On the physical interpretation of fractional diffusion

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Enrique NADAL, Emmanuelle ABISSET-CHAVANNE, Elias CUETO, Francisco CHINESTA - On

the physical interpretation of fractional diffusion - Comptes Rendus - Mecanique - Vol. 346(7),

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Model reduction, data-based and advanced discretization in computational mechanics

On

the

physical

interpretation

of

fractional

diffusion

Enrique Nadal

a

,

∗

,

Emmanuelle Abisset-Chavanne

b

,

Elias Cueto

c

,

Francisco Chinesta

d

,

∗

a_{ResearchCentreonMechanicalEngineering(CIIM),UniversitatPolitècnicadeValència,CaminodeVeras/n,46071Valencia,Spain} b_{HighPerformanceComputingInstitute&ESIGROUPChair,ÉcolecentraledeNantes, 1,ruedelaNoë,BP92101,44321Nantescedex3,France} c_{I3A,UniversityofZaragoza,MariadeLunas/n, 50018Zaragoza,Spain}

d_{PIMMLaboratory&ESIGROUPChair,ENSAMParisTech,151,boulevarddel’Hôpital,75013Paris,France}

a

b

s

t

r

a

c

t

Keywords: Fractional calculus Anomalous diffusion Non-local models

Evenifthe diffusionequationhasbeenwidelyusedinphysics andengineering,and its physicalcontentiswellunderstood,somevariantsofitescapefullyphysical understand-ing.Inparticular,anormaldiffusionappearsintheso-calledfractionaldiffusionequation, whosemainparticularityisitsnon-localbehavior,whosephysicalinterpretationrepresents themainpartofthepresentwork.

1. Introduction

1.1. Fromstandarddiffusiontoanomalousdiffusion

Themicroscopicnatureofdiffusionisevident fromthepioneeringworksofEinstein,who consideredtheincrementin theparticleposition



(assumeddefined,forthesakeofsimplicity,intheunboundedone-dimensionalaxisx)asarandom variable,withaprobabilitydensitygivenby

φ ().

Thus,theparticlesbalancecanbeexpressedbyboth

ρ

(

x

,

t

+

τ

)

=

ρ

(

x

,

t

)

+

∂

ρ

∂

t

τ

+ (

τ

2

₎

₍₁₎ and

ρ

(

x

,

t

+

τ

)

=



R

ρ

(

x

+ ,

t

)φ ()

d



(2) Developing

ρ

(

x

+ ,

t

),

ρ

(

x

+ ,

t

)

=

ρ

(

x

,

t

)

+

∂

ρ

∂

x



+

1 2

∂

2

ρ

∂

x2



2

_{+ (}

3

₎

₍₃₎

*

Corresponding author.

E-mailaddresses:[email protected](E. Nadal), [email protected](E. Abisset-Chavanne), [email protected](E. Cueto),

[email protected](F. Chinesta).

https://doi.org/10.1016/j.crme.2018.04.004

which,substitutedintotheright-handsideofEq. (1),andtakingintoaccountthenormalityandexpectedsymmetry

 

R

φ ()

d



=

1



R

φ ()

d



=

0 (4)

leads,afterequatingEqs. (1) and(2),to

ρ

(

x

,

t

)

+

∂

ρ

∂

t

τ

=

ρ

(

x

,

t

)

+

1 2

∂

2

ρ

∂

x2



R



2

φ ()

d



(5) or

∂

ρ

∂

t

τ

=

1 2

∂

2

ρ

∂

x2



R



2

φ ()

d



(6)

Ifwedefinethediffusioncoefficient D as D

=

1

2

τ



R



2

φ ()

d



(7)

theparticlebalance,alsoknownasthediffusionequation,isgivenby

∂

ρ

∂

t

=

D

∂

2

ρ

∂

x2 (8)

Theintegrationofthisequation,assumingthatalltheparticlesarelocalizedattheoriginattheinitialtime,

ρ

(

x

,

t

=

0)

=

δ(

x

),

leadsto

ρ

(

x

,

t

)

=

√

1

4

π

Dte −x2

4Dt ₍₉₎

whosesecond-ordermoment(variance)scaleswithtime



x2

 =

2Dt (10)

thatis,themeansquareddisplacementscaleswiththeelapsedtimet,andthediffusioncoefficientD.

