Table Of ContentModeling and Simulation in Science,
Engineering and Technology
Alberto d'Onofrio
Editor
Bounded Noises
in Physics,
Biology, and
Engineering
ModelingandSimulationinScience,EngineeringandTechnology
SeriesEditor
NicolaBellomo
PolitecnicodiTorino
Torino,Italy
EditorialAdvisoryBoard
K.J.Bathe P.Koumoutsakos
DepartmentofMechanicalEngineering ComputationalScience&Engineering
MassachusettsInstituteofTechnology Laboratory
Cambridge,MA,USA ETHZu¨rich,Zu¨rich
Switzerland
M.Chaplain
H.G.Othmer
DivisionofMathematics
DepartmentofMathematics
UniversityofDundee
UniversityofMinnesota
Dundee,Scotland,UK
Minneapolis,MN,USA
P.Degond K.R.Rajagopal
InstitutdeMathe´matiquesdeToulouse DepartmentofMechanicalEngineering
CNRSandUniversite´ PaulSabatier TexasA&MUniversity
Toulouse,France CollegeStation,TX,USA
A.Deutsch T.Tezduyar
CenterforInformationServices DepartmentofMechanicalEngineering&
andHigh-PerformanceComputing MaterialsScience
TechnischeUniversita¨tDresden RiceUniversity
Dresden,Germany Houston,TX,USA
A.Tosin
M.A.HerreroGarcia
IstitutoperleApplicazionidelCalcolo
DepartamentodeMatematicaAplicada
“M.Picone”ConsiglioNazionaledelle
UniversidadComplutensedeMadrid
Ricerche
Madrid,Spain
Roma,Italy
Forfurthervolumes:
http://www.springer.com/series/4960
Alberto d’Onofrio
Editor
Bounded Noises in Physics,
Biology, and Engineering
Editor
Albertod’Onofrio
DepartmentofExperimentalOncology
EuropeanInstituteofOncology
Milan,Italy
ISSN2164-3679 ISSN2164-3725(electronic)
ISBN978-1-4614-7384-8 ISBN978-1-4614-7385-5(eBook)
DOI10.1007/978-1-4614-7385-5
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013944907
MathematicsSubjectClassification(2010):60-Gxx,60-H10,82-C31,37-Hxx,60-H15,82-Cxx,92-XX,
92-C40,34-K18,34-A08,93-XX
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Preface
SincethehallmarkseminalworksonBrownianmotionbyEinsteinandLangevin,
Gaussiannoises(GNs)havebeenoneofthemainconceptsusedinnon-equilibrium
statistical physics and one of the main tools of its applications, from engineering
to biology. The later, and quite dichotomic, mathematical works by Ito and
Stratonovich laid a firm theoretical basis for the mathematical theory of stochas-
tic differential equations, as well as a long-lasting—and currently unresolved—
controversy on which of the two approaches is best suited for describing mathe-
maticalmodelsoftherealworld.Otherhallmarksinstochasticphysicswereinthe
1970sthebirth,intheframeworkoftheIlyaPrigogineschool,ofthetheoryofnoise-
inducedtransitionsbyHorsthemkeandLefever;intheearly1980s,intheframework
of the Rome school, the introduction of the concept of stochastic resonance, first
introducedbyBenzi,Parisi,Sutera,andVulpianitomodelclimaticchanges.Finally,
in nonlinear analysis, starting from 1990s of the past century a rigorous theory of
stochasticbifurcations—bothphenomenologicalanddynamics—hasbeenanditis
beingdeveloped.
Asfarasthemanyapplicationsofstochasticdynamicalsystemsareconcerned,
in biology and biochemistry noise and noise-induced phenomena are acquiring a
(somewhat unforeseen) fundamental relevance, due to recent discoveries that are
showing the constructive role of noise for some biological functions, for example
cellulardifferentiation.Theincreasingimportanceofnoiseinunderstandingintra-
and intercellular mechanisms can indeed be summarized with the motto “noise is
notanuisance.”
Theabove-summarizedbodyofresearchisessentiallybasedontheuseofGNs,
which is backgrounded in the Central Limit Theorem, and which is, it must be
clearly said here, the best approximation of reality in many cases. However, since
1960s an increasing number of experimental data motivated theoretical studies
stressing that many real-life stochastic processes do not follow white or colored
Gaussian laws, but other densities such as “fat-tail” power laws. Although this is
notthetopicofthisbook,itisimportanttorecallthepioneeringstudiesbyBenoit
MandelbrotandhisintroductionoftheconceptsoffractalBrownianmotion.
vii
viii Preface
More recently, a vast body of research focused on another important class of
non-Gaussian stochastic processes: the bounded noises. Previously, in the above-
summarizedhistoricalframework,thestudiesonboundednoises,apartfromsome
sporadic exception, were mainly confined to the applications of telegraph noise,
nowadaysbetterknownasdichotomousMarkovnoise(DMN).Inthelast20years,
together with a renewal of theoretical interest for DMN, other classes of bounded
noises were defined and intensively studied in the statistical physics and in the
stochasticengineeringcommunities,and—toalesserdegree—inmathematicsand
quantitativebiology.
