Table Of ContentMemory function onturbulent fluctuations insoft-mode turbulence
Takayuki Narumi,1,∗ Junichi Yoshitani,1 Masaru Suzuki,1 Yoshiki Hidaka,1,†
Fahrudin Nugroho,1,2 Tomoyuki Nagaya,3 and Shoichi Kai1
1DepartmentofAppliedQuantumPhysicsandNuclearEngineering,KyushuUniversity,Fukuoka819-0395, Japan
2Physics Department, Gadjah Mada University, Yogyakarta 55281, Indonesia
3Department of Electricaland ElectronicEngineering, OitaUniversity, Oita870-1192, Japan
(Dated:December27,2012)
Modalrelaxationdynamicshasbeenobservedexperimentallytoclarifystatistical-physicalpropertiesofsoft-
mode turbulence, thespatiotemporal chaos observed inhomeotropically aligned nematic liquidcrystals. We
2
found a dual structure, dynamical crossover associated with violation of time-reversal invariance, the corre-
1
spondingtimescalessatisfyingadynamicalscalinglaw.Tospecifytheoriginofthedualstructure,thememory
0
functionduetonon-thermalfluctuationshasbeendefinedbyaprojection-operatormethodandobtainednumer-
2
icallyusingexperimentalresults. Theresultsofthememoryfunctionsuggestthatthenon-thermalfluctuations
c canbe divided intoMarkov and non-Markov contributions, thelatteriscalledtheturbulent fluctuation (TF).
e
Consequently,therelaxationdynamicsisseparatedintothreecharacteristicstages:bare-friction,early,andlate
D
stages. If the dissipation due toTFs dominates over that of the Markov contribution, the bare-friction stage
contracts;theearlyandlatestagesthenconfigurethedualstructure.ThememoryeffectduetoTFsresultsinthe
5
time-reversiblerelaxationattheearlystage,andthedisappearanceofthememorybyturbulentmixingleadstoa
2
simpleexponentialrelaxationatthelatestage.Furthermore,thememoryeffectduetoTFsisshowntooriginate
] fromcharacteristicspatialcoherencycalledthepatchstructure.
t
f
o PACSnumbers:61.30.-v,05.45.-a,47.54.De,05.40.-a
s
.
t
a I. INTRODUCTION other is homeotropic alignment in which the director aligns
m
perpendicularto substrates (z-direction). Rubbing along the
- Weaknonlinearitycangeneratespatialandtemporaldisor- x-directionofasubstrate’ssurface(x–y plane)producespla-
d
dersinsystemswherethenumberofeffectivedegreesoffree- narsystems,andintrinsicallybreakstherotationalsymmetry.
n
o dom increases with increasing system size. A phenomenon Inhomeotropicsystems,bycontrast,therotationalsymmetry
c triggeredbyweaknonlinearityinsuchhigh-dimensionalsys- inthex–yplaneremains.Withasufficientlystrongappliedac
[ tems is called spatiotemporal chaos, in contrast with chaos voltage,magnitudeV andfixedfrequencyf,theFre´edericksz
3 where unpredictable behavior emerges from few degrees of transition, occurringat a certain threshold voltage VF, spon-
v freedominadeterministicway. Theoreticalworkonthespa- taneously breaks the rotational symmetry [4]; the transition
5 tiotemporal chaos (Refs. [1, 2] and references therein) has is accompanied by the excitation of the Nambu–Goldstone
4 covered topics such as the complex Ginzburg–Landauequa- modes[5–7]. By furtherincreasingV, electrohydrodynamic
5 tion,theKuramoto–Sivashinskyturbulence(KST),theNiko- convectionoccursatVc. Thenonlinearcouplingbetweenthe
7 laevskiiturbulence,andcoupledmaplattices. Anoutstanding convectiveandNambu–Goldstonemodesgeneratesa pattern
0. featureofchaosandturbulenceisitsdualstructure,wherethe thatis bothspatially and temporallydisordered. Theexperi-
1 dynamicsisseparatedintoaninitialregimecorrespondingto mentallyobservedphenomenon,calledsoft-modeturbulence
2 deterministicshortorbitsandafinalregimecorrespondingto (SMT),isanexampleofspatiotemporalchaos[8–10].
1
stochasticlongorbits. MoriandOkamura,forexample,have
: Inour precedingstudy, we haveobservedSMT relaxation
v theoreticallystudied1D-KSTandrevealedthedualstructure
by measuring a temporal autocorrelation function and re-
i inturbulentmixingbynumericalanalysis[3].
