analog [x, p] = ihunder action of a zero point field, which is able to introduce Planck’s constant intoSED via the spectrum of the zero-point field.

Since matter couples to this zero-point field, it is able to manipulate it. Thismay for example explain the wave-like nature of electrons observed in the doubleslit experiment. There waves from the zero-point field may interfere and thusinfluence the trajectory of such an electron, making the conclusion that the electronexhibits wavelike properties invalid. On the other hand it is hard to imagine howthis zero-point field may carry entanglement accros large distances. Thus there areboth arguments in favour for as well as against SED.

Finally, it is realised that the energy throughput due to SED keeps matterstable with the energy from stochastic fields going in to compensate for the effectfrom radiative damping. This can be seen as an arrow of time, called the subquan-


tum arrow of time, which is more fundamental than the entropic (second law ofthermodynamics) and cosmological (expansion) arrows of time [8]

Nevertheless SED is known to work well for harmonic (linear) problems such asthe harmonic oscillator, see e.g. [1]. However there is dispute about its correctnessfor non-linear problems such as the ground state of hydrogen. Claverie and Soto forexample put forward that the motion is non-recurrent for highly eccentric orbits,due to the plunging of orbits to lower angular momenta and higher eccentries [9].However this treatment may be deemed flawed since it doesn’t include relativisticcorrections, which according to T. Boyer 2004 [13] should suppress orbits withhigh eccentricities. Puthoff puts forward a comparison between energy gain andloss terms, concluding that it should be stable [10]. However calculating averageenergy gain and loss terms doesn’t properly take into effect orbital resonances,which can lead to runaway situations.

The theory has been tested numerically on the hydrogen ground state in a 2-dapproximation by Cole and Zou 2003, who observed close correspondence to the 1sground state without ionisation. However their simulations employed a so calledwindow approximation and insuffucient simulation time due to the unavailabil-ity of sufficient computational resources at that time. These simulations togetherwith the advent of GPUs, which eventually sped up our code more than two ordesrof magnitute, inspired us to redo these simulations with higher numerical preci-sion/longer runtimes (paper 1 [14]) and the inclusion of relativistic effects (paper2 [15]).

In this thesis I will first describe the classical theory of a hydrogen atom andshow how we included the zero point field inside our numerical simulations for ourfirst paper [14]. Then I will add the theory behind the relativistic corrections pro-posed in T. Boyer 2012, and implemented in our second paper [15]. Subsequently Iwill describe how the code was developed and give a short introduction about howGPUs benefited this project. Then I will present our results and compare them toa conjecture for the hydrogen ground state, before finishing of with a discussion.


3 The zero-point field

SED assumes the presence of a fluctuating vacuum field called the zero-pointfield (ZPF). This field is theorized to be present everywhere in our universe andcould be the origin of the universe’s dark energy. The origin of this ZP-field ispostulated in quantum field theory, which describes the quantisation of fields suchas the electromagnetic (EM) and Higgs fields in terms of quanta. We will limit ourdiscussion to the EM-field for this thesis.

QFT postulates that the EM-field at every point in space and time has to bequantised, with the the excited states (quanta) being the photons propagating asEM-waves through vacuum. Each of these EM-waves has a wavelength of λ = 2π

kand complex waveform in the radiation gauge similar to:

A = A0eikx−ωt, E = −∂A

∂t, B = ∇×A (1)

The most suitable quantum mechanical system describing this quantised EM-fieldis the quantum harmonic oscillator. It has the following Hamiltonian for a 2Dsystem with complex displacement z = x+ iy and unit mass m = 1:

H =p2

2+ω2z2

2(2)

Compare this to the energy density ρ of an electromagnetic wave and you see thecorrespondence that A becomes the displacement in the Hammiltonian assumingω = ck and c = 1:

ρ =EE∗ +BB∗

2=

AA∗

2+k2 ∗AA∗

2(3)

The total energy per mode becomes like for every quantum harmonic oscillator:

En = (n+1

2)hω (4)

The number of photons occupying each state is given by n. For the ZPF ourchosen value of A0 dictates n = 0, which gives a zero point energy of hω

2 for thecase where no photons occupy the relevant mode. The origin of this discrepancywith classical mechanics lies in the Heisenberg uncertainty principle since a photonoccupying a harmonic oscillator state has either to have uncertainty in momentum(kinetic energy) or position (potential energy) which leads to it having an averageminimum energy of hω

2 .Now we can represent each possible wave with wavevector k = 2π

L (nx, ny, nz)as occupying one quantum harmonic oscillator state with ω = c

λ . The total EM-field at a point in space would be the summation of all harmonic oscillators statesin ω-space with each point contributing E = hω

2 .This zero point field has been observed in different experiments. The most

famous experiment studied the Casimir effect, where because two conducting platesare seperated by a finite distance, the ZP-field is able to exert a force on them. Sincethese plates are conducting, they serve a boundary conditions for electromagneticwaves, ie an electro magnetic wave can’t traverse the plates. Since their seperationdistance is finite, less modes are accepted between the plates than outside in the


vacuum, because the maximum wavelength beteween the plates seperated by λbecomes 2λ. This radiation overpressure causes the plates to be drawn together.

Another example is the shift in energy levels between 2P an 2S states, called theLamb shift. The Dirac equation predicts no energy shift between these levels, whilein reality there is a shift whose leading terms may be explained by the inclusion ofan electromagnetic field. Explaining the non-linear terms of this Lamb shift usingSED simulations would constitute a strong point for its validity. Though numericalsimulations didn’t reach the point where they are even able to replicate the 2Pand 2S states fading this hope in the near term.

There is also a big problem with zero point field, since the number of possiblemodes ω is infinite in a finite sized box. Let’s consider the EM-field energy densityρ, which can be obtained by integrating the density of states function g(ω), givingthe number of modes per unit volume between ω and ω + dω:

g(ω) =4πω2

(2πc)3dω (5)

multiplied by the average ZP-energy of each mode E = hω2 over ω-space by:

ρ =

∫ ωmax

0g(ω)Edω =

∫ ωmax

0

2πhω3

(2πc)3dω =

hω4max

4π2c3(6)

We thus see that the energy density diverges by the fourth power of the cutofffrequency ωmax. One partial remedy is to define a frequency cutoff at the planckscale, though this still gives us a value 100 orders of magnitude bigger than themeasured bounds on the cosmological constant. This is called the vacuum catas-trophe, the biggest discrepency with reality arrising from quantum field theoryand also present in SED.

3.1 The zero-point field: Physical representation

Thus the bottom line of the above story is that while a ZP-field is able to explainsome things in nature, the theory also has serious flaws as evident from the stillunresolved vacuum catastrophe. Also the exact shape of this ZP-field along witheffects such as its interaction with charges remains unresolved.