Theriseofscientificinterestonboundednoisesismotivatedbythefactthatin
manyapplicationsbothGNsand“fat-tailed”stochasticprocessesareaninadequate
mathematicalmodelofthephysicalworldbecausetheyareunbounded.Thisshould
precludetheirusetomodelstochasticfluctuationsaffectingparametersoflinearor
nonlineardynamicalsystems,whichmustbebounded.Moreover,inmanyrelevant
cases, especially in biology, the parameters must also be strictly positive. As a
consequence,nottakingintoaccounttheboundednatureofstochasticfluctuations
may lead to unrealistic model-based inferences. For example, in many cases the
onset of noise-induced transitions depends on trespassing of a threshold by the
varianceofnoise.InthecaseofGNthisoftenmeansmakingnegativeorexcessively
large a parameter. To give an example taken from real life, a GN-based modeling
oftheunavoidablefluctuationsaffectingthepharmacokineticsofanantitumordrug
deliveredbymeansofcontinuousinfusionleadstotheparadoxthattheprobability
that the drug increases the number of tumor cells may become nonzero, which
is absurd. The problems sometimes induced by the scientific artifacts caused by
a bona fide but acritical use of GN-based models of noises may go beyond the
purelyscientificframework,speciallyinengineeringandotherapplications,where
economical side effects of the bad modeling is a relevant issue. For example, in
probabilistic engineering, the use of unbounded noises leads to overconservative
design, which induces a remarkable increase in the costs. In order to avoid these
problems,thestochasticmodelsshouldinthesecasesbebuiltonboundednoises.
Thedeepeninganddevelopmentoftheoreticalstudiesonboundednoisesledto
the attention of a vast readership on new phenomena, such as the dependence of
thetransitionsorofthestochasticresonanceonthespecificmodelofnoisethathas
been adopted. This means that, in the absence of experimental data on the density
andspectrumofthestochasticfluctuationsfortheprobleminstudy,ascientificwork
shouldoftencomparemultiplekindsofpossiblestochasticperturbations.Moreover,
currentlytheboundednoiseapproachalsoimpliesthatthepossibilityofobtaining
analytical results is remarkably reduced or sometimes annihilated. Indeed, for
example,inthisfieldmodelsareneverbasedonasinglescalarstochasticequation,
sincetheproblemsinstudyareinmostsimplecasesatleastbidimensional,oneor
moreadditionalequationsbeingdevotedtothemodelingoftheboundedstochastic
processes.
Preface ix
Theaimofthiscollectivebookistogive,throughaseriesofcontributesbythe
world-leading scientists, an overview of the state of the art and of the theory and
applicationsofboundednoises,andofitsapplicationsinthedomainsofstatistical
physics,biology,andengineering.
Quite surprisingly, given that in the last 15 years an increasingly large body of
research has been and is being published on the subtle effects of bounded noises
ondynamicalsystems,thisvolumeisprobablythefirstbookreallydevoted tothe
generaltheoryandapplicationsofboundednoises.Itisapleasuretoremindtothe
reader that a single monographic volume was published in 2000, by Springer, in
a similar topic: Bounded Dynamic Stochastic Systems: Modeling and Control by
Prof. Hong Wang, which was focused on industrial applications, and was mainly
devotedtosomeinnovativeapproximationmethodsintroducedbyitsauthor.Onthe
contrary,ourcollectiveworkisabasicsciencebook.
Thisvolumeisorganizedintofourparts.
The first part is entitled Modeling of Bounded Noises and Their Applications
in Physics, and it includes both contributes on the definition of the main kinds
of bounded noises and their applications in theoretical physics. Indeed, in this
moment, the theory of bounded stochastic processes is intimately linked to its
applications to mathematical and statistical physics, and it would be extremely
difficult and unnatural to separate theory from physical applications. In the first
contributeofthebook,ZhuandCaiillustratetwomajorclassesofboundednoises—
therandomizedharmonicmodelandthenonlinearfiltermodel—andtheirstatistical
properties,aswellaseffectivealgorithmstonumericallysimulatethem.Thesecond
contribute is written by the pioneer of the theory of bounded stochastic processes,
Prof. Dimentberg, who first introduced the randomized harmonic model in 1988
as a representation of a periodic process with randomized phase modulation. In
his contribute, Prof. Dimentberg focuses on the dynamics of the classical linear
oscillator under external or parametric bounded excitations, with an excursus in
an important nonlinear case. Another major example of bounded noise is the one
based on Tsallis statistics (aka η-process or Tsallis–Borland noise). This noise is
introduced here in the contribute by Wio and Deza, who also illustrate its effects
inthemostimportantnoise-inducedphenomena,suchasstochasticresonanceand
noise-induced transitions. Properties of dynamical systems driven by DMN are
investigated in the third contribute by Ridolfi and Laio, who also focus on the
applicationofDMNinenvironmentalsciences.
Stochasticoscillatorsareacentraltopicinstatisticalphysics,whichisconfirmed
bythenexttwochapters.Thefirst,byGitterman,isdevotedtothestudyofBrownian
motionwithadhesion,i.e.,anoscillatorwitharandommassforwhichtheparticles
ofthesurroundingmediumadheretotheoscillatorforsomerandomtimeafterthe
collision. The second, by Bobryk, is devoted to the numerical study of energetic
stability for a harmonic oscillator with fluctuating damping parameter, where the
stochasticperturbationismodeledbymeansofthesine-Wienernoise,aparticular
case of the above-mentioned randomized harmonic model. In the next chapter
Hasegawaappliesamomentmethod(MM)totheLangevinmodelforaBrownian
particlesubjectedtotheabove-mentionedTsallis–Borlandnoise.
