Table Of Content

Anomalous transport in the crowded world of biological cells REVIEWARTICLE FelixHöfling1andThomasFranosch2 1Max-Planck-InstitutfürIntelligenteSysteme,Heisenbergstraße3,70569Stuttgart,andInstitutfürTheoretischePhysikIV, 3 UniversitätStuttgart,Pfaffenwaldring57,70569Stuttgart,Germany 1 2InstitutfürTheoretischePhysik,Friedrich-Alexander-UniversitätErlangen–Nürnberg,Staudtstraße7,91058Erlangen, 0 Germany 2 Submittedto:Rep.Prog.Phys.;Received25June2012;Revised7December2012;Accepted19December2012 n a Abstract J Aubiquitousobservationincellbiologyisthatthediffusivemotionofmacromoleculesandorganellesisanomalous,andade- 9 scriptionsimplybasedontheconventionaldiffusionequationwithdiffusionconstantsmeasuredindilutesolutionfails. Thisis 2 commonlyattributedtomacromolecularcrowdingintheinteriorofcellsandincellularmembranes,summarisingtheirdensely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square ] t displacementasfunctionofthelagtime,butthereareothermanifestationslikestronglyreducedandtime-dependentdiffusion f o coefficients, persistent correlations in time, non-gaussian distributions of spatial displacements, heterogeneous diffusion, and s a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we . t a summarisesomewidelyusedtheoreticalmodels: gaussianmodelslikefractionalBrownianmotionandLangevinequationsfor m visco-elasticmedia,thecontinuous-timerandomwalk(CTRW)model,andtheLorentzmodeldescribingobstructedtransport - inaheterogeneousenvironment. Particularemphasisisputonthespatio-temporalpropertiesofthetransportintermsoftwo- d pointcorrelationfunctions,dynamicscalingbehaviour,andhowthemodelsaredistinguishedbytheirpropagatorsevenifthe n mean-squaredisplacementsareidentical.Then,wereviewthetheoryunderlyingcommonlyappliedexperimentaltechniquesin o thepresenceofanomaloustransportlikesingle-particletracking,fluorescencecorrelationspectroscopy(FCS),andfluorescence c [ recoveryafterphotobleaching(FRAP).Wereportonthelargebodyofrecentexperimentalevidenceforanomaloustransportin crowdedbiologicalmedia: incyto-andnucleoplasmaswellasincellularmembranes,complementedbyinvitroexperiments 1 whereavarietyofmodelsystemsmimicphysiologicalcrowdingconditions. Finally,computersimulationsarediscussedwhich v 0 playanimportantroleintestingthetheoreticalmodelsandcorroboratingtheexperimentalfindings. Thereviewiscompleted 9 byasynthesisofthetheoreticalandexperimentalprogressidentifyingopenquestionsforfutureinvestigation. 9 6 PACSnumbers: 05.40.-a87.16.-b . 1 0 3 1 Contents : v i X 1. Preface 2 4.2. Fluorescencecorrelationspectroscopy(FCS) 19 4.3. Fluorescencerecoveryafterphotobleaching(FRAP) 22 r a 2. BasicsofBrownianmotion 3 2.1. Simplediffusion 4 5. Anomaloustransportincrowdedbiologicalmedia 24 2.2. Anomalousandcomplextransport 5 5.1. Crowdedcellularfluids 24 3. Theoreticalmodels 8 5.2. Crowdedmembranes 33 3.1. Gaussianmodels 8 5.3. Reactionkinetics 42 3.2. Continuous-timerandomwalks(CTRW) 10 3.3. Obstructedmotion:Lorentzmodels 14 6. Conclusion 43 3.4. Othersourcesofsubdiffusion 18 Acknowledgments 46 4. Experimentaltechniques 18 4.1. Single-particletracking 18 References 46 2 1. Preface widely used mean-square displacement of a tracer and more generally the entire two-time conditional probability density. Transport of mesoscopic particles suspended in simple sol- Thereby, we introduce some notation that will be employed ventsisgovernedbyBrownianmotionandisoneofthepillars throughout the review and exemplify the important concept ofbiologicalandsoftcondensedmatterphysics. Thepioneer- of scaling. Then we contrast complex transport from normal ing works of Einstein and von Smoluchowski identified the transport by requiring that deviations from normal transport randomdisplacementsastheessenceofsingle-particlemotion arepersistentwaybeyondthenaturaltimescalesofthesystem. and—employingideasofthecentral-limittheorem—suggesta The most celebrated indicator is the subdiffusive increase of gaussianprobabilitydistributionwithamean-squaredisplace- the mean-square displacement. A second example which has mentthatincreaseslinearlyintime. Thisparadigmconstitutes drawnsignificantinterestinthestatisticalphysicscommunity, thebasisofnumerousapplicationsrangingfromcolloidalsus- althoughalmostentirelyignoredbybiophysicists,isgivenby pensions,emulsions,andsimplepolymericsolutionstomany the velocity autocorrelation function. We recall that even the biologicalsystems. caseofasingleparticleinasimplesolventencodesnon-trivial Fromthisperspective,theemergenceofanomaloustrans- correlationsandlong-timeanomaliesduetotheslowdiffusion port in more complex materials, typically characterised by a oftransversemomentum. Further,weemphasisethatanoma- sublinear increase of the mean-square displacement, appears loustransportcannolongerbedescribedbyasinglediffusion as exotic. Yet experiments from many different areas reveal constant, rather the transport properties depend on the con- that anomalous transport is ubiquitous in nature, signalling sidered length and time scales. Thus, it is essential to make thatslowtransportmaybegenericforcomplexheterogeneous testable predictions for the spatio-temporal behaviour, which materials;examplesarecrowdedbiologicalmedia,polymeric weelucidateintermsofthenongaussianparameterand,more networks, porous materials, or size-disparate mixtures. One generally,theshapeofthedistributionofdisplacements. prominentingredientcommontomanyofthesesystemsisthat Thetheoreticaldescriptionofthephenomenonofanoma- theyaredenselypacked,knownasmacromolecularcrowding loustransportisreviewedinSection3. Wefocusonthethree inthebiophysicscommunity.Thepresenceofdifferentlysized most widely used concepts each of which has been corrobo- proteins,lipids,andsugarsinthecellcytoplasm,aswellasthe rated from some experiments and excluded from others. The filamentousnetworkspermeatingthecell,isbelievedtobeat first one consists of