Noise, Oscillators and Algebraic Randomness: From Noise in Communication Systems of Number Theory Lectures of a School Held in Chapelle des Bois, France, April 5–10, 1999

417 Pages·2000·2.72 MB·English

by  Michel Planat (auth.)

Lecture Notes in Physics
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Michel Planat (Ed.)
Noise, Oscillators
and Algebraic Randomness
From Noise in Communication Systems
to Number Theory
Lectures of a School Held in Chapelle des Bois,
France, April 5–10, 1999
1 3
Editor
MichelPlanat
LaboratoiredePhysiqueetMetrologie
desOscillateursduCNRS
32Avenuedel’Observatoire
25044Besanc¸onCedex,France
LibraryofCongressCataloging-in-PublicationData.
Noise,oscillators,andalgebraicrandomness:fromnoiseincommunicationsystemsto
numbertheory:lecturesofaschoolheldinChapelledesBois,France,April5-10,1999/
MichelPlanat(ed.).
p.cm.–(Lecturenotesinphysics,ISSN0075-8540;vol.550)
Includesbibliographicalreferences.
ISSN3540675728(alk.paper)
1.Electronicnoise–Mathematicalmodels–Congresses.2.Oscillators,
Electric–Congresses.3.Numericalanalysis–Congresses.4.Algebra–Congresses.5.
Telecommunication–Mathematics–Congresses.I.Planat,Michel,1951-II.Lecture
notesinphysics;550.
TK7867.5.N6272000
621.382’24–dc21
00-032966
ISSN0075-8450
ISBN3-540-67572-8Springer-VerlagBerlinHeidelbergNewYork
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Preface
This volume presents most contributions given at the School “Noise of fre-
quenciesinoscillatorsandthedynamicsofalgebraicnumbers”whichwasheld
in Chapelle des Bois, Jura (France) from 5 to 10 April 1999. The event was
madepossiblebythefullsupportofthethematicschoolprogramoftheCen-
tre National de la Recherche Scientifique in France.
Noise is ubiquitous in nature and in man-made systems. For example, noise
inoscillatorsperturbshightechnologydevicessuchastimestandardsordig-
ital communication systems. The understanding of its algebraic structure is
thus of vital importance to properly guide the human activity.
The book addresses these topics in three parts. Several aspects of classical
and quantum noise are covered in Part I, both from the viewpoint of quan-
tum physics and from the nonlinear or fractal viewpoint. Part II is mainly
concerned with noise in oscillating signals, that is phase or frequency noise
and 1/f noise. From a careful analysis of the experimental noise attached to
the carrier the usefulness of the number theoretical based method is demon-
strated.ThisviewisexpandedinPartIII,whichismathematicallyoriented.
In conclusion, the noise concept proved to be a very attractive one gathering
peoplefromatleastthreescientificcommunities:electronicengineering,the-
oretical physics and number theory. They represented an original mixture of
talentsandthepresenteditoracknowledgesallauthorsfortheirpatienceand
open-mindednessduringtheschool.Mostmanuscriptsarecomprehensibleto
a large audience and should allow readers to discover new bridges between
the fields. We ourselves have identified but a few.
ThemeetingwasfollowedbyasmallworkshopsponsoredbyM.Waldschmidt
attheInstitutHenriPoincar´einParison3and4December1999:‘Th´eoriedes
nombres,bruitdesfr´equencesett´el´ecommunications’.Thepurposeherewas
to emphasize the newly discovered link between the Riemann zeta function
and communication systems. Some papers and related material are available
at the address: http://www.archetypo.web66.com, a new URL site built by
Matthew Watkins and devoted to the relationship between prime number
theory and physics.