X ported that the relaxation is well fitted by a compressed ex-
Convectionsystemshavebeenexperimentallyinvestigated
r ponential function [11]. Because the compressed exponen-
a to study nonlinearity. An example is electrohydrodynamic
tialisemployedtodescribethedynamicsofjammedsystems
convection observed in the nematic liquid crystal and con-
[12], we remarkedon the similarity between SMT and glass
trolled by an ac voltage applied to system. One can easily
formingliquids. Thenonlinearityinthedynamicsofthelat-
achieveaccessiblecharacteristiclengthandtimescalesinex-
ter originates from dynamic coherency in some regions. In
perimentsofelectroconvectioncomparedwiththeRayleigh–
fact,spatiotemporalfluctuatingcooperativeregionshavebeen
Be´nard convection. Another advantage of investigating ne-
observed as dynamical heterogeneities near the glass transi-
maticsistheeaseincontrollingtheanisotropytostudysym-
tion point [13–16]. A characteristic length for these coop-
metry properties. There are two types of layer alignment
erative regions increases as the glass transition point is ap-
in nematic liquid crystals; one is planar alignment in which
proached. InSMT,ontheotherhand,patchdomainsexistin
thedirectoralignsparalleltosubstrates(x-direction),andthe
whichconvectiverollsalign in a uniqueorientation[17–19].
The characteristic size ξ of a patch domain is several times
longerthanthetypicalsize ofconvectiverollsanddecreases
∗[email protected] with the distance from SMT onset; ξ ε−1/2 with control
†[email protected] parameterε.Wehavethereforeconclud∼edthattheSMTpatch
2
domainsbehavelikethecooperativeregionsinglassforming (a)
liquidsandthecoherentmotioninthedomainsgeneratesnon-
exponentialrelaxation. TostudySMTdynamicsindetail,the
temporalcorrelationsofeachwavenumber,modalautocorre-
lationfunctions,aresuitable. Ourpreviousstudyfocusedon
thenetautocorrelationfunctionconsistingoftheentirewave-
number information. Here, we investigate the modal relax- y
ationdynamicstospecifystatisticalphysicsofSMT.
x
k
II. EXPERIMENT y
(b) (c)
We study a 2D pattern dynamics of SMT observed in a
homeotropicalignmentofnematicliquidcrystals. Thisstudy nit)
k u
follows a standard setup [8, 20, 21]. The space between x b.
ar
twwiothpthaeranlleemlagtliacslsiqpuliadtecsr,ysstpala,cNed–(247–Mµmethoapxayrbt,enwzialsidefinllee)d– Pk (
4–buthylaniline(MBBA).Theplatesurfaceswerecoatedwith
transparentelectrodes,madeofindiumtinoxide(ITO)witha
circularcross-sectionofradius13mm.Toobtainhomeotropic 0.0 0.2k 0.4 0.6
alignment, the surfaces was coveredby a surfactant, N, N– ( µ m - 1 )
dimethyl–N–octadecyl–3–aminopropyl-trimethoxysilylchlo-
ride50%(DMOAP).Thevaluesofthedielectricconstantand
electric conductivity parallel to the director were ǫ = 6.25 FIG. 1. Staticinformation of our experiment at ε = 0.1. (a) A
k
andσ = 1.17 10−7 Ω−1m−1, respectively. Denotingthe typicalsnapshotofSMTwiththewhitescalebarindicating100µm.
k × Each bright line indicates upward flow. (b) Magnitude tints of the
dielectricconstantperpendiculartothedirectorbyǫ ,thedi-
⊥ spatialpowerspectrumPkintheFourierspace,whereshadingcor-
electricconstantanisotropyǫ = ǫ ǫ isfoundtobeneg-
a k− ⊥ responds to the Pk value. (c) Plot of the power spectrum Pk as a
ative. AnacvoltageV(t) = √2V cos(2πft)wasappliedto functionoftheradialwavenumberk=|k|.Thepeak,markedbyan
the sample. For a control parameter, we employ a normal- arrow,correspondstothefundamentalperiodinaconvectiverolling.