Nevertheless for this thesis we represent the ZP-field as a classical EM-fieldin a finite volume of dimensions V = LxLyLz with an exponential cutoff at theplanck scale (Compton time) of τ = h

mc2 . We have chosen V such that it is a goodapproximation for an infinite universe (see later sections).

In the Coulomb gauge the SED vector potential of a cube of volume V is asum of plane waves with random (for this work Gaussian) coefficients,

A(r, t) =∑k,λ

√h

2ϵ0ωkVe−ωkτc/2εkλ [ Akλ sin(k · r− ωkt)

+ Bkλ cos(k · r− ωkt)] (7)


E(r, t) =∑k,λ

√hωk

2ϵ0Ve−ωkτc/2εkλ [ Akλ cos(k · r− ωkt)

− Bkλ sin(k · r− ωkt)] (8)

B(r, t) =∑k,λ

√µ0hωk

2Ve−ωkτc/2k× εkλ [ Akλ cos(k · r− ωkt)

− Bkλ sin(k · r− ωkt)] (9)

The wave vector components ka = 2πna/V1/3 involve integer na = 1, 2, · · ·∞,

(a = 1, 2, 3). The εkλ with λ = 1, 2 are polarisation vectors.The Akλ and Bkλ are independent random Gaussian variables with average

zero and unit variance. This is an often used assumption for SED since the exactdistribution of these random variables is not know SED, but also a fixed Amplitudewith a random phase was used in past research (Pena and Cetto, 2005). The onlyrequirement is that for each (k, λ) term the average energy∫

Vd3r(

ϵ02⟨E2⟩+ 1

2µ0⟨B2⟩) = 1

2hωk exp(−ωkτc) (10)

becomes equal to the photon zero point energy combined with an exponential

cutoff. This proves that the average amplitude A0 =√

h2ϵ0ωkV

is well chosen.

3.2 The zero-point field: Numerical representation

Numerically calculating the above 3D sum over the k− values is presently impos-sible, even on a GPU cluster. But one nice property of using a sum of Gaussianvariables is that it can be changed with a simpler sum of Gaussian variables, aslong as the correlators of these sums remain the same. If these correlators re-main the same than the field will look similar at time t independent of the exactrepresentation of the Gaussian sum.

We adopt a uniform grid in ω-space with ∆ωn = 1/N with N ≫ 1, so that

ωn =n

N, (n = 1, 2, · · · ), N =

L

2π(11)

which corresponds to (n/N)ω0 in physical units. Next we assume for each n andfor each direction a = x, y, z, two independent Gaussian random variables Aa

n andBa

n, with average 0 and variance 1, and consider the 1d sum

E(t) =∞∑

n=0

√∆ωn ω3

n

πe−

12Z2α2ωn(−An cosωnt+Bn sinωnt). (12)

Its two-point correlation function reads

< Ei(t)Ej(0) >= δijCEE(t) = δij1

8πN4ℜ 3 + sinh2[(Z2α2 + it)/2N ]

sinh4[(Z2α2 + it)/2N ]. (13)


At fixed t it reproduces in the limit N → ∞ the autocorrelation function for theautocorrelator of the original field in equation 8 using Bohr units:

< Ei(t)Ej(0) >= δijCEE(t) = δij6

πℜ 1

(t− iZ2α2)4. (14)

Plotting both autocorellation functions shows that for for finite N , the discretiza-tion will be reliable for times up to t ≃ N . Ideally N is bigger than the totalsimulation time, which was (almost) satisfied for our first article.

In our second article we applied the same idea, but then for an position depen-dent autocorrelator. Due to the mathematical complexity we refer the interestedreader to our second article for the specifics. We however encountered the problemthat the alternative EM-field needed 16 times as much memory, which precluded usfrom satisfying this condition (t ≃ N) for a whole run. Though since the shape ofthe field changes as the electron evolves, we don’t think this necessarily invalidatesour results.

3.3 The zero-point field: Fixed vs moving cuttoffs

Numerically it is impossible to calculate a ZPF with a cutoff either at infinityor the Planck scale. Previous work (Cole and Zou 2004b [12]) proved that themost important frequencies for the electron’s dynamics lie around the Keplerianfrequency or a multiple of it (1 < n < 12). Let’s call these multiples of the Kep-lerian frequency ’harmonics’. SED simulations of the hydrogen atom in 2D (Coleand Zou 2003[11]) defined a small window of around 3 percent below and abovethe first harmonic. Since we had much more computational power we removed thewindow and instead defined a cutoff at a frequency of 1.5 harmonics at an energy ofE = −1.6. This corresponds to 52 harmonics at an energy of Eion = −0.05, whereEion is defined as the ionisation energy. We thus state that the electron ionisedif it stays for a time totally inconsistent with the quantummechanical predictionabove Eion

Since these simulations introduced noise in the sense that the 50th harmonic is2700 times stronger (sum of equation goes as n2) than the first harmonic, we alsoworked with a moving cutoff during this project. We did cutoff the frequency ateither 2.5, 4.5 or 6.5 harmonics. To prevent discontinuous jumps in the EM-fieldwe did change the cutoff if it differed more than 20 percent from the previouslyused cutoff frequency, which happens on the order of 102 electron orbits. Otherwisewe would introduce a lot of noise in the field strength due to adding or removingmodes every timestep.


4 Deriving the equations of SED

In this section we will derive the SED equations and introduce Bohr units. Wewill start with the non-relativistc, zero spin equations since they are simple andrepresent the physics correctly. We will also discuss the canonical equations ofmotion, implemented to reduce numerical errors. Later in this section we willderive the relativistic equations of motion with spin-orbit coupling, which wereimplemented to remedy the problem where the atom ionized, though unsuccesfull.

4.1 Deriving the equations of SED: The coulomb force

Let’s start with only the Coulomb force assuming a stationary nucleus with chargeZ:

Fc =−Ze2r4πϵ0r3

(15)

In this case we get an orbit according to Kepler’s laws with energy E , angularmomentum L and eccentricity ε:

E =p2

2m− Ze2

4πϵ0r, L = r× p, ε =

1

mp× L− Ze2

4πϵ0r. (16)

We can prove that the expression for the eccentricty vector reduces to a Kep-lerian orbit by noting that:

ε · r =1

mr · (p× L)− Ze2

4πϵ0r =

1

m(r × p) · L− Ze2

4πϵ0r =

1

mL2 − Ze2

4πε0r (17)

Rearranging gives the well known Keplerian orbit:

1

r=

Ze2m

4πϵ0L2(1 + εcos(θ)) (18)

Taking the dot product of ε gives the relationship between eccentricty, angularmomentum and energy:

ε · ε = ε2 = 1 + 2EL2

m(19)

4.2 Deriving the equations of SED: The Abraham-Lorenz force

We however know from the theory of electromagnetism that an accelerating chargedparticle will radiate energy through self interaction with its EM field making astable Keplerian orbit impossible. This force is called the Abraham-Lorentz forceFL. Since the light travel time between the accelerating charge and observer isnon-negligable, we introduce the concept of retarded time:

t′r = t− r − r′

c(20)