Besan¸con, April 2000 Michel Planat
Contents
Introduction
Michel Planat .................................................. 1
Mathemagics (A Tribute to L. Euler and R. Feynman)
Pierre Cartier .................................................. 6
Part I Classical and Quantum Noise
Thermal and Quantum Noise in Active Systems
Jean-Michel Courty, Francesca Grassia, Serge Reynaud .............. 71
Dipole at ν =1
V. Pasquier .................................................... 84
Stored Ion Manipulation Dynamics of Ion Cloud
and Quantum Jumps with Single Ions
Fernande Vedel ................................................ 107
1/f Fluctuations in Cosmic Ray
Extensive Air Showers
E. Faleiro, J.M.G. Go´mez ....................................... 125
Stochastic Resonance and the Benefit of Noise
in Nonlinear Systems
Franc¸ois Chapeau-Blondeau ...................................... 137
Time is Money
Marcel Ausloos, Nicolas Vandewalle, Kristinka Ivanova .............. 156
Part II Noise in Oscillators, 1/f Noise and Arithmetic
Oscillators and the Characterization
of Frequency Stability: an Introduction
Vincent Giordano, Enrico Rubiola................................. 175
Phase Noise Metrology
Enrico Rubiola, Vincent Giordano................................. 189
VIII Contents
Phonon Fine Structure in the 1/f Noise
of Metals, Semiconductors
and Semiconductor Devices
Mihai N. Mihaila ............................................... 216
The General Nature of Fundamental 1/f Noise
in Oscillators and in the High Technology Domain
Peter H. Handel ................................................ 232
1/f Frequency Noise in a Communication Receiver
and the Riemann Hypothesis
Michel Planat .................................................. 265
Detection of Chaos in the Noise of Electronic Oscillators by
Time Series Analysis Methods
C. Eckert, M. Planat ............................................ 288
Geometry and Dynamics of Numbers
Under Finite Resolution
Jacky Cresson, Jean-Nicolas D´enari´e ............................. 305
Diophantine Conditions
and Real or Complex Brjuno Functions
Pierre Moussa, Stefano Marmi ................................... 324
Part III Algebraic Randomness
Algebraic and Analytic Randomness
Jean-Paul Allouche.............................................. 345
From Symbolic Dynamics to a Digital Approach
K. Karamanos.................................................. 357
Algebraic Dynamics
and Transcendental Numbers
Michel Waldschmidt............................................. 372
Dynamics of Some Contracting Linear Functions Modulo 1
Yann Bugeaud, Jean-Pierre Conze ................................ 379
On the Modular Function
and Its Importance for Arithmetic
Paula B. Cohen................................................. 388
On Generalized Markoff Equations
and Their Interpretation
Serge Perrine................................................... 398
Introduction
Michel Planat (cid:1)
Laboratoire de Physique et M´etrologie des Oscillateurs du CNRS, 32 Avenue de
l’Observatoire, 25044 Besan¸con Cedex, France
In the classical realm the best known type of noise is thermal noise. Due to
thermal agitation, free electrons in a metallic conductor are moving around
continuously causing collisions with the atoms and a continuous exchange of
energybetweenthemodes.ThiswasfirstinvestigatedexperimentallybyJ.B.
Johnson and H. Nyquist in 1928. Nyquist’s theorem states that the power
spectraldensity(psd)ofvoltagefluctuationsthrougharesistorR attemper-
ature T is S (f) = 4kRT, with k the Boltzman’s constant. The quantum
V
approachisalsoaveryefficientwaytounderstandthelimitationintheaccu-
racy of measurements performed in thermal equilibrium. The paper by J.M.
Courty et al. examines the fluctuations which are expected in an operational
amplifier from a quantum network approach. The ultimate sensitivity of a
cold damped accelerometer designed for space applications is calculated as
well.
QuantumphysicsisalsousedtounderstandthequantumHalleffect,that
is the behaviour of charged interacting electrons in the presence of a strong
magnetic field. The indivisibility of the electron charge e may be demon-
strated from a measurement of the current power spectral density (psd)
S (f)=2eI whichisknownasshotnoise(Schottky,1918).Suchnoiseresults
I
from the random emission of electrons from the cathode to the anode in a
diode or a semiconductor. Similarly, partition noise is added to the measure-
mentwheneveracurrentisdistributedrandomlybetweentwoelectrodes.Its
net effect is to introduce an extra multiplicative factor in the relation for the
shot noise psd. Partition noise measurements in small size quantum conduc-
tors have recently revealed the existence of fractional charges ep (p and q
q
integers) associated to quasi particle tunneling states. The paper presented
attheschoolbyC.Glattliwaspublishedelsewhere(Phys.Rev.Lett.79,2526
(1997)andin“Topologicalaspectsoflowdimensionalsystems”,Proceedings
ofLesHouches1998SummerSchool).Thetheoryofthefractionalquantum
Hall effect is still very open to debate, as shown in the paper by V. Pasquier
whichadressestheproblemofbosonicparticlesinteractingrepulsivelyatthe
filling factor ν =1.