izedvoltageε=(V/V )2 1,whereV denotesthethreshold
c c
−
voltageforelectroconvectionhavingvalue7.78 0.05V.We
±
showresultsforε=0.025,0.050,0.075,0.10,0.20,0.30,and We employ the Fourier transform of the fluctuation
0.40,whereεhasamarginoferrorof 0.013. Anothercon- ∆I(x,t)=I(x,t) I(x,t) asthegrossvariableuk(t);
± −h i
trol parameter was the frequency f of the ac voltage. Two
patternsofSMTarise;obliquerollinf <fL andnormalroll uk(t):= dx∆I(x,t)eik·x, (2)
inf >f ,wheref denotestheLifshitzfrequency[8,9]. We Z
L L
setthefrequencyf =100Hzwellbelowf .Thetemperature
L
wasregulatedto30.00 0.05◦C. Beforeeachsampling,we wherei = √ 1andtheintegralrangeisovertheentiresys-
waitedfor10minatfixe±dvoltageVwandthenafurther10min temdomain.I−nisotropicsystems,itissufficienttostudyuk(t)
at the set V to get a desired ε value, where V < V < V . [23], wherek denotestheradialwave number;k = k. We
F w c | |
The waiting time is sufficiently long for systems to achieve focusonthenormalizedmodaltime-correlationfunctionofk;
steadystate.
Uˆ (τ):= u (t+τ)u∗(t) P−1, (3)
The electroconvection pattern was observed by a micro- k h k k i k
scope (ECLIPSE E600POL,Nikon Corporation,Tokyo)and
where the asterisk denotes the complex conjugate operation
was captured by a high-speed camera (HAS220, DITECT
and P = u (t)2 denotes the spatial power spectrum.
Co.Ltd.,Tokyo)thatcansuccessivelytake4200frames. The k | k |
numbersofframespersecondwere5forε<0.1and10oth- Note that Uˆk(cid:10)(τ) is a(cid:11)real number due to translational sym-
erwise. A typical two-dimensionalimage is shownin Fig. 1 metry and isotropy. The spatial power spectra as functions
(a).Themeasurementareawas830 830µm2(450 450pix- of the wave number k and k = k are illustrated in Fig. 1
els). Thetransmittedlightintensity×I(x,t)ateach×pixelwas (b)and(c),respectively. SMTiso|tro|pyisclearlyreflectedin
digitizedinto8-bit(i.e.,256-level)information,whereaseries Fig.1(b). AlthoughexperimentalparametersinRef.[23]are
ofpatternanalyzingwasprocessedaccordingtoRef.[22]. slightly different from ours, the Pk profiles agree. A clear
Withanglebracketssignifyingthelong-timeaverageinthe peakexistsatkpeak 0.321µm−1 forε = 0.10anditscor-
≃
steady state, the temporalcorrelationof two functionsis de- respondinglengthλpeak = 2π/kpeak is 19.6µm thatisa half
finedas ofλ0,whereλ0 = λ0(ε)denotesthelengthofapairofelec-
troconvections. Thepeakwavenumbersk forotherεval-
peak
1 T ues are similar. The wave number k is normalized by λ as
f(t+τ)g(t) = lim dtf(t+τ)g(t). (1) 0
h i T→∞2T Z−T kˆ =kλ0/2π.
3
III. RESULTANDDISCUSSION decayform;
2
Uˆ (τ) 1 τ/τ(a) . (4)
A. Dualrelaxation k ∝ −(cid:16) k (cid:17)
Incontrast,therelaxationinthelatestageiswelldescribedby
The modalautocorrelationfunctionUˆ (t) obtainedexper- exponentialdecay;
k
imentally is shown in Fig. 2. In the short-time regime, the
rIenlsatxeaadti,owneisfonuontddedsucarlibreedlabxyatiaonsimaspeleviedxepnotninenFtiiagl.f3u.nIcntiothne. Uˆk(τ)∝exph−τ.τk(e)i. (5)
earlystage[24],thecorrelationfunctionfollowsanalgebraic Figure4presentsthecharacteristictimescalesateachstage
forthedualstructure. Thetimescalesarealmostconstantat
smallwavenumbers. Notethattheresultsatsmallkˆ maybe
affected by limitations in sample averaging. At large wave
1.0 numbers, the time scales of the dual relaxation satisfy dy-
ε = 0.050 namicscalinglaws;
τ(a) kˆ−za , τ(e) kˆ−ze (6)
k ∝ k ∝
)
τ with dynamicalexponentsz 1.0and z 1.5regardless
U^k ( 0.5 naorafemεz.icAa=llthsco1aulaginnhdgMzexopr=ioann2edn[t3Os]ok,abawmt≃aeiuncreaadnhntahoveteodrrieerepetiocc≃traltyleldycotihnma1tpDathr-eeKdoSyuT-r
a e
resultswiththeirtheoreticalworkbecausethedynamicalex-
ponent depends on spatial dimensionality. Indeed, dynamic
0.0
scalingforseveralturbulencemodelsdependsonthedimen-
sionality[25–27].