This is the time at which the photons contributing to the measured electromagneticfield were emitted. Then we define scalar and vector Lienard-Wiechert potentials:

ϕ(r, t) =1

4πϵ0

∫ρ(r′.t′r)

|r− r′| d3r′, A(r, t) =

µ0

4π

∫J(r′.t′r)

|r− r′| d3r′ (21)

Now calculate the electric and magnetic fields:

E = −∇ϕ− dA

dt, B = ∇ ·A (22)

The Poynting flux vector becomes:

S =E×B

4π(23)

The power per unit area becomes along direction n:

dP

dΩ= (S · n)R2 (24)

With total power for the accelerating charge given by:

P =

∫dP

dΩdΩ =

µ0q2

6πca2 (25)

To get the Abraham-Lorentz force from the expression for an accelerating chargewe need to realize that:

W =

∫Pdt =

∫F · dt =

∫µ0q

2

6πc

dv

dt

dv

dtdt (26)

Assuming periodic motion of period T and integrating by parts gives:

W =µ0q

2

6πca · u|T0 +

∫ T

0

µ0q2

6πc

d2u

dt2· u = 0 +

∫ T

0

µ0q2

6πc

d2u

dt2· u (27)

Thus we get for FL:

FL =µ0q

2

6πc

da

dt(28)

4.3 Deriving the equations of SED: Final equation of motion

Adding the electric and magnetic force is trivial, which yields the final SED equa-tion of motion:

F = Fc + FE + FB = − Ze2r

4πϵ0|r|3+µ0q

2

6πc

da

dt− e(E+ u×B) (29)

Since solving for a would incur extra numerical noise we iterated the equationonce, so we only have to solve a second order differential equation. This is validsince the damping term is much smaller than the Coulomb term and the Lorentzterm. In most of the runs we did take this even a step further by only iteratingthe Coulomb term since it is far stronger than the Lorentz-term itself:

a =Fc

m(30)


4.4 Deriving the equations of SED: Switching to Bohr units

Working in SI units is not the way forward, since it leads to values much smaller orbigger than 1. For that we switch to Bohr units, where a0 gives the orbital radiusof a classic hydrogen atom in the ground state and orbitial period P0 = 2πτ0:

a0 =h

Zαmc, τ0 =

1

ω0=

h

Z2α2mc2, ϵ0 =

1

4π(31)

The equation of motions reads for these units:

r = − r

r3+ β2...r − β[E(Zαr, t) + Zαr×B(Zαr, t)], (32)

Using this form we see as well that the B-field and positional dependence of theE/B-field get suppressed by a factor α and are thus not very relevant. Furthermoreboth the fluctuations and the damping involve the small parameter: 1

β =

√2

3Zα3/2 =

Z

1964.71, α =

e2

4πϵ0hc≈ 1

137(33)

This parameter β sets the timescale of the simulation. We call 1β2 the damping

time, which is the timescale on which the electron falls into the nucleus. ChosingZ > 1 makes the electric and magnetic forces linearly stronger, while reducing thedamping time in units τ0 by β2. This implies that we can increase Z to make thesimulation reach n damping times with less Keplerian orbits, thus saving computa-tional time, which can be used to for example increase the precision and timespanof our simulations.

4.5 Deriving the equations of SED: Canonical equations of motion

We did solve standard Newtonian second order equation of motion numerically inmost of our runs. However, we did also built a code that solved the non-relativisticcanonical problem derived from Hamilton’s rule to give us extra confidence in ourresults:

p = f(r), q = p+ βAs + β2f(r), r = q+ βCg (34)

This implies a physical momentum of r = p + β(As + Cg) + β2f and reducescorrectly to the second order ordinary differential equation (ODE).The EM-fields were defined as:

A = As + Cg, (35)

As =N1∑n=1

√n

πN2(An sin

nt

N+Bn cos

nt

N), (36)

Cg =∞∑

n=N1+1

√n

πN2(An sin

nt

N+Bn cos

nt

N), (37)

(38)

Cg =∞∑

n=N1+1

√1

πn(−An cos

nt

N+Bn sin

nt

N) (39)

1 In order to have β also as prefactor of E, we absorb a factor√

3/2 in A, B and E.


This approach was chosen because for large ωn, the coefficients of the E field grows

as ω3/2n , which may cause numerical errors, especially if we define a cutoff quite

high. In that case the first harmonic, that contributes most to orbital resonances,can be 1000 times weaker than for example the 35th harmonic. This could causea small interpolation error (main source of error in the field strength) in the 35thharmonic to influence our results (see discussion). Thus we may want to integrate

the E field field twice numerically, because the resulting C field grow as ω−1/2n

and thus the higher modes get well behaved. But now the low frequency modesget too strong, thus we did chose for the low frequency modes up to N1 to justintegrate once by using the A field.

The present scheme is valid only whenN1 andN2 stay constant. We do howeverupdate N1 and N2 as the orbit of the electron changes. Let us assume that oursimulation covers nh + 1

2 harmonics of the orbit, with nh = 2 or 4, or · · · . At theinitial time we set N2 = (nh + 1

2 )k3N , next to N1 = k3N .

At some later time t′ where k has evolved to some k′ we may wish to update notonly N1 but also N2, to become N ′

1 = k′3N and N ′2 = (nh + 1

2 )k′3N . This change

is also covered in the above formulae, where now C involves limits N1 +1 and N2

before t′ while the update C′ involves limits N ′1 + 1 and N ′

2 after t′. Likewise, Ainvolves limits 1 and N1, and A′ involves limits 1 and N ′

1.All by all, this leads to discontinuous jumps in the position r and angular

momentum p. To compenstate for this we introduce δA and δC in the dynamics:

p = f(r) , q(t) = p+ β2f + β[A′(t) + δA] ,

r(t) = q(t) + β[C′(t) + δC] f(r) = − r

r3(40)

In the initial period, one just has δA = δC = 0. However after the first changeof N1 and N2 one works with the updates A′ and C′, which involve N ′

1 and N ′2,

rather than N1 and N2, respectively. Matching at t′ yields

δA = A(t′)−A′(t′) + C(t′)− C′(t′),

δC = C(t′)−C′(t′) (41)

For subsequent changes of N1, N2 one repeats this schedule. One must add thenew shifts to the previous ones, which amounts in total to:

δA =∑t′<t

[A(t′)−A′(t′) + C(t′)− C′(t′)]

δC =∑t′<t

[C(t′)−C′(t′)]. (42)

Note that this scheme, like the Newtonian scheme, still introduces discontin-uous jumps during field switchover in acceleration a. Nevertheless this schemegave us some extra confidence in our results, that they are robust when the highharmonics (up to 52th harmonic) introduce an error similar to 20 percent of the1st harmonic’s amplitude. Since this approach didn’t lead to better results forthe non-relativistic case, we decided that we didn’t need this extra ’test’ for therelativistic version (second paper).