relaxing the white-noise assumption in a theoriginoftheobservedslowingdownoftransport, render- Langevinapproach. Thentoinduceasublinearincreaseofthe ingdiffusionconstantsmeaninglessthatweremeasuredindi- mean-square displacement a power-law time correlation has lutesolution. Similarlyinheterogeneousmaterialsdisplaying to be imposed on the increments while the probability distri- pores of various sizes, the agents interact strongly with sur- bution is still assumed to be gaussian. In this simple model, faces of complex shape giving rise to an entire hierarchy of one easily derives two-time correlation functions such as the timescales,whichimpliesapower-lawincreaseofthemean- vanHovecorrelationfunctionoritsspatialFouriertransform, squaredisplacement. theintermediatescatteringfunction. Simplegeneralisationsto The goal of this review is to provide a pedagogical modelswherethecorrelationfunctionsencodeonlyapower- overviewoftheexistingtheoreticalframeworkonanomalous lawtailatlongtimesthenleadtothegaussianmodel. Asec- transport and to discuss distinguished experiments from the ond, complementaryapproachisgivenintermsofadistribu- different fields, thus establishing a common language for the tion of waiting times characteristic of the complex medium. observations coming from various experimental techniques. The class of these models is known as the continuous-time The focus here is on crowded biological systems in vivo and random walk model and allows for a simple solution after a invitrobothinthebulksystemssuchasthecytoplasmorbio- Fourier–Laplacetransformtothecomplexfrequencydomain. logicalmodelsolutionsaswellasinplanarsystems,forexam- Whilethesetwodescriptionsarephenomenologicalinnature, ple,crowdedcellularmembranes. Wealsowanttoemphasise thethirdconceptisamicroscopicapproachthatalsoprovides thesimilaritieswithcomplexheterogeneousmaterialsasthey insightintothemechanismhowsubdiffusioncanemergefrom naturally occur, e.g., porous rock, to technologically relevant simpledynamicrulesinacomplexenvironment. Inthisclass, materialssuchasmolecularsievesandcatalysts. whichwerefertoasLorentzmodels,theoriginofslowtrans- In Section 2, we provide some elementary background to portisfoundinanunderlyinggeometricpercolationtransition quantifythestochasticmotionofasingleparticleinacomplex of the void space to which the tracer is confined. We recall disorderedenvironment.Werecallthestandardargumentsthat how self-similar, fractal structures emerge and why transport underlie the theory of Brownian motion both on the level of inthesesystemsisnaturallysubdiffusive. Wecomparediffer- an ensemble average (Fickian diffusion) as well as in terms entmicroscopicdynamicsanddiscussthecrossovertohetero- of single-particle trajectories. In particular, we rederive the geneous diffusion at long time scales, scaling behaviour, and 3 theappearanceofanimmobilefractionofparticles. experiments provides clear evidence for subdiffusive motion, but there are notable experiments which report normal diffu- The section is concluded by delimiting the scope of the sion, and some findings appear to be specific to certain bio- reviewtosubdiffusionthatisrelatedtomolecularcrowdingor logical conditions. Important progress in the understanding the excluded volume effect. Polymer physics and single-file of anomalous transport was made by biologically motivated diffusion, where subdiffusive motion is well known too, will model systems, where in contrast to living cells key parame- not be covered here. In addition, while many experimental tersareadjustable. Thediscussioniscompletedbyaddressing data are interpreted in terms of subdiffusion, some of them computer simulations of simple and complex model systems, actuallydisplayonlydeviationsfromnormaltransportthatare which provide essential support for the interpretation of ex- not persistent over large time windows. Hence one has to be periments. Thepresentationisdividedintothree-dimensional carefulwhetheronespeaksofsubdiffusionorsimplecrossover transportincellularfluids,e.g.,inthecytoplasmoflivingcells, scenarios.Typicalexampleswheresuchapparentsubdiffusion and transport in cellular membranes, which may be approx- isobservedcomprisediffusionofmorethanonespecieseach imated as a (curved) two-dimensional manifold. Finally, the movingwithitsowndiffusioncoefficient. Inthesamespirita consequencesofanomaloustransportonreactionkineticsare particlecouldchangeitsconformationthusexhibitingseveral briefly sketched and some recent progress on this emerging internalstatesagaincharacterisedbyseveraldiffusiveregimes. topicisreported. Sometimes,thedataonlysuggestpowerlawswhichcouldbe interpretedassubdiffusion,yetthecorrelationsareduetothe InSection6,wefirstsummarisewhatthestateoftheartin measurementtechniqueandhavenorelationtotheunderlying thefieldofanomaloustransportcurrentlyisandprovidesome physicalprocesses. general conclusions on what has been achieved so far both theoretically and experimentally. We point out which ques- Sections 4 and 5 review anomalous diffusion in crowded tionsarestillunderdebateandgivesuggestionswherefuture biological systems from an experimental point of view. We research may go and what crucial issues still need to be ad- first introduce the experimental techniques that have proved dressed. themselves as useful tools in biophysics to measure transport properties in mesoscopic samples and on microscopic scales. An important technique that is both intuitive and powerful is 2. BasicsofBrownianmotion provided by single-particle tracking. Here the trajectory of some agent in a complex environment is recorded over suf- The naming of the observation of anomalous transport fre- ficiently long time, allowing for the evaluation of, in princi- quently found in complex systems already suggests that the ple, all correlation functions. The most widely used quantity phenomenaarefundamentallydifferentfromthestandardcase