0 10 20
1.0 To elucidate the dual structure of SMT theoretically, we
havetakenaccountofanexperimentalfactthatthespatiotem-
ε = 0.10
poraldisorderaccompaniedbythepatchstructurecanberep-
resentedasnon-thermalfluctuations[17,28].Theprojection-
operatormethodforturbulence[29]isoneofthemostsuitable
)
τ
U^k ( 0.5 metflhmueectatmhtuiocaedatmislootpnorrsyom(cesefaefdneeuacArgteepthdpsaeeutrnciidvhsiexensqoAaunn)i-.vtehaTvleheorunmlusta,ttiloionflnthutheecqetauuruaaetttsoiiotocnonosfr;crteohiltneasttapimoiannapitnehorg-f,
analysisforthememoryfunctionsplaysacentralroletostudy
arelationshipbetweenthedualstructureandthenon-thermal
0.0
fluctuations.
0 10 20
1.0
1.0
ε = 0.40
)
τ
U^k ( 0.5 U^τk () 0.5
0.0 0.0
0 1 2 3 4 5
0 10 20 τ (s)
τ (s)
FIG.3. (color online) Thenormalizedmodal autocorrelationfunc-
FIG.2. (coloronline)Plotofthemodalcorrelationfunctionsatε= tionofε = 0.1andkˆ = 2.0,whichisthesameasshowninFig.2.
0.050 (top), 0.10 (middle), and0.40 (bottom). Thewavenumbers Thealgebraicdecay(4)(brownsolidline)welldescribesthedynam-
kˆ are 0.76 (red square), 1.0 (brown inverted triangle), 2.0 (black icsintheearlystage,butasimpleexponential(5)(greendashedline)
circle),3.0(greentriangle),and4.0(bluediamond). isbetteratthelatestage.
4
2 2
10
ε = 0.025
ε = 0.10
)
101 -2
s
) ( 1
(a) τk (s 1100-01 0.1ε = 0.30 ε = 01.20 10 k Γτ ´ () 00 1 2 τ (s) 3 4 5
2
10
ε = 0.025
ε = 0.10
FIG.5. ThememoryfunctionΓ′(τ)characterizedbyEq.(7). This
1 k
10 istheresultofε=0.1andkˆ=2.0asanexample.
)
s
(
(e) τk 100 ε = 0.3ε0 = 0.20 contributions.Therefore,with
ε = 0.40 Γ′k(τ)=2γk(0)δ(τ)+Γk(τ), (9)
-1
10 0.1 k 1 10 whereγk(0)denotesthebarefrictionduetotheMarkovcontri-
butionofthenon-thermalfluctuationsandΓ (τ)thememory
k
function due to the non-Markov contribution, the evolution
equation(7)reducesto
τFkI(eG).(b4o.tt(ocmol)ofroronselivneer)alLcoogn-tlrooglppaloratsmoeftetrism;εe =sca0le.0s2τ5k(a()pi(ntko)p,)0a.1n0d ∂Uˆ∂kτ(τ) =−γk(0)Uˆk(τ)−Z τdτ′Γk(τ −τ′)Uˆk(τ′). (10)
(blue),0.20(green),0.30(brown),and0.40(red). Thedashedlines 0
plotthepowerlawwithexponentsz =1.0(toppanel)andz =1.5
a e Toemphasizethetransportduetotheturbulent-likedynamics,
(bottompanel).
the non-Markovcontribution to the non-thermalfluctuations
is here called the turbulent fluctuations (TFs) in SMT. One
candefineacharacteristictimescaleforthememoryfunction
B. Memoryeffectduetonon-thermalfluctuations
duetoTFsas
ofWmeacarsossucmoepitchevamrioadbalelselienmSenMtsT{u[3k0}].forAmccaocrdoimngpletotetsheet τk(Γ) := Γ 1(0)Z ∞dτΓk(τ), (11)
k 0
projection-operatorformalism,theevolutionequationinSMT
isrepresentedas withinwhichthememoryeffectsduetoTFsarealive.