5 Relativistic corrections in the hydrogen problem

We implemented the previous equation of motion in our first paper and achievednegative results for the stability of the hydrogen atom. We found it ionised in a fewthousand to a few million orbits depending on the zero-point field. We encounteredthe problem that the orbits of the electron became wider and wider with higherand higher eccentricities. This is the same mechanism by which asteroid can beejected by the orbital resonances due to the sun and Jupiter from our solar system.However while in classical mechanics the electron can have aribtrary low angularmomentum (corresponding to high eccentricities), this is not the case for specialrelativity, since this could imply that the speed at the perihelion would exceed thespeed of light (T Boyer 2004).

Thus the next step for this project was the inclusion of leading order relativisticterms that go as 1/c2, hence as α2, with α = 1/137 the fine-structure constant andare thus very weak, but may exert a force close to the nucleus, that could preventthe system from ionising. To derive these equations we follow a slightly differentapproach and start from the Schrodinger equation. The spin-orbit interaction readsfor an electron of mass m and charge −e in the field of a nucleus with charge Ze(e > 0), with Bohr magneton µB = eh/2m in SI units:

HSO =µB

hmec2r

dV

drL·S =

Ze2

8πϵ0m2c2 r3L·S (43)

The relativistic Hamiltonian including the Darwin term reads:

Hrel =√m2c4 + p2c2 − Ze2

4πϵ0r+

Ze2

8πϵ0m2c2L·Sr3

+h2

8m2c24π

Ze2

4πϵ0δ(r) (44)

Bohr units allow to introduce dimensionless variables r → a0r and p → (ma0/τ0)p,implying that L = r × p → hL, and to take consistently S → hS. Keeping

|S| =√

12 (

12 + 1) = 1

2

√3 one arrives at:

Hrel

mc2=√1 + Z2α2p2 − Z2α2

r+Z4α4

2

L·Sr3

+ Z4α4 π

2δ(r) (45)

So the dimensionless nonrelativistic HamiltonianH = (Hrel−mc2)/Z2α2mc2 picksup the leading relativistic corrections:

H =1

2p2 − 1

r− 1

8Z2α2p4 +

Z2α2

2

L·Sr3

+ Z2α2 π

2δ(r). (46)

This leads to corrections to the energy of order α2, which is of the same order asthe positional dependence of the E-field or the effect of the magnetic field B.


From Hammilton’s equations r = ∂pH and p = −∂rH the dynamics reads forr = 0:

r = (1− Z2α2

2p2)p+

Z2α2

2

S× r

r3, (47)

p = − r

r3+Z2α2

2

S× p

r3+

3Z2α2

2

L · Sr5

r, (48)

Note that the Darwin term describing the zitterbewegung of the electron closeto the nucleus disappeared in this treatment, which can’t properly handle a δ(r)term.

The spin progresses approximately as:

S =Z2α2

2

L× S

r3=Z2α2

2

(pr− rp)·Sr3

. (49)

It is trivial to see that the spin S is conserved and we verified that the total angularmomentum J = L+ S is conserved up to terms of order α4.

An external electromagnetic field is added by the minimal substitution p →p = p+βA(Zαr, t), with β given below and the spatial scale factor Zα expressingthe ratio of the Bohr radius to the wavelength of a photon with Bohr energy α2mc2.This represents a renormalization of the rest mass δmec

2 in quantum mechanics.From Eq. (49) it is confirmed that S2 remains conserved, as desired.

Stochastic Electrodynamics leads to a specific stochastic electric and magneticfield, as well as to an

...r damping term, see NL1 and Ref. [1]. When we neglect

terms of order α7/2 higher, the contribution ∇ip to (48) and the time derivativeof (47) lead to the Abraham-Lorentz or Brafford-Marshall equation for a particlewith spin:

r = − r

r3− β(E+ r×B) + β2...r + Z2α2 r

2r+ 2r·r r

2r3

+Z2α2

2r3(S× r− 3

r· rr2

S× r+ 3S · Lr2

r ), (50)


6 Conjecture for the ground state phase space density

For a dynamics with weak EM-noise and damping the stationary distributionin phase space must be a function of the conserved quantities, since only thesedetermine it’s dynamics.. Here these are the parameters E , L and/or ε.

A conjecture for the phase space density of several states of the relativisticH-atom has been made in a previous paper[16]. This was done by requiring thatmomentum space integral of the phase space density Ppr(r,p) reduces to the 1sground state.

Here we restrict ourselves to the ground state in the non-relativistic limit, whilethis conjecture is also valid for higher states and consistent with the relativisticHammiltonian discussed in the previous chapter. Nevertheless, the conjecture re-duces to:

Ppr(r,p) = f(E(r,p), L(r,p)) (51)

f(E , L) = 2Le2/E

π3|E|3 Φ(E) =2

π3LR3e−2RΦ(E) (52)

Here we included a relativistic correction factor Φ(E), which takes the value 1 inthe non-relativistic limit and an inverse energy R = − 1

E for notational convenience.The first task is to verify that the ground state density emerges after integrating

over momenta. At given r one can take the pz-axis along r, so that:

p = p(sinµ cos ν, sinµ sin ν, cosµ),

√p = 2E +

1

r=

√2(R− r)

rR, (53)

with r ≤ R ≤ ∞, 0 ≤ µ ≤ π, 0 ≤ ν ≤ 2π. The volume element reads

d3p = dpdµdν p2 sinµ = dRdµdν

√2(R− r)

rR5sinµ. (54)

Since L = pr sinµ, Eq. (51) indeed reproduces the QM result, viz.

Pr(r) =

∫d3pPpr(r,p) =

4

π

∫ ∞

rdR(R− r)e−2R =

e−2r

π. (55)

This can indeed be written as:

Pr(r) = ψ20(r)Y

200(θ, ϕ), ψ0(r) = 2e−r, Y00(θ, ϕ) =

1√4π, (56)

and leads to Pr(r) = r2ψ20(r) = 4r2e−2r with normalisation

∫∞0 dr Pr(r) = 1.

For PEL(E , L) we have the definition

PEL(E , L) =∫

d3r

∫d3p δ(E − E)δ(L− L)Ppr(p, r)

= 4πf(E , L)∫

dr r2∫ 2π

0dν

∫ π

0dµ

√2(R− r)

rR

sinµ δ

(r

√2(R− r)

rRsinµ− L

)(57)


Hence, taking into account the contributions from µ = µ < 12π and from µ = π−µ,

PEL(E , L) = 16π2f(E , L)∫ r+

r−

drrL√R/2√

rR− r2 − 12L

2R

Expressing κ = kL, that lies between 0 and 1, as

κ =L

Lmax=

L√R/2

= kL =√1− ε2, (58)

and using that r± = 12R(1± ε), this reduces to

PEL(E , L) = 8√2L2

|E|9/2e−2/|E| , (59)

where L ≤ Lmax. Because the latter depends on E , the result does not factorize.However, since both ε and κ lie between 0 and 1, the weight PEL(E , L)dEdL canbe factored in the forms PE(E)dE Pε(ε)dε and PE(E)dE Pκ(κ)dκ, where

PE(E) =4

3|E|6 e−2/|E| , (−∞ < E < 0),

Pε(ε) = 3ε√1− ε2, (0 ≤ ε < 1), (60)

Pκ(κ) = 3κ2, (0 < κ ≤ 1).