InhisearlystudyofthermalnoiseJ.B.Johnsonalsofoundalargeamount
of voltage noise S (f) ∼ KV2/f at low Fourier frequencies f. From many
V
experiments it was found that the noise intensity is inversely proportional
to the number of carriers N in the sample, that is K ∼ γ/N, with γ in
(cid:1) [email protected]
M.Planat(Ed.):LNP550,pp.1–5,2000.
(cid:1)c Springer-VerlagBerlinHeidelberg2000
2 Michel Planat
the range 10−3 to 10−8. This was generally attributed to different scatter-
ing mechanisms, by the crystal lattice or the impurities, leading to mobility
fluctuationsoftheelectrons.Thesefindingspointtoanonlinearoriginofthe
1/f noise.Finestructuresrevealingtheinteractionofelectronswithbulkand
surface phonons in several solid-state physical systems are given in Mihaila’s
paper in Part II. On the theoretical side, a quantum electrodynamical the-
ory was developed by P. Handel in the seventies based on infrared divergent
coupling of the electrons to the electromagnetic field in the scattering pro-
cess.Thebasicresultfortheγ parameterinvolvesthefinestructureconstant
α ∼ 1/137 times the square of the ratio between the change of the velocity
oftheacceleratedchargeoverthelightvelocity(seetheintroductionofHan-
del’spaper).Thisso-called(byhim)conventionalquantum1/f effectapplies
to small solid-state devices with K of the order 10−7. For large samples K
increases to 2α/π ∼4.6×10−3 which is the value predicted by the coherent
state approach of the quantum 1/f effect (see Fig. 2 in Handel’s paper).
Chargedparticlescanbekeptfreeandinterrogatedforverylongtimesin
aminiaturetrapasshowninthepaperbyF.Vedel.Synchronizedandchaotic
states of the oscillating ions are studied in such a set-up. Using laser cooling
with a few ions, the technique also allows the study of quantum jumps and
the design of a very accurate clock.
Highenergyparticlesandnucleireachingearthfromspacearecalledcos-
mic rays. Their energy spectrum is very broad (from 109 to 1020 eV); they
are emitted from multiplicative cascades (cosmic showers) with a variety of
disintegrationsasshowninthepaperbyJ.Gomez.Atthegroundlevelparti-
cle densities show fluctuations with a 1/f power spectrum in the polar angle
coordinate.Thisanewexampleofthedeeprelationshipbetweennonlinearity
and 1/f noise.
The paper by F. Chapeau-Blondeau studies the intrinsically nonlinear
link between signal and noise in a variety of systems. The word stochastic
resonancehasbeencoinedtodescribesituationsinwhichthenoisecanbenefit
the information-carrying signal. The ability to increase the signal to noise
ratiofromnoiseenhancementinanonlinearinformationchanneloranimage
is conclusively demonstrated.
It is not so well known that the first mathematical study of Brownian
motion, which is due to Bachelier (prior to Einstein) in 1900 concerned the
pricing of options in speculative markets. Anomalous (non-Gaussian) distri-
butionsaretheruleinstockmarketdataasshowninthepaperbyM.Ausloos.
ThepapersbyV.Giordano(G),E.Rubiola(R),M.Planat(P),C.Eckert
(E) and J. Cresson (C) in Part II are closely related. They mainly concern
the understanding of the building block of an electronic oscillator, that is
a resonant cavity (a quartz crystal) and a sustaining amplifier. Time and
frequency metrology was born in an effort to improve the design and perfor-
manceofsuchoscillatorsusedasaccurateclocks(G).Instrumentstomeasure