AsseeninFig.6,thememoryfunctionΓ (t)causedbyTFs
k
∂Uˆ (τ) τ hasnon-negligiblemagnitudesandtimescalesoverarangeof
k = dτ′Γ′(τ τ′)Uˆ (τ′), (7)
∂τ −Z k − k wavenumbersintheearlystage;therefore,SMTdynamicsis
0
regarded as being non-Markoviandue to TFs. The memory
where Γ′(τ) denotes the memory function that results from has a small peak around τ 2 s. We believe that this is
k ≃
thenon-thermalfluctuations.Here,translationalsymmetryre- due to a residual of the mechanical coefficient appearing in
ducesthetemporalcorrelationofthemodalelementsto the projection-operatormethod, despite the coefficientbeing
theoreticallyzero.
uk(t+τ)u∗k′(t) =δk,k′ uk(t+τ)u∗k(t) (8) Each equation of the dual relaxation is elucidated by the
h i h i
evolutionequation(10). First,letusconsiderthedynamicsat
andthemechanicalcoefficientsωkk arezerobydefinition. the late stage, τ(Γ) τ. The frequency-dependentfriction
k ≪
Usingexperimentalresultsforthemodalcorrelationfunc- Γ caused by TFs is defined as the Fourier-Laplace trans-
k,ω
tion,onecannumericallysolveEq.(7)toobtainthememory formationofthememoryfunction;
function. The memory function Γ′(τ) has a sharp peak at
k
τ = 0, as evident in Fig. 5, implying that the non-thermal ∞
Γ := dτΓ (τ)eiωτ. (12)
fluctuations can be separated into Markov and non-Markov k,ω Z k
0
5
2 1.5
ε = 0.050
2 ) 1.0 ε = 0.20
k Γτ- () (s 1 Γ() τk(s) 0.5 ε = 0.40
00 1 τ (s) 2 3 0 1 2 k^ 3 4 5
FIG.6. (coloronline)Time-dependenceofthememoryfunctionfor FIG.7. (coloronline)Wave-numberdependences ofthetimescale
ε=0.10atseveralfixedwavenumbers: kˆ=0.76(redsquare),1.0 τ(Γ) forseveralvaluesofε; ε = 0.050 (purple),0.20(green), and
k
(browninvertedtriangle),2.0(blackcircle),3.0(greentriangle),and 0.40(red). Eachdottedcurve isobtainedbyapeak-fittingmethod
4.0(bluediamond). andtheverticalarrowpointswherekΓis.
Atthelatestage,frictioncanberegardedasstatic, C. Characteristictimescales
γ(Γ) :=Γ ; (13) A characteristic time scale for the modal autocorrelation
k k,ω=0 functioncanbedefinedby
hence,thememoryeffectisapproximatedbyadeltafunction ∞
τ(U) := dτUˆ (τ). (18)
k Z k
Γ (τ) 2γ(Γ)δ(τ). (14) 0
k ≃ k
This is the relaxation time due to non-thermal fluctuations.
The modal relaxation dynamics is thus obtained as a simple Incontrast,thetimescale1/γ(0) denotestherelaxationtime
k
exponentialfunction(5)withthecharacteristictime duetoMarkovfluctuationsonly.Fromtheevolutionequation
(10),τ(U) isexactlyrepresentedbythefrictioncoefficients,
k
−1
τk(e) =(cid:16)γk(0)+γk(Γ)(cid:17) . (15) τ(U) = γ(0)+γ(Γ) −1. (19)
k k k
(cid:16) (cid:17)
Next,atτ τ(Γ),anapproximatesolutionofEq.(10)is
≪ k Therefore,τ(U) isnearlyequalto τ(e) accordingto Eq.(15);
k k
itsaccuracyhasbeenconfirmedwithinthemarginsofnumeri-
Uˆ (τ)=1 γk(Γ) τdτ′ γ˜ τ(Γ)+τ′ + (τ3) (16) calerror,indicatingthattheapproximation(14)isreasonable.
k − τ(Γ) Z0 (cid:16) k k (cid:17) O Thecharacteristictime scale τ(Γ), plottedasa functionof
k k
normalizedwavenumberinFig.7,hasaweakpeakinkˆ < 2
with γ˜ = γ(0)/γ(Γ). Thus, if γ˜ 1, the time range can for each ε. Let k denote the wave number at which τ(Γ)
k k k k ≪ Γ k
be separated into τ γ˜ τ(Γ) and γ˜ τ(Γ) τ τ(Γ). In hasapeak,wherekΓ wasdeterminedbyamethodformulti-
≪ k k k k ≪ ≪ k peakfitting so that roughnessof curveswas smoothened. In
the shortertime regimeτ γ˜ τ(Γ), called thebare-friction
≪ k k expectationthatthe coherencyin patch domainsleadsto the
stage, modal relaxation decays linearly with slope γk(0). In non-Markov memory effect, the time scale τ(Γ) should in-
contrast, it reduces to time-reversible algebraic decay in the k
cludefeaturesrelatingtothepatchdomains. Thelengthscale
longertimeregimeγ˜kτk(Γ) ≪ τ ≪ τk(Γ), i.e., theearlystage λΓ = 2π/kΓ is several times larger than the diameter of an
ofthedualstructurewithcharacteristictime electroconvectiveroll.Inaddition,thepowerlawλ ε−1/2
Γ
∝
isquantitativelyreasonableasindicatedinFig.8. Therefore,
τ(a) = 2τ(Γ) γ(Γ). (17) it follows that the dual relaxation caused by TFs originates
k r k k fromthepatchstructure. Thetimescalerapidlydecreasesat
.