An often made mistake is to take some value out of the conjecture as initialconditions for the simulation. This will lead to a convolution with the conjectureif the simulation didn’t run long enough, which is often hard to tell by lookingmerely at the data. Fluctuations in radius than may for example suggest a stablesolutions in the quantum regime while fluctuations in for example eccentricty occuron longer timescales.

This was exactly the problem with simulations of Cole and Zou 2003, sincethey claimed a stable ’quantum’ solution for long run times. However they summedmultiple runs with the same initial conditions (a circular orbit at E = −0.5), whichled them to miss variations in for example angular momentum occuring on longertimescales and eventually miss the fact that the atom ionised and in fact doesn’teven form a ’quantum solution’ on shorter timescales.

For our initial non-relativistic run we chose reasonable values out of the con-jecture, while for the relativistic results we chose values, which were around 2σ offfrom the conjecture. Both led to similar results showing that we reached a stablesolution up until ionisation (more on this later).


7 Numerics

7.1 Numerics: Limitations of previous simulations

Authors like Cole and Zou tried to solve the hydrogen problem for SED in 2003using simulations. These simulations though were on some points flawed due to thefact that lack off computational forced them to make some unphysical assumptions.

Firstly due to the available computational power at that time they were onlyable to simulate the electron for short times, which is flawed, since angular mo-mentum evolves on longer timescales. Instead they summed 11 runs with the sameinitial conditions, giving nothing more than a random fluctuation around the initalconditions. This accidently reduced to something similar to an 1s state since thesame initial condictions were chosen every run. The difference with our simulationsis that they found circular orbits, while we found eccentricites even above ϵ > 0.9

Secondly they also had to assume a tight window about the n = 1 harmonic,which probably led to these circular orbits with maximum angular momentum. Byimplementing a tight window abot the n = 1 harmonic, we did reproduce theseresults.

Finally they simulated the problem in 2D which has a different radius distri-butions than our conjecture:

2D : P (r) =16

2πexp(−4r) 3D : P (r) =

1

πexp(−2r) (61)

However they used the 3D result for their ’semi’-2D simulations, which probablyis invalid!

7.2 Numerics: Our code

Our simulation code was developed from scratch in C/C++, while the data analy-sis part was trivially implemented in Mathematica. The summation of the field wasdone on a GPU. This is described in the next section. Furthermore the code wasthoroughly hardware profiled, to extract maximum performance on the CPU/GPUand convergence testing was performed to guarantee valid results. It is able to sim-ulate a system with 107 modes in a reasonable amount of time on a single 250 euroGPU and any CPU (is not the bottleneck).

Here we give a pointwise summary of the main features/caveats of our self-developed code:

1) The considered frequencies are ωn = n/N . Due to the presence of a vastamount of random variables we had to set N = 105 instead of N = 106 as wasdone in our non-relativistc version of the code [12]. Up to t = N the autocorrelatorof equation (13) reduces to the autocorrelator of the exact E field in equation(14). For larger t, we may thus not simulate a genuine 3D problem anymore inthe relativistic case. This could be a problem since it is shorter than the dampingtime 1/β2 and possibly also shorter than other timescales relavant to the electron’sdynamics.

2) We used the ’classical’ Runge Kutta fourth order integration scheme (RK4)with full energy and angular momentum conservation up to eccentricities of 0.99and higher. We did also test other ODE schemes like several of the Adam-Bashforth


type and the simpler Euler method. Though still very accurate they were lessaccurate than the RK4 method and thus disbanded. On top of that we did vary thenumber of iterations per orbit from run to run and achieved consistent results. Thuswe can conclude that our results are ODE scheme independent, with the biggestnumerical error induced due to the other points mentioned in this paragraph.

3) We used 4000 iterations per orbit, while 600-2000 where found to be enoughfor high eccentricities using the RK4 method. This was done because solving theODE was not the computational bottleneck and we wanted to make sure to havethe maximum achievable precision. The EM-field (bottleneck), though, was onlyupdated 25-30 times per period of the highest EM-mode with the other pointsdetermined by interpolation using 4th order Lagrange Polynomials. We calculatedthis EM-field 1.5 to 2 times as often with respect to the non relativistic version ofthe code [12], since higher numerical precision was required so to assure that theeffects of the relativistic corrections, the magnetic field, the positional dependenceof the EM-field and the spin-orbit coupling were fully accounted for.

4) We updated the moving cut-off frequency when the orbital period of theelectron changed more than 20%. This was done to minimise the presence of dis-continuities in the equation of motion (9). Note that this field switching everey102 orbits could still induce errors in our solution.

5) Since the electron’s energy can drop below E = −1.6 at which point ourGPU runs out of memory and/or the code becomes very slow due to presence ofmany more modes than at higher energies, we artificially increase the electron’senergy by giving it a ’push’. For this code we chose a scheme in which we randomlygave the electron a push either parallel or perpendicular to its velocity axis, whichshould give less bias towards higer angular momenta than a previous version ofour code where we only gave it a push parallel to it’s velocity axis. The advantageof this push scheme is also that its very simple, while the results are robust againstthe exact implementation. This push scheme shouldn’t influence our final results,since according to the conjecture of previous section the electron should stay outof this regime 99%+ of time. We also tested if the electron recovers from very lowenergies like E = −4.0, and it indeed seems so.

6) The code does the first 8 summation steps of the EM-field in single precisionon the GPU. The rest of the summation steps as well as solving the second orderNewtonian ODE are done in double precision. Using single precision in this firstsummation step can speed up the code by a factor 2, while the error in this EM-field remains dominated by interpolation noise. Thus this so called hybrid-precisionscheme works really well.


8 Computation

8.1 Computation: CPUs not becoming faster

We tried to improve on the previously discussed simulations more than a decadelater. The only problem was that since approximately 2009 single core non-vectorizedperformance of CPUs didn’t scale up as predicted by Moore’s law. The problem isthat CPUs have a clock frequency, which is fixed around 3-4 GHz for more thana decade. During each clock cycle most CPUs can handle at most two addition(ADD) or multiply (MUL) instructions, at least if these intructions are indepen-dent of each other. This is called instruction level parallelism and can be exploitedin practice with differing succes per code. Our code performs relatively well in thatregard.

The first explicit optimisation we implemented was to include multiple coresusing OpenMP. This managed to speedup the code by almost a factor 4 on aquad-core CPU, which is a very good number.