is of course the mean-square displacement, since it is rather thatthereforequalifiesasnormaltransport. Beforereviewing robustwithrespecttostatisticalfluctuationsandappearseasy how anomalous transport can be addressed theoretically and tointerpret. Asecondpowerfulmethodthathasbeenwidely measured experimentally, we discuss the framework for nor- applied in the biophysical context is fluorescence correlation mal transport connecting macroscopic diffusion with fluctua- spectroscopy. Herethebasicideaistolabelfewmoleculesby tionsatsmallscales. a fluorescent dye and record the fluctuating fluorescent light The erratic movement of a mesosized particle suspended uponilluminatingasmallpartofthesamplewithalaser. We in a simple solvent is referred to as Brownian motion, after briefly introduce the theory underlying the measurement and theScottishbotanistRobertBrown,whoobservedthecontin- discusshowanomaloustransportmanifestsitselfinthecorre- uously agitated motion of minute particles ejected from cer- sponding correlation function. Complementary to these two tain pollen grains under a light microscope. The first theo- single-particle methods is fluorescence recovery after photo- retical description has been achieved independently by Ein- bleaching, which detects the diffusion front of fluorophores stein [1] and Smoluchowski [2] in terms of a probabilistic after depleting a small spot by an intense laser pulse. The approach. These ideas have been rephrased shortly after by techniqueisapttomeasureveryslowtransportandimmobile Langevin[3,4]intermsofstochasticdifferentialequationsby particles, and we outline how anomalous transport becomes separatingtheforcebalanceintoadeterministicandarandom manifestintherecoverycurveasfunctionoftime. part. Thecharacterisationoftherandomforcesislargelydue Section 5 is devoted to the plethora of experiments on to Ornstein [5] thereby laying the foundations of the modern anomaloustransportinthecellinteriorandrelatedmodelsys- calculusofLangevinequations. tems and their interpretation in terms of the three theoretical Theexperimentaldemonstrationoftheprobabilisticroute frameworks introduced before. We have compiled and dis- toamacroscopiclawhasbeenachievedbyPerrin[6]andhis cussthemostsignificantexperimentsinthefield,focusingon studentsaroundthesametimebymeticulouslyanalysingsin- thepastdecade. Almostallexperimentsagreethattransportis gle trajectories of colloidal particles observed under a micro- hinderedandsloweddownbymolecularcrowding,manifested scope. His contribution was awarded with the Nobel prize in a suppression of the diffusion constant. A large subset of in 1926 as a breakthrough in proving the physical reality of 4 molecules. verifiesthatthegaussianpropagatorfulfilsthediffusionequa- TheimpactofEinstein’stheoryonBrownianmotion,i.e., tion thenormalcase,canhardlybeunderestimatedandconstitutes ∂P(r,t)= D 2P(r,t), (5) t ∇ oneofthemilestonesinphysics. Recently,ontheoccasionof andsatisfiestheinitialconditionofaspatiallylocaliseddistri- 100th anniversaryofEinstein’sannusmirabilis1905, aseries bution,P(r,t)=δ(r). ofreviewshavebeenpublishedhighlightingthenewconcepts For applications, a representation in terms of spatial andfuturedirectionsinthefieldofBrownianmotion[7–10]. ∫︀ Fourier modes is advantageous, P(k,t) = ddre ik·rP(r,t), − andP(k,t)isknownasthe(self-)intermediatescatteringfunc- 2.1. Simplediffusion tion[11,12].Itcanbemeasureddirectlybyneutronscattering employing the spin-echo technique [13] or, on larger length In the molecular kinetic approach advocated by Einstein and scales,byphotoncorrelationspectroscopy[14]. Themomen- Smoluchowskithesuspendedparticleexperiencesrapidcolli- tum transferred from the sample to the photon or neutron is sions with the solvent molecules. These events occur at the thensimply(cid:126)k. Forthediffusionpropagator,onereadilycal- timescaleoftheliquiddynamics,typicallyinthepicosecond culates regime,andateachencounteratinyamountofmomentumis P(k,t)=exp( Dk2t), (6) exchanged. Simultaneously,thecollisionsareresponsiblefor − themacroscopicfrictionforcecounteractingtherandomkicks. implying that density modulations decay with a rate 1/τ = k Atthetimescaleswherethesuspendedparticlemovessignifi- Dk2.Then,long-wavelengthperturbationsarelong-livedsince cantly,theincrements∆R(t)=R(t+t′) R(t′)afteranelapsed by particle conservation the relaxation involves transport of − time t are considered as random variables that are identically particlesoverlargedistances. and independently distributed. By the central-limit theorem, Forfuturereference,wealsoprovidethevanHovecorre- the total displacement, being the sum of many independent lationfunctioninthecomplexfrequencydomain, tinyincrements,thenisgovernedbyagaussiandistribution ∫︁ P(r,t)=[︀2πδr2(t)/d]︀−d/2exp(︁ r2d/2δr2(t))︁ , (1) P(r,ω)= 0∞eiωtP(r,t)dt, Im[ω]≥0. (7) − The one-sided Fourier transform reduces to the standard whered isthedimensionoftheembeddingspace. Theprob- abilitydistributionP(r,t)isknowningeneralasthepropaga- Laplace transform, provided one identifies s = −iω; the ad- vantage of introducing complex frequencies as above is that tororthevanHoveself-correlationfunction[11,12]. Forin- ⟨ ⟩ dependent increments the variance δr2(t) := [R(t) R(0)]2 thetransformcanbereadilyinvertednumericallybyevaluat- − ingtheintegral growslinearlywiththenumberofsteps,implyingalinearin- crease of the mean-square displacement, δr2(t) = 2dDt. The ∫︁ 2 ∞ onlytransportcoefficientcharacterisingtheBrownianmotion P(r,t)= Re[P(r,ω)]cos(ωt)dω. (8) π is then the diffusion constant D, completely specifying the 0 propagator, Forthediffusivepropagatoronefinds P(r,t)= (4πD1t)d/2 exp(︃4−Dr2t)︃ . (2) P(r,ω)= (2πD1)d/2 ⎛⎜⎜⎜⎜⎝√−riωD⎞⎟⎟⎟⎟⎠d/2−1Kd/2−1⎛⎜⎜⎜⎜⎜⎝r√︂−Diω⎞⎟⎟⎟⎟⎟⎠ , (9) Theself-similarityoffreeBrownianmotionbecomesevident whereK ()denotesthemodifiedBesselfunctionofthesecond ν