Thesameequationcanbederivedbydescribingdiffusionasarandomwalk.Again,forthesakeofsimplicity,werestrict ourdiscussiontothe1Dcase,withthex-axisequippedwithagridofsize



x.Inadiscretetimestep



t,thetestparticleis assumedtojumptooneofitsnearestneighborsites,withrandomdirection.Suchaprocesscanbemodeledbythemaster equationthatwrites,atsite j,

Wj

(

t

+ 

t

)

=

1

2Wj+1

(

t

)

+

1

2Wj−1

(

t

)

(11)

where Wj

(

t

)

representstheprobability ofhaving theparticleatsite j attime t andthe pre-factor1/2 accounts forthe

directionisotropyofthejumps. BytakingclassicalTaylorexpansions,

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

Wj

(

t

+ 

t

)

=

Wj

(

t

)

+

∂ Wj(t) ∂t

_t



t

+ (

t2

)

Wj+1

(

t

)

=

Wj

(

t

)

+

∂W_∂x(t)

_j



x

+

12 ∂2_W(t) ∂x2

j



x 2

_{+ (}

_x3

₎

Wj−1

(

t

)

=

Wj

(

t

)

−

∂W_∂x(t)

j



x

+

1 2 ∂2_W(t) ∂x2

j



x 2

_{− (}

_x3

₎

(12)

that,injectedintoEq. (11),yield

∂

W

∂

t

=

D

∂

2W

∂

x2 (13)

withD definedinthelimitof



x

→

0 and



t

→

0 by

D

=



x 2

2



t (14)

whichleadstothediffusionequationpreviouslyderived.

In complex fluids, micro-rheological experiments often exhibit anomalous sub-diffusion or sticky diffusion, in which the mean square displacement of Brownian tracer particles is found to scale as



x2

_

_∝

_{tα , 0}

_<

_α

_<

_{1 (see [1] and the}

Fig. 1. Schematic cartoon of the derivative of order 1.7.

referencestherein). Inthesecases,the useofnon-integer derivativescanconstitute an appealingalternative,asit allows onetocorrectlyreproducetheobservedphysicalbehaviorwhilekeepingthemodelassimpleaspossible.Moreover,froma physicalpointofview,theuseofnon-integerderivativesintroducesadegreeofnon-localitythatseemstobeinagreement withtheintrinsicnatureofthephysicalsystem,asdiscussedlater.

Asprovedin[2,3] fractionaldiffusionmodelscanbederived fromcontinuoustimerandomwalks–CTRW–leadingto afractional diffusionequationinthesamemannerasstandardrandomwalksleadtotheusual diffusionequation,where subdiffusion isassociatedwithlongrest,whereas superdiffusionisrelatedwithlongjumps. Adetailedderivationcan be foundin[4].

In [5], we reinterpreted,by usingfractional diffusion,the rheological findings reportedin [6], which were difficultto explain using standardmodels. However, that purely phenomenological route,which assumednon-standard randomizing mechanisms, failed to give a physical picture of the subjacent reality. Thus, despite the fact that a surprising excellent agreementbetweenthepredictionsandtheexperimentalmeasurements,bothinlinearviscoelasticityandrelaxationafter astepstrain,wasfound,nophysicalmechanismswereproposedtojustifytheconsideredanomalousdiffusion.

Aphysically-based explanation was advancedin [7] whereparticles involved inthesuspension were subjected tothe bombardmentcomingfromtheneighbormoleculesofsolvent,aviscousdragandanelastictermrelatedtotheinteraction withneighboringparticles.Thestochasticinertia-freesimulationprovedthatthemeansquaredisplacementdoesnotscale with the time, but witha power of it, lower than one. The analysis of the covariance reveals that the motion can be assimilatedto afractional Brownianmotion,that allowed ustoassociatesuch a motionwiththesolutionto afractional Fokker–Planckdiffusionequation.

Other anomalies have been reported and explained through involving fractional models. One of them concerns the fact that theapparent conductivity inalmost one-dimensional systems(e.g., carbonnanotubes)or two-dimensional (e.g., graphene)dependsonthesystemssize,beingthisdependencemoreintensewhenthesystemdimensionalitydecreases[8]. Theseworksclaimedtheneedofassociatingfractionalmodelswithfractalmedia[9,10].

1.2. Fractionalderivatives

Inthissection,we proposea generaldiffusiveflux definitionbasedontheuseoffractional derivatives.Thisfractional fluxwillbecoupledwiththeusual balanceequation(whichweassume ofintegernature).Beforemaking theseproposals anddevelopments,westartbyrevisitingsomekeyresultsoffractionalcalculusconsideredallalongthepresentwork.

There existmanygood bookson fractional calculus andfractional differentialequations (e.g.,[11,12]).Wesummarize herethemainconceptsneededtounderstandthedevelopmentscarriedoutbelow.