Another approach to speed up the code we considered was to use vectorizeddatatypes (float4 ,double4, int4), since modern CPUs can execute for examplea double4 (double x, double y, double z, double w) in a single step, giving atheoretical speedup by a factor 4. However writing vector code is difficult, since assoon as only one component of a double4 needs to be manipulated, all componentshave to be copied back from the vector registers, acted on by the vector ’SIMD’units, to scalar registers, acted upon by the aritmetric logical and floating pointunits. In practice we only managed to get a modest speedup of less than 30 percentdue to vectorization of the code.

Thus it seemed that having long run times and 3D, even with the smart dis-cretization was impossible on CPUs, since the frequency of CPUs is not expectedto increase untill semiconductor manufacturers start to move away from silicon.

8.2 Computation: The tremendous speed of GPUs

However since approximately 2008 graphics card manufacturers also brought theirGPUs to to the professional market. Besides graphics rendering in single precision,modern GPUs can also do double precision scientific calculations. Simply said theyare much faster because they trade control and cache area on their dies for ALU(arithmetic logic unit) and FPU (floating point unit) space.

To discuss how GPUs deliver such amazing performance for parallel algorithmsof more than two orders of magntiude vs a single CPU core we first have to considerwhat determines the speed of CPU/GPU assuming sufficient memory bandwidth.Actually there are two factors: Instruction throughput and instruction latency

Instruction throughput determines the theoretically maximum number of acertain instruction executed per second. The famous FLOP for example showshow many floating point operations a CPU/GPU can execute per second. Forthis the benchmark usually used MAD instructions that do a MULL and ADDinstruction in a single cycle through some clever tricks. FMA instructions occur alot in codes and are thus a good metric.


Fig. 2 a): GPUs trade control flow and cache area for ALUs and FPUs. This makes themmuch faster in a heavily parallised environment.

The second factor determining CPU/GPU performance is instruction latency. Ifa CPU for example can read 100 values from the LLC (last level/L1 data cache)with a latency of 12 cycles it can only reach its maximum intruction throughput ifall instructions are independent. If the first instruction depends on the second, thesecond depends on the third etc, the CPU will only achieve 1/12th of its statedperformance. There are two remedies for this, clever flow control and thread levelparallelism (TLP).

CPUs usually choose for clever flow control, which includes features such asbranch prediction, where a CPU tries to guess the branch taken in an if-else state-ment and instruction reordering, where a CPU looks for independent instructionsin a code with the intent of changing the order in which instructions are executed.This flow control costs lots of die area. On top of this CPUs have a large part ofthe die area reserved for the L3 cache which can be seen as a low latency RAM,since it has a factor 20 less latency. It is allways a challenge for the programmerfor making sure that all relevant data is loaded in this cache and for the CPUmanufactures to make the cache big enough.

GPUs however chose a different approach for latency handling. They havea modest sized register, which is the fastest memory available (only 4 cycles oflatency). The registers file can often accomodate (except in very complex code)multiple processes. The idea of this mutli-threading/thread level parallism is thatthe GPU can run multiple threads in parallel. If thread A is waiting at for examplea memory read of 4 cycles to complete, the other threads can do calculationd duringthat time.

This makes GPUs massive parallel systems, since they are built up out of3000 of stream processors running each up to 40 threads. Each of these streamprocessors is much weaker than a single CPU core (usually 2 to 4 cores per CPU),but taken together they are tens to hundreds of times faster than a CPU. Toaccomodate this performance also the memory bus and speed is higher than fora normal CPU. In total GPUs are on a per core basis much slower than CPUcores, but as a whole much faster faster in both performance as well as memorybandwidth.

On paper the speedup is a factor 25-100 (GPUs have a smaller double to singleprecision performance ratio), but sometimes much higher speedups are attainablesince GPUs don’t rely on difficult to achieve vectorization or intruction level par-allelism for their performance. Instead they have scalar registers and use threadlevel parallelism to achieve their tremendous speedups. Also their (and OpenCL’s)


clever usage of the caches and registers can reduce the requested memory band-width to speed up memory bandwidth limited codes.

So a program has to be able to run on 120000 independent threads to exploit thefull power of a modern GPU. This is three orders more than required for a modernCPU with 4 cores, which can only run up to 8 threads with HyperThreading (HT)enabled.

To run 10 threads per stream processors the program has to optmize the reg-ister file usage in its code, otherwise less threads can be run and the full power ofa GPU may not be exploited. This entails in practice to write an efficient code,splitting up kernels, reusing variables and making good use of the memory hier-archy in GPUs. In our case we managed to achieve an occupancy of 60 percent,which can be called pretty good.

8.3 Computation: Implementing our algorithm on a GPU

OpenCL is an extension to C++ that makes it possible to parallelise the sum-mation in equation (12) on a GPU (GPUs don’t support normal C/C++ code).Normally the summation of the modes is performed within for loops, where allelements are summed serially on a single CPU.

In the OpenCL paradigm our GPU is called a compute device. Since we possessa single GPU, we utilise only one compute device. This compute device posseses44 compute units, which are subdivided over 4 16-wide SIMDs (Single instruction,multiple data). Thus there are 44 × 4 × 16 = 2816 processing elements. Each ofthese compute units can process up to 40 wavefronts of GPU specific size 64 simul-taneously, only limited by the register (GPR) and local memory size (processingmultiple workgroups of data is done to hide memory latency).

The sum in equation 12 is then summed by all of these processing elementsusing parallel reduction. The global work size is defined by the total number ofelements to sum. These elements are summed in workgroups of size 256, i.e., 4times the wavefront size for an AMD GCN (Graphics core next) GPU. Theseworkgroups share local memory, such that every work item reads in its value fromthe global memory and copies it to the local memory, where it is summed in 8steps (28 = 256) within a workgroup. During the first step the first 128 work itemsare summed with the last 128 work items in pairs of two. In this way the sum isreduced by a factor of 2 each step. This reduces the total sum by a factor of 256,after which the remainder is copied to pinned RAM memory via a PCI Expressbus and summed by the CPU using double precision.

8.4 Computation: Used system and performance

It’s an annoying fact that people in science often don’t state the speed of theircodes or the specs of the hardware used. This makes it very difficult to judgethe feasibility of performing the same or similar simulations in the future usinghardware available to them.

We ran the simulations on a state of the art PC, consisting out of an IntelCore i7 2600k overclocked to 4.6 GHz (Core i7 4770k+ equivalent in performance),together with 16 GB of RAM. In our earlier simulations we used an AMD HD6970


GPU, which was later upgraded to an AMD R9-290X GPU. This GPU delivers5.6 TFLOPS of single precision floating point performance and 350 Gigabytes persecond of memory bandwidth.