bywritingthepropagatorinascale-freeform, · kind. Theexpressionsimplifiesforthedimensionsofinterest, P(r,t)=r−d𝒫gauss(rˆ), rˆ∝rt−1/2, (3) ⎛ √︂ ⎞ irn=trordu,cainndgaasdciamlinengsfiuonncletisosns,calingvariable,rˆ := (2Dt)−1/2r, P(r,ω)= 2√1iωDexp⎜⎜⎜⎜⎜⎝−r −Diω⎟⎟⎟⎟⎟⎠ (d =1), (10a) | | − ⎛ √︂ ⎞ 𝒫gauss(rˆ)=(2π)−d/2rˆdexp(−rˆ2/2). (4) P(r,ω)= 2π1DK0⎜⎜⎜⎜⎜⎝r −Diω⎟⎟⎟⎟⎟⎠ (d =2), (10b) ⎛ √︂ ⎞ Oretvhieerws.caling forms will be encountered in the course of this P(r,ω)= 4π1Dr exp⎜⎜⎜⎜⎜⎝−r −Diω⎟⎟⎟⎟⎟⎠ (d =3). (10c) Theconnectiontothemacroscopicdescriptionariseswhen consideringmanyparticlesperformingBrownianmotionsuch In scattering techniques, where the energy transfer to the thattheprobabilitycloudP(r,t)displaysthesamespace-time sampleisalsorecorded,thecentralquantityisthefrequency- dynamicsasthemacroscopicconcentration. Inparticular,one and wavenumber-dependent scattering function P(k,ω) = 5 ∫︀ ∞eiωtP(k,t)dt. The scattering cross section corresponding obeys again a gaussian distribution. Thus it suffices to cal- to0 anenergytransfer(cid:126)ωandamomentumtransfer(cid:126)kisthen culate the first two cumulants. Since the mean of the noise ⟨︀ ⟩︀ essentially given by Re[P(k,ω)], known as the (incoherent) vanishes,oneinfers ∆R(t) = 0,andthecorrelationfunction dynamicstructurefactor[11]. Inthecaseofsimplediffusion, ofthedisplacementsfollowsfromthedelta-correlatednoiseas the dynamics is represented by a simple pole on the negative ⟨ ⟩ imaginaryaxis, ∆Ri(t)∆Rj(t) =2Dtδij. (14) ⟨ ⟩ 1 Inparticular,onerecoversδr2(t) = ∆R(t)2 = 2dDt,andthe P(k,ω)= . (11) iω+Dk2 probability distribution is determined by the diffusion propa- − gator,Eq.(2). Anequivalentwaytocharacterisethedynamicsofatracer is to give a prescription on how individual trajectories are 2.2. Anomalousandcomplextransport generated as a result of the stochastic fluctuations in the medium [15, 16]. The propagator P(r,t) is then the result Theprobabilisticreasoningpresentedintheprevioussubsec- of a suitable average over the possible individual realisations tion suggests that normal diffusion emerges as a statistical of the jittery motion for the tracer. The equations of motion law essentially by the central-limit theorem. In particular, naturally become stochastic differential equations referred to themean-squaredisplacementisexpectedtoincreaselinearly as Langevin equations, which incorporate the randomness of in time for time scales much larger than microscopic ones. the kicks by the medium as “noise”. The modern formula- In simple systems such as normal liquids [11, 12] one ob- tionintermsofaLangevinequationismostlyduetoOrnstein, servesdiffusionalreadyattimescalesexceedingthepicosec- whoshapedthenotionofwhatisnowknownasrandomgaus- ond scale. The phenomena of anomalous or complex trans- sianwhitenoise. Amathematicalrigorousintroductiontothe portdealwithdynamicswherethisdiffusiveregimeisnotvis- stochasticdifferentialequationsandBrownianmotioncanbe ibleevenontimescalesthatarebymanyordersofmagnitude foundintheexcellenttextbookbyØksendal[17]. largerthanpicoseconds. Conventionally, anon-lineargrowth For overdamped motion, the displacements R(t) are as- of the mean-square displacement δr2(t) is taken as indicator sumedtoobeythestochasticdifferentialequation of such unusual behaviour. Typically, the mean-square dis- placementisproportionaltoapowerlaw,δr2(t) tα,withan ∂R(t)=η(t), (12) ∝ t exponent 0 < α < 1. Hence the mean-square displacement increases slower than for normal diffusion, formally the dif- with noise terms η(t) = (η (t),...,η (t)) that are consid- 1 d fusioncoefficientbecomeszero,neverthelessthetracerisnot eredasindependent,randomquantitiesonsufficientlycoarse- localised. Thiskindofbehaviourisreferredtoassubdiffusion grained time scales. In fact, these η(t), i = 1,...,d, rep- i oranomaloustransport1. Theoretically,thephenomenonthen resentalreadyaveragesovermanyindependentprocessesoc- callsforreasonswhythecentral-limittheoremdoesnotapply curring on even shorter time scales such that the central- atthetimescalesofinterest.Rephrasingtheargumentinterms limit theorem applies. The probability distribution then cor- ofincrementsrevealsthatpersistentcorrelationsarehiddenin responds to a multivariate gaussian, symbolically written as (︁ ∫︀ )︁ thedynamicsonmeso-ormacroscopictimescales. P[η(t)] [η(t)]exp dtη(t)2/4D ,whichischaracterised ∝𝒟 − Wewouldliketomakeadistinctionbetweenasimplevi- completelybytheonlynon-vanishingcumulant[15], olationofthecentral-limittheoreminsomeintermediatetime ⟨ ⟩ window and mechanisms leading to subdiffusive behaviour ηi(t)ηj(t′) =2Dδijδ(t t′), i, j=1,...,d. (13) − thatcaninprinciplepersistforever. Inthefirstcasesomedy- Such a noise displays only short-time correlations and corre- namicprocessesareunusuallyslowthatspoilthecentral-limit spondstoapowerspectraldensitythatisflatatthefrequencies theorem on these scales, yet ultimately normal transport sets ofinterest,commonlyreferredtoaswhitenoise. Wehaveim- in. This scenario of complex transport occurs generically by posedthatdifferentCartesiandirectionsη(t)areuncorrelated having constituents of the medium of different sizes or soft i and, invoking isotropy, the strength of the noise, 2D, is iden- interactions, e.g., polymers. Then the mean-square displace- tical for all directions. The idea of coarse-graining and the ment displays only a crossover from some short-time motion seemingly innocent assumption of independence then neces- to long-time diffusion. Since the crossover can extend over sarilyleadstogaussianwhitenoiseastheuniversallawforthe several decades (due to a series of slow processes occurring statistics of the displacements at small times. Any deviation inthemedium),fitsbypowerlawsareoftenasatisfactoryde- from this law indicates the existence of non-trivial persistent scription. In the second case, the correlations in the incre- correlationsinthesystem. mentsdecayslowlyandupontuningsuitablecontrolparame- The