largerlengthscales k < k , indicatinginter-patchdynamics
Γ
Therefore,asmallratioγ˜ ofthefrictioncoefficientsisanec- does not affect the memory effect due to TFs. Some work
k
essary requirement for the appearance of dual relaxation in doessupporttherelationshipbetweenthepatchstructureand
SMT.ThecharacteristicstagesaresummarizedinTableI. thedualstructure[17,28,31],wherethedynamicsofatagged
6
TABLEI.Characteristicstagesofmodalrelaxationdynamicsinsoft-modeturbulence.
stage timerange decayform
Bare-frictionstage τ ≪γ˜ τ(Γ) LineardecaycausedbytheMarkovfluctuations
k k
Earlystage γ˜ τ(Γ) ≪τ ≪τ(Γ) Time-reversibledecaythatoriginatesfrommemoryduetoTFs[Eq.(4)]
k k k
Latestage τ(Γ) ≪τ Simpleexponentialdecayafterturbulentmixing[Eq.(5)]
k
0
10
0.04 ε = 0.40
k 2 -2 (µ m ) Γ 00..0020 γ k(0) -1 (s ) 1100--21 ε = 0.20 ε = 0.050
-3
0.0 0.1 0.2 0.3 0.4 10
0 1 2 3 4 5
ε
1
10
ε = 0.40
FIG.8.ε-dependenceofk atwhichτ(Γ)hasalocalmaximum.The
Γ k ) 0
gathnreadyScdM1aTs=hpe0adt.c0lih5n2es.tmruTachrtkeusreetx:hpeξoln∼ineneεt−fiot1,f/k2thΓ2.e=pocw0e+rlca1wεwisisthugcg0e=ste0d.0fr0o4m5 Γ) −1k (s 10
particleintheSMTdisordercanbedividedintotwotypesof γ ( 10-1 ε = 0.20 ε = 0.050
mode: onedominatedbyconvectiverollingwithinpatchdo-
-2
mains(i.e.,intra-patchdynamics)andtheotherdominatedby 10
transferswithslowpatchmovements(i.e.,inter-patchdynam- 0 1 2 k 3 4 5
ics).
Figure 9 shows corresponding dispersion relations. The
frictionγ(Γ)isoneorderlargerthanthebarefrictionγ(0)near
k k
k ,butconvergesatlargewavenumbers.Acharacteristicfea- FIG.9. (coloronline)Dispersionrelationsforseveralεvalues;ε=
Γ
tureisadipappearingnearkˆ 1inγ(0)butnotinγ(Γ). The 0.050(purple),0.20(green),and0.40(red).Thetoppanelgivesthe
bare friction γ(0) and the stat≃ic frictiokn γ(Γ) causedkby TFs barefrictioncoefficientγk(0),andthebottompanelgivesfrictionγk(Γ)
k k causedbyTFs.
areexplicitlyrelatedtotheaveragerateinentropyproduction
S˙ [29]. Overmacroscopictimescale,itisanalyticallyrepre-
k
sentedas
clearly appears in SMT at intermediate length scales, where
S˙k =kB(cid:16)γk(0)+γk(Γ)(cid:17)=kB.τk(U) (20) τekf(fiΓc)iiesnntodtuetotoosThFosrtiscommupcahreladrgweirththτak(nU)thaenbdatrheefrfircictitoionn,ic.eo.-,
wherek istheBoltzmannconstant. Thedipmightreflecta γ˜k 1.
B ≪
lawofminimumentropyproductionrateinelectroconvection. Mori and Okamura have proposed the affinity hypothesis
The time scales, τ(U), τ(Γ), and γ˜ τ(Γ), are compared in betweenthecorrelationandmemoryfunction[3],inwhichthe
k k k k correlationandmemoryfunctionshavethesameformofde-
Fig.10forseveralεvalues.Thecorrelationtimeτ(U)ismuch
k cay with differentmagnitudesand characteristic time scales.
longer than τk(Γ) at large length scales, where the relaxation The main prediction from the hypothesis is that, when τk(U)
dynamicsisapproximatelyrepresentedbyasimpleexponen- isalmostthesameasτ(Γ),decayofthecorrelationfeaturesa
tial. Theratioτ(U)/τ(Γ) approachesunitywithincreasingkˆ, k
k k long-timetailwithexponent 3/2overallspatialdimensions.