WestartwiththeformulaattributedtoCauchytoevaluatethen-thintegration,n

∈ N

,ofafunction f

(

t

):

Jnf

(

t

)

:=



· · ·



f

(

τ

)

d

τ

=

1

(

n

−

1

)

!

t



0

(

t

−

τ

)

n−1f

(

τ

)

d

τ

(15)

Thiscanberewrittenas

Jnf

(

t

)

=

1

(

n

)

t



0

(

t

−

τ

)

n−1f

(

τ

)

d

τ

(16)

where

(

n

)

= (

n

−

1)

!

isthegammafunction.Thelatterbeing,infact,definedforanyrealvalue

α

∈ R

,wecandefinethe fractionalintegralfrom

Jαf

(

t

)

:=

1

(

α

)

t



0

(

t

−

τ

)

α−1f

(

τ

)

d

τ

(17)

Ifweconsiderthefractionalderivativeoforder

α

andselectanintegerm

∈ N

suchthatm

−

1

<

α

<

m,thenitsufficesto consideranintegerm-orderderivativecombinedwitha(m

−

α

)fractionalintegral(thisisdepictedinsketchforminFig.1).

Obviously,we couldtakethederivative oftheintegralortheintegralofthederivative,resultingintheRiemann–Liouville andCaputodefinitionsofthefractionalderivativeusuallydenotedby

D

α f

(

t

)

and

D

α_∗f

(

t

),

respectively,

D

αf

(

t

)

=

⎧

⎪

⎨

⎪

⎩

dm dtm

1 (m−α) t



0 f(τ) (t−τ)α+1−md

τ



,

m

−

1

<

α

<

m dm_f(t) dtm

,

α

=

m (18) and

D

α_∗f

(

t

)

=

⎧

⎪

⎨

⎪

⎩

1 (m−α)

t



0 dm f(τ) dτm (t−τ)α+1−md

τ



,

m

−

1

<

α

<

m dm_f(t) dtm

,

α

=

m (19)

Becausetheseapproachestothefractionalderivativebeganwithanexpressionfortherepeatedintegrationofafunction, one could consider a similar approach forthe derivative. This was the route considered by Grunwald andLetnikov (GL) definingtheso-called“differintegral”,whichleadstothefractionalcounterpartoftheusualfinitedifferences.

Whenconsideringspacedifferentialoperators,theleft- andright-hand-sideformsoftheRiemann–LiouvilleandCaputo definitionsareused[13].Thusweconsiderthedefinitionsofbothderivatives

x 0

D

αf

(

x

)

=

dm dxm

⎛

⎝

1

(

m

−

α

)

x



0 f

(ξ )

(

x

− ξ)

α+1−md

ξ

⎞

⎠

(20) and L x

D

αf

(

x

)

=

dm dxm

⎛

⎝

1

(

m

−

α

)

L



x f

(ξ )

(ξ

−

x

)

α+1−md

ξ

⎞

⎠

(21)

inthecaseoftheRiemann–Liouville,and

x 0

D

α∗f

(

x

)

=

1

(

m

−

α

)

⎛

⎝

x



0 dmf(ξ ) dξm

(

x

− ξ)

α+1−md

ξ

⎞

⎠

(22) and L x

D

α∗f

(

x

)

=

1

(

m

−

α

)

⎛

⎝

L



x dm_{f(ξ )} dξm

(ξ

−

x

)

α+1−md

ξ

⎞

⎠

(23)

whenconsideringtheCaputofractionalderivative.

Oneimportantpropertythatwillbeusedallalongthisworkisthefactthat,whenthederivativeapproachesaninteger value,theassociatedexpressionreducestoitsstandardintegerexpression.Forthesakeofcompleteness,wereproducethe proofitinthecaseoftheRiemann–Liouvillederivative.Forthatpurpose,weconsider

t a

D

αf

(

t

)

=

dm dtm

⎛

⎝

1

(

m

−

α

)

t



a f

(

τ

)

(

t

−

τ

)

α+1−md

τ

⎞

⎠

(24)

that,byintegratingbyparts,results

t a

D

αf

(

t

)

=

dm dtm

⎛

⎝(

t

−

a

)

m−αf

(

a

)

(

m

−

α

+

1

)

+

1

(

m

−

α

+

1

)

t



a

(

t

−

τ

)

m−αd f

(

τ

)

d

τ

d

τ

⎞

⎠

(25)

Bytakingthelimitwhen

α

→

m leadsto

t a

D

α→mf

(

t

)

=

dm dtm

(

f

(

a

)

+ (

f

(

t

)

−

f

(

a

)))

=

dmf

(

t

)

dtm (26)

where,otherthantheinterestofprovingthatfractionalderivativesapproachwithcontinuitytheonesofintegerorder,we justprovedthat,foranyfunction f

(

t

),

when

α

→

m

Lim_α→m 1

(

m

−

α

)

t



a

(

t

−

τ

)

m−α−1f

(

τ

)

d

τ

=

f

(

t

)

(27)

1.3. Paperoutline

Theapproach basedontheuseofCTRW remainstoo microscopictoallow asimpleandphysicalinterpretationofthe resulting macroscopic fractional equations. In thispaper, we focusin superdiffusion andwe are following two different routesto extract its physical content,based on the useof theRiemann–Liouville andCaputo fractional derivatives.Next section defines fractional fluxes, the associated fractional diffusion equation, and illustrates the main implications on a simplenumericalcase.Then,Section3proposesaphysicalviewoftheRiemann–Liouville–RL–fractionaldiffusion,while Section4addressesasimilaranalysisontheCaputofractionaldiffusion,toconcludewithashortdiscussionontheproposed physicalinterpretations.