displacement after a finite lag time follows from for- ters the window of subdiffusion can become arbitrarily long. mally integrating the Langevin equation, ∆R(t) = R(t) 1Indifferentcontextsonefindsalsosuperdiffusivetransportcorresponding R(0) = ∫︀tdt η(t ), and being a sum of gaussian variables,−it toα>1,whichisbeyondthescopeofthisreview. ′ ′ 0 6 Thefirstgeneralisationofsimplediffusionconsistsofas- sumingthattheincrements∆R(t)followagaussianprobability distributionwithzeromean, ~ Dt on usi 1 (︃ r2 )︃ D) diff Pgauss(r,t)= [︀2πδr2(t)/d]︀d/2 exp 2δr−2(t)/d . (15) S M ~ tα log( s u b diff u sio n Hweidreththoef mtheeand-issqtruiabruetiodnis;plfaocremoerdnitnaδrry2(td)ifcfhuasrioacnt,eriitsehsotlhdes ~ D0t on δr2(t)=2dDt,ofcourse. Thehighermoments, usi ∫︁ ff di δrn(t):=⟨︀∆R(t)n⟩︀= ddr rnP(r,t), (16) | | | | log(τ) log(t ) x log(time) are obtained by performing gaussian integrals, for example, one finds for the mean-quartic displacement, δr4(t) = [(d + Figure1. Schematicmean-squaredisplacement(MSD)forinterme- 2)/d][︀δr2(t)]︀2. diatesubdiffusivetransport. Freediffusionatmicroscopicscalesis Equivalently to the van Hove function, one can study followedbysubdiffusivetransportatintermediatetimescales. Ina thesingle-particledynamics bymonitoringthedecayof den- physicalsystem, thesubdiffusivegrowthendstypicallyatasecond sity fluctuations in the wavenumber representation ρ(k,t) = crossover, wheretheMSDgrowslinearlyagainwithreduceddiffu- exp(ik·R(t)).Thecorrespondingcorrelationfunction,P(k,t)= sionconstant,D≪ D0,orwhereitsaturates,e.g.,duetoboundaries ⟨︀ρ(k,t)*ρ(k,0)⟩︀, is the (self-)intermediate scattering function likethecellmembrane. andmerelythespatialFouriertransformofthevanHovefunc- tion. Again by isotropy, P(k,t) depends only on the magnitude k = k of the wavenumber. The explicit representation Henceinawell-definedlimitthesubdiffusionpersistsforever ⟨︀| | ⟩︀ P(k,t)= exp[ ik·∆R(t)] permitsaninterpretationofP(k,t) andthe central-limitlimittheorem neverapplies. We reserve − as the characteristic function of the random variable ∆R(t), the term anomalous transport for the latter scenario. Typi- such that the lowest order moments can be obtained from a cally,themean-squaredisplacementisexpectedtodisplaytwo seriesexpansionforsmallwavenumbers,k 0, crossover time scales, see Fig. 1 and Ref. 18, which is also → foundinexperiments[19]. Thelong-timediffusioncoefficient k2 k4 (︀ )︀ isthenstronglysuppressedcomparedtothemicroscopicmo- P(k,t)=1 δr2(t)+ δr4(t)+ k6 . (17) − 2d 8d(d+2) 𝒪 tionatshorttimes,forexample,subdiffusionwithα=0.6over 4decadesintimeyieldsareductionof Dover D byafactor 0 Here we used that the orientational average over a d- of(t /τ)1 α = 104 0.4 40. Wepostponethediscussionwhat x − × ≈ dimensionalsphereyields2cos2ϑ=1/d,cos4ϑ=3/d(d+2). physicalmechanismscanleadtosuchdrasticchangesandfo- The logarithm of the characteristic function generates the cushereonthemeasurablequantitiessuitedtorevealcomplex cumulants, andanomaloustransport. 2.2.1. VanHoveself-correlationfunction lnP(k,t)=−2kd2δr2(t)+ k4[︀δ8rd22(t)]︀2 ⎛⎜⎜⎜⎜⎜⎝d+d 2[︀δδrr24((tt))]︀2 −1⎞⎟⎟⎟⎟⎟⎠ Thebasicobservableisthefluctuatingsingle-particledensity, + (k6), k 0. (18) ρ(r,t) = δ(r R(t)), and the corresponding correlation func- 𝒪 → − ⟨︀ ⟩︀ tion,P(r r′,t t′) = V ρ(r,t)ρ(r′,t′) ,isreferredtoasvan In the case of gaussian transport, the Fourier transform of − − (︀ )︀ Hove (self-)correlation function [11]. Here V is the volume Eq. (15) yields P (k,t) = exp k2δr2(t)/2d , and all cu- gauss − of the container and the thermodynamic limit V is an- mulants apart from the second one, δr2(t), vanish identi- → ∞ ticipated. Furthermore by translational invariance and for a cally. Thusasimple,dimensionlessindicatorfortransportbe- stationary stochastic process, P depends only on the elapsed time t t and the accumulated displacement r r. Then 2LetnibetheCartesiancomponentsofaunitvector.Thenbysymmetryone − ′ − ′ arguesthat the van-Hove correlation function can be cast into the form P(r,t) = ⟨︀δ(r [R(t) R(0)])⟩︀ which is interpreted directly ninj=Aδij as probability d−ensity−for an observed displacement r after a ninjnknl=B(δijδkl+δikδjl+δilδjk), lagtimet. Furthermoreforisotropicsystems,towhichwere- where...indicatesasphericalaverage. Contractingtheindicesiand jin strictthediscussion,onlythemagnituder = r entersthevan the first relation reveals A = 1/d, contracting in the second shows B = Hovefunction. | | 1/d(d+2).Thusn4z =3/d(d+2). 7 yondthegaussianapproximationisthenon-gaussianparame- that Z(ω) is the one-sided Fourier transform of the velocity ter[11,12], autocorrelationfunction(VACF), d δr4(t) α2(t):= d+2[︀δr2(t)]︀2 −1. (19) Z(t)= 1⟨︀v(t)·v(0)⟩︀= 1 d2 δr2(t). (23) d 2ddt2 Thesubscriptindicatesthatthereisanentireseriesofsimilarly Reversely, themean-squaredisplacementisobtainedbyinte- defined quantities involving higher moments of the displacements. The inequality ⟨︀X2⟩︀ ⟨︀X⟩︀2 for the random variable gration, ∫︁ ≥ t X = ∆R(t)2 implies a lower bound on the non-gaussian pa- δr2(t)=2d dt′(t t′)Z(t′). (24) | | − rameter,α (t) 2/(d+2). 