signifyingthat the memory effectpersists for relatively long ThosetimescalesforSMTa−realmostthesameforkˆ & 3.0,
times at short length scales. Meanwhile, γ˜kτk(Γ) approaches asrepresentedinFig.10. Ourmodalcorrelationfunctionsin
τ(Γ) at large kˆ, implying that the duration of the early stage thelong-timeregionhoweverhavealargemarginoferror;we
k
is shorter at small length scales. Therefore, dual relaxation cannotevaluatewhethersuchlong-timetailsexistornot.
7
2 theprojection-operatorformalisminwhichthememoryfunc-
10
tionnaturallyarisesastheautocorrelationfunctionofthenon-
ε = 0.050
thermal fluctuations. Indeed, we have verified the expecta-
)
s
es ( 101 tion;thepeakwavenumberkΓ (Fig.8)ofthetimescaleτk(Γ)
m (Fig.7)suggeststhatthedualrelaxationiscausedfromintra-
c ti 0 patchdynamics.
sti 10 Solving the evolution equation [Eq. (7)] derived in
eri the projection-operator formalism, we revealed two well-
ct -1 separated contributions for the non-thermal fluctuations in
a 10
r
a Fig. 5: rapidly-varying Markovian fluctuations and non-
h
c -2 MarkovTFs. Theformerrelatestothebarefrictionγk(0) and
10 the latter to the memory function Γ (t) [Eq. (9)]. SMT dy-
0 1 2 3 4 5 k
2 namicswere shownanalytically to separate into three stages
10
(Table I): the bare-friction stage, τ γ˜ τ(Γ), where the
es (s) 101 ε = 0.10 rτe(lΓa)x,awtiohnereisthlieneraerladxeactiaoyn;itshetimeaer-lryevse≪tarsgieb,lekγ˜akklτgk(eΓb)ra≪ic dτec≪ay
m k
c ti 0 [Eq.(4)];andthelatestage,τk(Γ) ≪τ,wheretherelaxationis
sti 10 simpleexponentialdecay[Eq.(5)]. Further,bythevariables
ri definedin thetheory(i.e., γ(0), γ(Γ), andτ(Γ)), we havean-
racte 10-1 alyticallyspecified timescalkes obktainedin ekxperiments: τk(a)
ha [Eq.(17)],τ(e) [Eq.(15)],andτ(U) [Eq.(19)].
c k k
-2 Comparisonbetweenthecharacteristictimescales(Fig.10)
10
0 1 2 3 4 5 clarified the nature of the dual structure in SMT. It clearly
2 appearsatintermediatelengthscaleswherethecharacteristic
10
time τ(U) of the modal relaxation is not too short compared
mes (s) 101 ε = 0.40 fwriicthtiothnkatcoτek(fΓfi)coiefnttheγk(mΓ)emduoerytofuTncFtsioinsdmuuectholTaFrgse.rAthlsaon, tthhee
c ti 0 barefrictionγk(0).
sti 10 ThephysicaloriginoftheMarkoviannon-thermalfluctua-
eri tionsis still an open question. The dip near kˆ 1 in Fig. 9
act 10-1 suggeststhattheMarkovcontributioncontainssp≃atialfeatures
r
ha relating to electroconvection. That is, the bare friction con-
c
-2 sists of notonlythermalfluctuationsbut also rapidvariation
10 in non-thermal ones. Note that the Markovian non-thermal
0 1 2 3 4 5
k fluctuationsseemnottoaffecttheSMTpatterndynamics,but
actsasatriggerfortheonsetofSMT.Ifnon-thermalMarkov
fluctuationsplayedaroleinpatterndynamics,themodalcor-
relationwouldbreaktime-reversalsymmetryevenattheearly
stage; nevertheless, early-stage relaxation is invariant under
FIG. 10. Plot of the characteristic times τ(U) (dashed line), τ(Γ)
k k timereversal[Eq.(4)].