Overall, the performance improvement amounts to a factor of 270 againsta double precision vectorized single core C++ implementation using the IntelC/C++ compiler, or a factor 160 against a vectorized single core CPU IntelOpenCL hybrid precision implementation. We compare in this case against a 2.8GHz Intel Sandy Bridge CPU, since these are widely used in CPU clusters. Thisallowed us to simulate on the order of 107 modes in real time for several millionof orbits in a few days, allowing us to tackle the problem in 3D without the previ-ously mentioned deficiencies of previous simulations (Cole&Zou 2003). Note thatthese simulations sometimes ran for several days. Getting so much time on a CPUcluster would have been difficult for SED simulations.

Using a GPU performance profiler we found out that the code is memorybandwidth limited, despite clever usage of the local memory and efficient code. Itreaches 70 percent of it’s peak memory bandwidth while the instruction through-put stays at a modest 13 percent of it’s peak value, which makes an total utili-sation factor of around 80 percent, suggesting together with an occupancy of 60percent that the high level of parallelism of the GPU is well used. This further-more suggests an average performane of around 350-370 GFLOP/s and a memorybandwidth (excluding register, local memory and caches) of 245 GB/s.


9 Results

During this project we conducted more than 25 simulations, ranging from simple3D runs to fully relativistic 3D runs with spin dependence. Nevertheless all runsgave similar distributions, so we only present the results for the moving and fixedcutoff cases. The longer runs took 50 to 200 GPU hours to complete.

9.1 Results: Moving cutoff

We ran the code several times using different values for the cut-off frequency. Insome runs we chose Z = 1 as proton charge, while in other Z = 3 was chosento reduce the number of orbits per damping time. However no difference in thedistributions for different Z was found, only in the ionisation time of the electron.

Here we present the results of our most promising run up to the point thatthe electron energy went above E=-0.05, what we define as ionisation. The movingcutoff was set at 2.5 times the electron’s Keplerian frequency. In figures 2a, 2b weshow the time series for the energy and radius while in figures 3a, 3b we show thecorresponding probability distributions compared to the 1s ground state and theearlier published conjecture [13]. In table 1 we give the run time of this simulationbefore ionisation and we compare it to the damping timescale of 1/β2. In table 2we do this for simulations of Cole and Zou 2003 as comparison.

These results suggested that the electron remained stable and had a probabilitydistribution closer to the conjecture [12, 13] for a longer time than was presented inour first article [12]. However, we did find out that the time up until the electron’sionisation varied quite drastically between t = 106t0 to t = 107t0 for different runsusing either the relativistic or (non-)relativistic code. Thus we can conclude thatthere is no statistically significant difference between this run and the idealisedrun in our first article [12]. Runs with the moving cutoff set at 4.5 or 6.5 times theorbital frequency showed ionisation on even shorter timescales (105 − 106t0).

We furthermore observed a clear lack of low angular momenta/high eccentricty.In the next chapter we *try* to give an explanation for this behaviour in the non-relativistic limit.

property value duration (s)

ttotal 2.05 107 t0 5.5 10−11 s

tdamp 4.28 105 t0 1.15 10−12 s

Norbit 3.26 106

Ndamp 48

Table 1: Time and number of orbits for our simulation with 2.5 harmonics and Z = 3.

property value duration (s)

ttotal 11× 6.21 105t0 11 ∗ 1.5 10−11 s

tdamp 3.8 106t0 9.36 10−11 s

Norbit 11× 1.0 105

Ndamp 11× 0.16

Table 2: Duration and number of orbits for the Cole-Zou simulation with Z = 1and a 5% window around the first harmonic in 2D. The factor 11 is the numberof different runs.


5.0´106 1.0´107 1.5´107 2.0´107 t

0.5

1.0

1.5

ÈEÈ

5.0´106 1.0´107 1.5´107 2.0´107 t

2

4

6

8

r

Fig. 3 a): Time series for the energy, in Bohr units. b): Time series for the radius.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 ÈEÈ

0.5

1.0

1.5

2.0

2.5

P

1 2 3 4 5 r

0.1

0.2

0.3

0.4

0.5

P

Fig. 4 a): Histogram for the energy data of Fig. 1a. The red curve is the conjecture. b):Histogram for the radius data of Fig. 1b. The red curve is the conjecture.

0.5 1.0 1.5 2.0L0.2

0.4

0.6

0.8

1.0

1.2

1.4

P

0.2 0.4 0.6 0.8 1.0¶

0.5

1.0

1.5

2.0

2.5

3.0

P

Fig. 5 a): Histogram for the angular momentum data. We see a clear discrepancy at lowerL with the conjecture given in red. b): Histogram for the eccentricity data. We see a cleardiscrepancy at high ϵ with the conjecture given in red

9.2 Results: Fixed cutoff

Multiple runs with a fixed cutoff set at 1.5 times the Kepler frequency for E =−1.6 did even show ionisation within t = 104t0. We see in figure 7 an increasingeccentricty until the electron picks up so much energy that it ionizes (figure 6).

Though for these fixed cut-off runs the accuracy of the interpolation scheme(limiting factor) may be brought into question, especially when we want to accountfor the higher order effects discussed in this article. This is because for high electronenergies very strong EM-modes are included, the frequency of which is 52 times theKeplerian frequency of the electron, while we expect only the first few ‘harmonics’to have a significant contribution [11, 12]. The integrated strength of all modes(assuming a Gaussian sum) of all modes till the first harmonic is for example morethan 2700 times weaker than the integrated strength of all modes till n = 52. The


50 000 100 000 150 000 200 000 250 000t

0.2

0.4

0.6

0.8

ÈEÈ

Fig. 6 Energy as function of time for Z = 1 with a fixed cutoff exposing the trend towardsionisation at E = 0, ε = 1.

50 000 100 000 150 000 200 000 250 000t

0.2

0.4

0.6

0.8

¶

Fig. 7 Eccentricity as function of time for Z = 1 with a fixed cutoff exposing the trendtowards ionisation at E = 0, ε = 1.

numerical error in the value of the EM-field was measured by comparing the outputof the CPU code with the GPU calculated and Lagrange interpolated EM-fields. Itwas found to be as large as 20 percent of the first harmonic between interpolationpoints. It is quite difficult to argue that this noise averages out over multipleorbits, especially if we want to properly take into account the previously discussedrelativistic effects. However the fact that the ionisation times become shorter withmore and more harmonics (when the numerical error is quite small) does suggesta real physical and not numerical ionisation problem.

We tried to suppress this numerical error first with exponential cutoffs rangingfrom value 1 at the lowest frequency to value 0.1 at the cuttoff frequency. Thisdidn’t help so we tried switching toward the canonical formulation of the dynamics,which also didn’t improve our results.


10 Other work

10.1 Other work: Theoretical prediction about ionisation

Thus we did observe ionisation at high eccentricites in simulations, because some-how the damping term doesn’t compensate for the energy gained at high velocitiesclose to the nucleus. In the non-relativistic regime this is not totally unexpectedtheoretically and this can be shown using perturbation theory, where we directlyintegrate the equations of motion over one period to obtain the change in energy,angular momentum, eccentricty and position every orbit n analytically.