0 2 ≥− Thefactthatprobabilitytheoryimposescertainconstraints Forstochasticprocesseswherethederivativeoftheincrements on P(k,t) as a function of wavenumber k naturally poses the ∆R(t)doesnotexist, e.g., foraBrownianparticle, theVACF question if additional conditions apply if P(k,t) is consid- may be defined via the mean-square displacement and can ered as a function of time t. More generally, what class of be shown to be a negative and completely monotone func- functions are permissible for correlation functions? Decom- tion[22]. pose the fluctuation density into Fourier modes ρ (k,ω) = Inthecaseofordinarydiffusion,δr2(t) = 2dDt,thediffu- T ∫︀T/2 dte iωtρ(k,t) for real frequencies ω and a long but fi- sionkernelsimplyassumesaconstant, D(k,ω) Z(ω) = D, n−itTe/2obse−rvation time T > 0. Then the corresponding power andthevelocitydecorrelatesinstantaneously, Z(↦→t) Dδ(t ↦→ − 0). Furthermore, thenon-gaussianparametervanishesidenti- spectral density is obtained via the Wiener–Khinchin theo- cally,α (t) 0,asdoallhighercumulants. rem[15,16], 2 ≡ lim 1 ⟨⃒⃒⃒ρT(k,ω)⃒⃒⃒2⟩=2Re[P(k,ω)]. (20) 2.2.2. Distributionofsquareddisplacements T T →∞ The measurements of the squared displacements ∆R(t)2 for Thus Re[P(k,ω)] 0 and inversion of the one-sided Fourier a single particle along its trajectory often does not represent transformyields ≥ wellthemean-squaredisplacementδr2(t),ratherasignificant scattering of the data is observed. This observation suggests ∫︁ 2 ∞ tointroducethedistributionfunctionforthesquareddisplace- P(k,t)= Re[P(k,ω)] cos(ωt)dω, (21) π mentsu, 0 ⟨ (︁ )︁⟩ p(u,t):= δ u ∆R(t)2 . (25) i.e., the propagator is the cosine transform of a non-negative − function. Transforming again to complex frequencies by T∫︀heprobabilitydistributionisobviouslyproperlynormalised, one-sided Fourier transform one derives relations of the ∞p(u,t)du = 1, its mean reproduces the mean-square dis- 0 Kramers–Kronigtype[20,21];inparticular,oneobservesthat placement, and higher moments yield the even displacement Re[P(k,ω)] 0notonlyforrealbutforallcomplexfrequen- moments, ≥ ciesintheupperhalf-plane,Im[ω]>0. ∫︁ ⟨ ⟩ Since the particle is to be found somewhere a parti- δr2k(t)= ∆R(t)2k = ∞ukp(u,t)du. (26) | | cle conservation law holds, and the intermediate scatter- 0 ing function approaches unity in the long-wavelength limit, Thefluctuationsinthesquareddisplacementsaregivenbythe limk 0P(k,t) = 1. For the one-sided Fourier transform, the second cumulant and are via Eq. (19) already encoded in the → particleconservationlawsuggeststherepresentationP(k,ω)= non-gaussianparameter, 1/[ iω+k2D(k,ω)],whereD(k,ω)isknownasthefrequency- and−wavenumber-dependent diffusion kernel [11, 12]. From Var[︁∆R2(t)]︁=⟨∆R(t)4⟩ ⟨∆R(t)2⟩2 (27) | | − Re[P(k,ω)] 0forIm[ω] > 0thesamepropertyisinherited [︃ ]︃ forthediffus≥ionkernel,Re[D(k,ω)] 0. = d+2α (t)+ 2 [︀δr2(t)]︀2. (28) ≥ d 2 d Ofparticularinterestisthelong-wavelengthlimitZ(ω) = D(k 0,ω), which encodes the spatial second moment of Rather than dealing with the probability distribution it is → the tracer motion. Note again Re[Z(ω)] 0 for Im[ω] > 0. oftenfavourabletoworkwithamoment-generatingfunction, ≥ Expandingforsmallwavenumber, whichistheLaplacetransformoftheprobabilitydistribution, 1 1 k2Z(ω) ∫︁ ⟨ (︁ )︁⟩ P(k,ω)= iω+k2D(k,ω) = iω − ( iω)2 +𝒪(k)4, (22) M(w,t):= ∞due−u/w2p(u,t)= exp −∆R(t)2/w2 , (29) − − − 0 and comparing to Eq. (17), one finds Z(ω) = where the second representation follows from Eq. (25). The ∫︀ (ω2/2d) ∞dteiωtδr2(t). Integration by parts reveals conventionischosensuchthat1/w2 isthevariableconjugate − 0 8 to ∆R(t)2, and w carries the dimension of a length. Since leadtosubdiffusion. Herewefocusonthethreemostwidely the exponential is approximated by unity for small displace- usedframeworks. Theperhapssimplestapproachisbasedon ments, ∆R(t)2 w2, and rapidly approaches zero for large stochasticdifferentialequationswherethenoisetermdisplays ≪ ones,∆R(t)2 w2, M(w,t)essentiallyconstitutestheproba- persistent correlations which then transfer to the increments. ≫ bilityfortheparticletobestilloragainwithinadistancewof Since usually the statistics of the noise is still assumed to be its initial position. In Section 4.2, it will be shown how this gaussian, they differ essentially only in the form of the tem- quantitycanbedirectlymeasuredbyfluorescencecorrelation poralcorrelationstheyincorporateandwesummarisethemas spectroscopy (FCS), where w corresponds to the beam waist gaussian models. The second category, the continuous-time oftheilluminatinglaser[23]. random walk (CTRW), consists of jump models where parti- ThevanHovecorrelationP(r,t)istheprobabilitydistribu- clesundergoaseriesofdisplacementsaccordingtoadistribu- tionofallvectordisplacementsr,andtheprobabilitydistribu- tion with large tails. Here the central-limit theorem does not tionofthesquareddisplacements p(u,t)followsbymarginal- applysincethemeanwaitingtimeforthenextjumpeventto ising, occurbecomesinfinite. ThelastclassofLorentzmodelsrelies ∫︁ (︀ )︀ onspatiallydisorderedenvironmentswherethetracerexplores p(u,t)= ddrδ u r2 P(r,t). (30) − fractal-likestructuresthatinduceanomalousdynamics. Fortheimportantcaseofstatisticallyisotropicsamples,P(r,t) depends only on the magnitude of the displacement r = r 3.1. Gaussianmodels | | andtheintegralcanbeevaluated. Insphericalcoordinates,the Herewecollectpropertiesofaclassofmodelsthatgiveaphe- angularintegrationyieldsthesurfaceareaofthed-dimensional nomenologicaldescriptionofcomplexandanomalousdynam- unitsphere, Ω , asafactor. Theradialintegralcollapsesdue d totheDiracdeltafunction,δ(︀u r2)︀= δ(︀√u r)︀/2u,andone icswhichallresultingaussianpropagators. − − obtains p(u,t)= Ωdu(d−2)/2P(︀r= √u,t)︀. (31) 3.1.1. FractionalBrownianmotion 2 A simple model for subdiffusion is fractional Brownian mo- For gaussian transport, solely characterised by the time- tion, introduced rigorously by Mandelbrot and van Ness [24] dependentmean-squaredisplacement,δr2(t),Eq.(15)yields assuperpositionofBrownianprocesseswithpower-lawmem- 1 ud/2 1 (︃ u )︃ ory. Here we follow a heuristic approach [25–27] that sum- p(u,t)= Γ(d/2)[︀2δr2(t)−/d]︀d/2 exp 2δr−2(t)/d ; (32) marises the essence of fractional Brownian motion. Assum- ingthestochasticdifferentialequation(12),∂R(t) = η(t),we t have already seen that if the noise η(t) is delta-correlated in the gamma function evaluates to Γ(d/2) = √π,1, √π/2 for i time, the mean-square displacements increase linearly. If ad- d =1,2,3. Fromtheseexpressionsonereadilycalculatesalso ∫︀ ditionally the noise obeys a gaussian statistics, this property U thecumulativedistributionfunctions, p(u,t)du. 0 is inherited also for the displacements ∆R(t) and transport is ThevanHovefunctionP(r,t)anditsspatialFouriertrans- completely characterised by the diffusion propagator P(r,t), form,theintermediatescatteringfunction,P(k,t),thedistribu- Eq.