(darksolidline), andγ˜ τ(Γ) (lightsolidline)forε = 0.050(top),
k k BesidesSMT,severaltypesofspatiotemporalchaosappear
0.10(middle),and0.40(bottom),respectively.
intheelectroconvectionoftheliquidcrystalsystems. Forex-
ample, instead of the homeotropicanchoring, one can apply
the memory-function procedure to the planner anchoring as
IV. CONCLUDINGREMARKS
the boundary condition in which the rotational symmetry is
notaliveinthex–y plane. Ourapproachproposedinthispa-
We experimentally observed the modal autocorrelation per can pave the way to understandthem universally from a
function Uˆ (t) of pattern dynamics in SMT. The modal re- viewpointofthenon-thermalfluctuationsrelatingto thepat-
k
laxationdynamicsfeaturedadualrelaxation(Fig.3);thecor- tern.Thiswillbediscussedelsewhere.
relation is well described by the time-reversible function at
the early stage and by a simple exponentialrelaxation at the
late stage. The correspondingtime scales τ(a) and τ(e) obey ACKNOWLEDGMENTS
k k
dynamicscalinglawsfortheintra-patchscales(Fig.4).
Withanexpectationthatthepatchstructuregeneratesmem- The authorsgratefullyacknowledgeDr. MakotoOkamura
oryeffectsduetothenon-thermalfluctuations,weemployed (KyushuUniversity)andDr.KazumasaA.Takeuchi(theUni-
8
versity of Tokyo) for productive discussions. This work termR(0)(t)satisfies
i
was partially supported by KAKEHNI (Nos. 21340110 and
24540408),aGrant-in-AidforScientificResearchonInnova- (0)
R (t+τ) (A(t)) =0
tiveAreas—”EmergenceinChemistry”(No.20111003),and D i F E
the JSPS Core-to-Core Program ”International research net-
for an arbitraryfunction (A) of a set of macroscopicvari-
workfornon-equilibriumdynamicsofsoftmatter.” F
ables, A .Thenonlinearityof A isincludedinv (t)and
i i i
{ } { }
J (t), and one can extractthe linear partof A from v (t)
i i i
{ }
andJ (t) usingthe linear projection-operatormethod. Mori
i
AppendixA:Projection-OperatorFormalism andFujisakaderivedtheevolutionequationforthecorrelation
functionU (τ):= A (t+τ)A∗(t) as[29,39]
ij i j
Physically, a macroscopic system can be divided into (cid:10) (cid:11)
∂U (τ) τ
slowly varying behavior described by a set of macroscopic ij = iω U (τ) dτ′Γ′ (τ τ′)U (τ′),
variables {Ai} and rapidly-varying terms [32, 33]. The ∂τ Xl il lj −Xl Z0 il − lj
projection-operator formalism mathematically allows us to (A2)
separatethesetwodynamics. where ω denotesthe mechanicalcoefficientand Γ′ (τ) the
ij il
Using the linear projectionoperator, Morimathematically memoryfunction.NotethatEq.(A2),whichrelatesthemem-
derived a generalized linear Langevin equation containing oryΓ′(τ)tothenon-thermalfluctuations,isofthesameform
a memory function [34]. The memory function is repre- as the evolution equation derived in the linear projection-
sentedastemporalcorrelationofthefluctuationsr (t),where operatorformalism;therefore,fromaphysicalpointofview,
i
r (t+τ)A (t) = 0. The generalized linear Langevin the obtained memory functions should be checked to see
i j
h i
equation can be employed for not only equilibrium but also whetherthememoryeffectoriginatesfromnon-thermalfluc-
non-equilibrium systems. However, in the linear projection tuationsornot. Ifthecharacteristictimescaleforthecorrela-
scheme, the fluctuation term r (t) is orthogonal only to the tionofR(0)(t)isextremelyshort,thenR(0)(t)satisfies;
i i i
linearfunctionsof A andcanconsistofnotonlythemicro-
i
{ }
scopic degreesof freedombutalso fluctuationsfromnonlin- R(0)(t+τ)R(0)(t) γ(0)δ(τ), (A3)
reealretvearnmtsinofn{oAneiq}u.iSliubcrhiunmonsy-tshteermmsa[l3fl5u,c3t6u]a.tionsarepossibly where γi(j0) deDnoties the barejfrictioEn≃coeifjficient. In addition,
if the fluctuations extracted from the nonlinear terms do not
Zwanzig formulated the generalized nonlinear Langevin correlate with R(0)(t), the memory functiondue to the non-
equationbythenonlinearprojectionoperator[37,38]as i
thermalfluctuationscanbedividedintotwodifferentterms,
A˙i(t)=vi(t)+Ji(t)+Ri(0)(t), (A1) Γ′ij(τ)=2γi(j0)δ(τ)+Γij(τ), (A4)
where v (t) denotes the streaming term including nonlinear where Γ (τ) is regarded as the temporal correlation of the
i ij
reversibletermsandJ (t)anirreversibleterm.Thefluctuation non-Markovcontributionofthenon-thermalfluctuations.
i
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