For this consider a stable Keplerian orbit with position vector r0(t), energyE0(t) and (scalar)angular momentum L0(t) perturbed by field E. The dynamicsbecome up to first order:

r(t) = r0(t) + βr1(t), E = E0(t) + βE1(t) L = L0(t) + βL1(t) (62)

The perturbed variables for the energy E1 and angular momentum L1 get updatedin the following way in the presence of an E-field:

E1(t) = Epar r + EperpL0(t)

rL1(t) = Eperpr (63)

The same can be done for the radiation damping. After several pages of algebraoutside the scope of this thesis we arrive at the following results after approximat-ing for small energy and high eccentricities:

<dEn>=<

dEfieldn

> + <dEradn

>= 3πβ2 f(0)− L

L6(64)

<dL

n>=<

dLfield

n> + <

dLrad

n>= −2πβ2 g(0) + L

L3(65)

The detailed derivation will be laid out in an upcomming paper [TMN, to ap-pear]. We can see that in this regime there is a possibility, independent of theelectron’s total energy E , that the angular momentum can shrink, while the en-ergy keeps growing for L < f(0). This is exactly what we see in our simulationduring ionisation. It is counterintuitive though that this mechanism still works inspecial relativity, since special relativity predicts a minimum angular momentumof L > α

c (T. Boyer 2004), which should supress plunging orbits.

10.2 Other work: A protocol for the electron’s conserved quantities

We did also extend the treatment of the previous paragraph to a second orderaccurate protocol for the energy, angular momentum and eccentricity evolution ofthe electron. In that case we calculate seperately the perturbation of these valuesevery orbit for each mode and thus don’t have to solve the Newtonian secondorder equation of motion, which could make the calculation faster. After 20 pagesof math, we end up with update values for the conserved quantaties in form ofBessel functions. This works well for modes that are not a multiple of the Keplerianfrequency of the electron, but for those that are close to this orbital fequency, the


numerical error will become so big that our simulations didn’t converge to a stablesolution with increasing time resolution.

Thus we had to abandon this approach and switch back to the divergences freeNewtonian equation. The thing we also learned from it on the computational sideis that doing calculations analytically doesn’t speed up the code very much, evenif on paper the calculation is simpler. This is due to the fact that the number oflines of code will increase and thus the demand on GPU register space will explode.This will lead to less performance due to a lower achievable level of thread levelparallelism (less threads fit in register), almost completely offsetting the expectedperformance gain. Thus the lesson to be learned from this is that GPUs are mostefficient in crunching simple code. Letting them handle difficult code is possible,but more complicated.


11 Discussion

In chapters 1-5 of this thesis we discussed the nonrelativistic SED theory, whichformed the basis for our first paper [14], while in chapter 6 we discussed therelativistic corrections, which were implemented in our second paper [15]. Thenwe discussed in chapters 7 and 8 how we numerically implemented this problemin a super-fast GPU based numerical code.

While we definitely found a strong correlation with the hydrogen ground state,we did find too few orbits with high eccentricities/low angular momenta to fullyexplain the hydrogen 1s ground state. The inclusion of relativistic effects and/orpositional dependence of the EM-fiels had no measurable effect.

On top of this we also found that the electron ionises for a cutoff at n =1.5,2.5, 4 5 and 6.5 harmonics of the orbital frequency within 107 Bohr times. Theresults were even worse for a fixed cutoff on the zero point field. In that case theelectron didn’t manage to remain stable for more than 104 Bohr times. This isclearly inconsistent with reality, even the first few excited states live longer thanthat (order τ0 = α−3 in nature). This was theorized to be due to runaway energygain at low angular momenta/high eccentricities.

We can not say that this is the end of SED in general. We for example didn’tinclude the possibility that the electron is no point particle, but a spherical chargedistribution with its Compton length as radius. This is because the compton lengthis only a factor 3-5 times smaller than the perihelia encountered before ionisationsets in. This charge distribution could smear out near the nucleus, so as to dissipateits energy on the hydrogen nucleus. This would enhance the damping, possiblyenough to compensate for the energy gain close to the nucleus, that is theorizedto cause the observed ionisation.

All in all, we feel that we have established the status of the present formulationof SED for the H atom.

12 Acknowledgements

Let me mention that this thesis has been the result of a continuing collaborationwith my supervisor dr. T.M. Nieuwenhuizen after a project in astronomy, whichresulted in a paper in 2012 [18]. It went from a request to integrate an ODE aftermy Bachelor to a full-fledged project featuring relativistic physics and high perfor-mance computing. While my primary interests and PhD topic lie in computationalastrophysics, I enjoyed a lot to work on this project and I am always open to newsuggestions regarding SED.

I also want to thank Erik van Heusden for the good discussions we had aboutSED.


References

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2. L de la Pena, A M Cetto and A Valdes Hernandez, The Emerging Quantum: The physicsbehind quantum mechanics, (Springer, Berlin, 2014).

3. Th M Nieuwenhuizen, Is the contextuality loophole fatal for the derivation of Bell inequal-ities? Found. Phys. 41, 580 (2010).

4. L de la Pena, A M Cetto and A Valdes-Hernandes 2012 Quantum behaviour derived as anessentially stochastic behaviour, Physica Scripta T 151 014008 (2012).

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10. H E Puthoff, Ground state of hydrogen as a zero-point-fluctuation-determined state, Phys.Rev. D 35, 3266 (1987).

11. D C Cole and Y Zou, Quantum mechanical ground state of hydrogen obtained from clas-sical electrodynamics, Physics Letters A 317, 14 (2003).

12. D C Cole and Y Zou, Simulation study ofaspects of the classical hydrogen atom interactingwith electromagnetic radiation, Journal of Scientific computing 20-3 (2004)

13. T Boyer, Unfamiliar trajectories for a relativistic particle in a Kepler or Coulomb potentialAm. J. Phys. 72, 992-997 (2004).

14. Th. M. Nieuwenhuizen and M. T. P. Liska, Simulation of the hydrogen ground state inStochastic Electrodynamics, (2015). Physica Scripta accepted 04-2015. arXiv:1502.06856.

15. Th. M. Nieuwenhuizen and M. T. P. Liska, Simulation of the Hydrogen Ground State inStochastic Electrodynamics-2: Inclusion of Relativistic Corrections , (2015). Foundations ofPhysics accepted 06-2015. arXiv:1502.06787.

16. T M Nieuwenhuizen, Classical phase space density for the relativistic hydrogen atom, AIPConf. Proc. 810, 198 (2006).

17. L de la Pena and A Jauregui, Stochastic electrodynamics for the free particle, J. Math.Phys. 24, 2751 (1983).

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numerical gpu simulations of the hydrogen 1s ground state ... · het nulpuntsveld moet dan genoeg...

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