(2). Theideaisnowtoincorporatepersistentcorrelations tionofthesquareddisplacements p(u,t),andthecorrespond- inthenoisesuchthattransportisdrasticallysloweddownwith ing moment generating function M(w,t) all encode spatio- respecttonormaldiffusion. Sincethenoiseplaystheroleofa temporalinformationonthemotionofthetracer.Forisotropic fluctuatingvelocity,weusethesamenotationasinSection2.2, systems they are all equivalent in principle, in practice they toexpressthenoisecorrelatorintermsofthevelocityautocor- aresensitivetodifferentaspectsoftransportoccurringondif- relationfunction, ferent length scales. Correlation functions that involve more ⟨ ⟩ thantwotimesprovideevenmoreinformationonthedynam- ηi(t)ηj(t′) =dZ(t t′)δij, i, j=1,...,d. (33) ics, and may hold the key to distinguish different theoretical | − | modelsthatyieldthesametwo-timecorrelationfunctions. Here the different Cartesian components are taken as uncorrelated, which certainly holds for an isotropic system. In the 3. Theoreticalmodels Fourierdomain,thisimplies ⟨ ⟩ Theparadigmofanomaloustransportistantamountwithavi- ηi(ω)*ηj(ω′) =4πdδijδ(ω ω′)Re[Z(ω)], (34) − olation of the central-limit theorem on arbitrarily long time ∫︀ scales. Modelling such processes requires including persis- where Z(ω) = ∞eiωtZ(t)dt is again the one-sided Fourier 0 tent correlations that manifest themselves as self-similar dy- transform of the velocity autocorrelation function. For ordi- namics in the mean-square displacements. Different models narydiffusion,Z(ω)= Disconstantandthenoisecorresponds andtheoreticalapproacheshavebeenpursuedthatgenerically towhitenoise. 9 The case of subdiffusion, δr2(t) = 2dK tα with an expo- where m denotes the mass of the particle, the deterministic α nent0<α<1andageneraliseddiffusioncoefficientK >0, friction force ζv(t) is merely the Stokes drag, and f(t) is a α − ⟨︀ ⟩︀ thenyieldsaspectraldensityZ(ω)=( iω)1 αK Γ(1+α)[24]. fluctuating force with zero mean, f(t) = 0. The friction − α i − Hencethestrengthofthenoiseapproacheszeroasthefrequen- constant constant, ζ = 6πηa, is directly connected to the sol- ciesbecomesmaller,whichexplainsthattransportslowsdown vent viscosity η and the particle radius a. The statistics of withincreasingcorrelationtime. Inthetemporaldomain,Z(t) therandomforcesf(t)ischaracterisedcompletelybytheonly isrepresentedbyapseudofunction3, non-vanishingcumulant[5], Z(t)=α(α−1)KαPf|t|α−2. (35) ⟨fi(t) fj(t′)⟩=2kB𝒯ζδijδ(t−t′), (39) Up to here only the mathematical frame has been set and no where is the temperature of the environment and k de- assumptions on the nature of the stochastic process has been 𝒯 B notes Boltzmann’s constant. Thus the Cartesian components made. f(t)oftheforcesaregaussiandistributedandindependentfor In fractional Brownian motion, the statistics of the noise i different times. The variance at equal times is again dictated correlator is assumed to be characterised by the only non- by the fluctuation–dissipation theorem, see, e.g., Ref. 15 for vanishing cumulant Z(t), Eqs.(33) and (35), i.e., a stationary details. The delta-correlation in the temporal domain for the gaussianprocessalthoughnotwhitenoise. Thenthestatistics forces translates to white noise for the corresponding power oftheincrements∆R(t)isagaingaussian,andthepropagator spectraldensity. Thevelocityautocorrelationthendecaysex- reduces to P (r,t), Eq. (15). Its scaling form corresponds gauss ponentially[3], tothatofsimplediffusion,Eq.(3), ⟨ ⟩ PFBM(r,t)=r−d𝒫gauss(rˆ), rˆ∝rt−α/2, (36) vi(t)vj(t′) =(kB𝒯/m)δij exp(−|t−t′|/τp), (40) sharing the gaussian scaling function, Eq. (4), but not the whereτ = m/ζ = m/6πηaisthemomentumrelaxationtime. p scaling variable, rˆ. In particular, the non-gaussian parameter Similarly,themean-squaredisplacementofthed-dimensional α (t) 0vanishesbytheconstructionoffractionalBrownian motioniscalculatedto 2 ≡ motion. [︁ (︁ )︁]︁ Incontrasttosimplediffusion,fractionalBrownianmotion δr2(t)=2dD t+τp e−t/τp 1 , (41) − isnotaMarkovprocess;inparticular,thevanHovecorrelation where D = k /ζ is the diffusion constant according to the functionisnotsufficienttocharacterisethestatisticalproper- B𝒯 Stokes–Einsteinrelation. ties completely. Multiple-time correlation functions encode The description can be easily generalised for the case non-Markovianbehaviour,forwhichfractionalBrownianmo- of visco-elastic media [30]. Here the response of the com- tionmakesdetailedpredictions. Asanexample, plex solvent to shear is encoded in the complex frequency- ⟨ ⟩ [R(t) R(0)]2[R(t+T) R(T)]2 dependentviscosity,η(ω).Intheconventionsemployedinthis − − review, Re[η(ω)] 0 corresponds to the dissipative part and =4d2Kα2t2α+2dKα2(︀|t+T|α+|t−T|α−2Tα)︀2, (37) Im[η(ω)]encodes≥thereactivepart.Equivalently,onemayem- ploythecomplexshearmodulusG(ω):= iωη(ω). Forexam- whichhasbeenderivedrecentlytostudytheergodicproperties − ple, in the Maxwell model, G(ω) = iωτ G /(1 iωτ ), offractionalBrownianmotion[29].ForT =0,thisexpression − M ∞ − M the modulus is characterised by a high-frequency elastic re- reproducesthequarticmoment,δr4(t),andiscompatiblewith sponse G and a crossover time scale τ . The Stokes drag avanishingnon-gaussianparameter,Eq.(19). M ∞ inavisco-elasticmediumthendependsonthefrequencyand theLangevinequationisdiscussedconvenientlyintheFourier 3.1.2. Langevinequationsforvisco-elasticmedia domain[31], Theerraticmotionofasphericalparticleimmersedinacom- plex medium can be described quite generally by a Langevin iωmv(ω)= ζ(ω)v(ω)+f(ω), (42) − − equation. Rather than directly addressing the displacements one may base the description on the velocity v(t) = R˙(t) and whereζ(ω)=6πη(ω)areplacestheStokesfrictioncoefficient. By the fluctuation–dissipation theorem, the force correlator formulate a force balance equation. The paradigm has been hastobemodifiedaccordingly, givenbyLangevin[3]himself, ⟨ ⟩ mv˙(t)=−ζv(t)+f(t), (38) fi(ω)* fj(ω′) =4πkB𝒯 Re[ζ(ω)]δijδ(ω−ω′). (43) 3The noise correlator corresponds to a distribution, and pseudofunction Since the fluctuations arise in the surrounding solvent as a means that integrals with test functions ϕ(t) extract only Hadamard’s fi- ∫︀ ∫︀ nitepart[28], ϕ(t)Pf|t|α−2dt:= [ϕ(t)−ϕ(0)]|t|α−2dt.Inparticular,one sum over uncorrelated regions, the forces are again assumed easilyverifiesthattheone-sidedFouriertransformofZ(t)yieldsZ(ω). to be gaussian. Then it is clear that the van Hove correlation 10 functionfortheparticlecorrespondstoagaussianpropagator vortextodiffuseoverthedistanceoftheradiusoftheparticle. andthedynamicsisspecifiedentirelybythemean-squaredis- Bythefluctuation–dissipationtheorem,Eq.(43),thespectrum placement,δr2(t). Ratherthansolvingforδr2(t),wesolvefor oftherandomforcesisnolongerwhitebutdisplaysacoloured theone-sidedFouriertransformofthevelocityautocorrelation componentthatincreasesasasquarerootwithfrequency. Re- function[30,31], cently,thepowerspectraldensityofthethermalnoisehasbeen measured experimentally for a single bead by ultra-sensitive k Z(ω)= B𝒯 . (44) high-bandwidth optical trapping [40] in excellent agreement imω+ζ(ω) withtheoreticalpredictions. − Thevelocityautocorrelationfunctioninthefrequencydo- Relyingontherelationζ(ω)=6πη(ω)a,thelocalvisco-elastic main,Eq.(44),actsasanadmittanceorfrequency-dependent responseofacomplexmediumisinferredfromthemotionof mobilityanddisplaysanon-analyticlow-frequencyexpansion, tracerparticlesinmicrorheologyexperiments[30]. [︁ ]︁ Z(ω)= D 1 √ iωτ + (ω) . Anexplicitexpressioninthe Subdiffusion at long times is obtained if Z(ω) = − − f 𝒪 ( iω)1 αK Γ(1+α)forω 0[25],i.e.,theelasticmodulus temporaldomainisachievedintermsoferrorfunctions[41], − α d−isplayspower-lawbehavio→ur,G(ω) ( iω)α,whichappears herewefocusonthelong-timeanomaly, ∼ − tobegenericinmanybiologicalmaterialsandsoftmattersys- √︂ D τ temsforintermediatefrequencies. Thisempiricalobservation Z(t) f t−3/2, t , (46) ≃ 2 π →∞ isformulatedinthesoftglassyrheologymodel[32]. Themi- croscopic mechanism remains in general unspecified, yet for which is a direct consequence of the non-analytic terms in the case of a solution of semiflexible polymer networks, the ζ(ω). ThemoststrikingfeatureisthatZ(t)encodespersistent bendingrigidityofasinglefilamentsuggestsanelasticpower- correlationsmanifestedbyaself-similartailinstrongcontrast law response, G(ω) ( iω)3/4 [33–36]. Similarly, by cou- totheexponentialdecayofLangevin’soriginaltheory. These ∼ − plingtotheelasticdegreesoffreedomofamembrane, effec- long-timetailshavebeendiscoveredfirstincomputersimula- tive fractional friction kernels can be generated in the same tionsforfluids[42,43]and,onlyrecently,havedirectlybeen waywithvariousexponentsdependingonthelevelofdescrip- observed for colloidal particles in suspension [44–47]. The tionofthemembrane[37,38] mean-squaredisplacementfollowsdirectlybyintegration, [︁ √︀ (︀ )︀]︁ 3.1.3. Long-timeanomalies δr2(t)=6Dt 1−2 τf/πt+𝒪 t−1 . (47) The assumption of an instantaneous friction term in the Thealgebraictailinthevelocityautocorrelationmanifestsit- Langevinequation(38)isinfactincorrectevenatlongtimes, selfinaslowapproachtonormaldiffusivetransport. as has been pointed out already by Hendrik Antoon Lorentz. The persistent correlations in the mean-square displace- The reason is that the Stokes formula applies for steady moment, buried under the leading linear increase, show that the tion of the particle only and the friction is accompanied by assumptionofindependentincrementsisnotsatisfiedandthat a long-ranged vortex pattern in the velocity field of the en- the regime of truly overdamped motion is never reached due trainedfluid. Forunsteadymotion, theparticleexcitesinces- tothehydrodynamicmemory,evenatlongtimescales. Nev- santly new vortices diffusing slowly through the fluid. As a erthelessthecentral-limittheoremremainsvalid,althoughthe consequence the friction force depends on the entire history convergence is slow due to the persistent power-law correla- of the particle’s trajectory, an effect known as hydrodynamic tionsinducedbyvortexdiffusion. memory. The theoretical description is achieved most conve- nientlyin thefrequency domain. The dragforce fora sphere 3.2. Continuous-timerandomwalks(CTRW) performingsmall-amplitudeoscillationsofangularfrequency ω has already been calculated by Stokes [39] and leads to a A different class of models that is widely discussed is the frequency-dependentfrictioncoefficient[15], continuous-timerandomwalk(CTRW)[48–51],originallyin- troduced by Montroll and Weiss [52] for hopping transport (︁ √︀ )︁ ζ(ω)=6πηa 1+ iωτ iωm/2. (45) on a disordered lattice. The particles spend most of the time − f − f bound to a trap with an escape time that depends sensitively For steady motion, ω = 0, the formula reduces to the Stokes on the depth of the trap. Rather than dealing explicitly with drag. The last term appears as an acceleration force for half the quenched disorder on the lattice, the medium is treated of the displaced fluid mass, m = 4πρa3/3, and it is natural as homogeneous with the new ingredient of a waiting-time f f to absorb this contribution by introducing an effective mass distribution for the next hopping event to occur. Anomalous for the particle, m = m + m/2. The second modification transport can be generated within this framework by assum- eff f isanon-analyticcontributionduetotheslowvortexdiffusion ingwaitingtimedistributionssuchthatthemeanwaitingtime in the liquid as the particle performs unsteady motion. The becomes infinite. The central-limit theorem does not apply characteristictimescale,τ = ρa2/η,isthetimeneededfora since longer and longer waiting times are sampled. It turns f f