2. Fromfractionalfluxestothefractionaldiffusionequation TheFourierorFickequationforstandarddiffusionwrites

Q

(

x

)

= −K

du

(

x

)

dx (28)

where

K

isthediffusioncoefficient.

Thisequationcanbegeneralized,fornon-integerderivatives,inthefollowingway

Q

α

(

x

)

= −

1 2

K



x 0

D

αu

(

x

)

+

Lx

D

αu

(

x

)



(29) and

Q

α_∗

(

x

)

= −

1 2

K



x 0

D

α∗u

(

x

)

+

Lx

D

α∗u

(

x

)



(30)

fortheRiemann–Liouville andCaputoderivativesrespectively;itcanbenoticedthatthespatialgradientoperatorhasbeen symmetrizedtorepresentthesymmetric(spatial)physicsofdiffusion.

Thefractionaldiffusionequationresultsfromthemassorenergybalances,

∂

u

∂

t

=

d

dx

F

(

x

)

(31)

where

F

(

x

)

referstothefluxatpositionx

∈ (

0,L

)

⊂ R

,inparticularthestandardoranomalousfluxes,

Q

(

x

),

Q

α

(

x

)

or

Q

α_∗

(

x

).

Inwhatfollows,werestrictouranalysistothecasewherem

=

1 and0

<

α

≤

m

=

1.

2.1. Numericalresults

Tocomparetheheatfluxbehaviorinthecaseofstandardandfractionalfluxes,wesolvetheheatequationinthedomain

(0,

L

),

forL

=

1 and L

=

5,whileenforcingasboundaryconditionsu

(

x

=

0)

=

0 and u

(

x

=

L

)

=

L,withconductivity

K

=

1,

α

=

1 and

α

=

0.5 inthestandardandfractionalflux,respectively.

Inthestandardcase, thesolutionbecomes u

(

x

)

=

x,andconsequentlytheassociatedflux u

(

x

)

= −

1.Inthefractional case(when usingtheCaputofractionalderivative),theexpectedsolutionisagainu

(

x

)

=

x,howeverfluxesareexpectedto dependonthedomain’slength.Figs.2and3corroboratetheseexpectations.

3. Riemann–Liouville–RL–fractionaldiffusion

WhenconsideringtheRLflux

Q

α ,theleftcontributiontothefluxinvolvestheintegral

x 0

D

αu

(

x

)

=

d dx

⎛

⎝

1

(

1

−

α

)

x



0 u

(ξ )

(

x

− ξ)

αd

ξ

⎞

⎠

(32)

We justproved that when

α

→

1,it becomes ₀x

D

α→1u

(

x

)

=

du_dx(x),that is,the standard integerderivative, leading tothe standarddiffusive flux.However, theprevious equation reflectsthefact that,foranychoiceof

α

,realorinteger,theflux canbe viewedastheresultofanintegral, evenif,inthe integercase,the integralcollapsestothe localgradient(inany caseanalternativeintegralformcanbeassociatedwithit).

To better apprehend its physicalcontent, we proceed by takingits derivative [14], and even ifthe derivative will be introducedintotheintegral,theresultingexpressionwilldifferfromtheCaputoformulation,asprovedlater.

ByapplyingtheLeibniz’sruletotheright-handmemberofEq. (32),itresults(aftertakingintoaccountthat,asproved in[14],thetermatx cancelswiththeright-fluxcontribution)

Fig. 2. Standard flux model withK=1: (top) temperature field; (bottom) flux field; (left) L=1 and (right) L=5.

Fig. 3. Fractional flux model withK=1 andα=0.5: (top) temperature field; (bottom) flux field; (left) L=1 and (right) L=5.

d dx

⎛

⎝

1

(

1

−

α

)

x



0 u

(ξ )

(

x

− ξ)

αd

ξ

⎞

⎠ = −

α

(

1

−

α

)

x



0 1

(

x

− ξ)

α+1u

(ξ )

d

ξ

(33)

where the derivative applies on the kernel without affecting the field u

(ξ ),

making the difference with respect to the Caputo’sexpressionaddressedlater.

Thus,theresultingfluxatpositionx,

Q

α

(

x

),

reads

Q

α

(

x

)

=

1 2

K

α

(

1

−

α

)

⎧

⎨

⎩

x



0 1

(

x

− ξ)

α+1u

(ξ )

d

ξ

−

L



x 1

(ξ

−

x

)

α+1u

(ξ )

d

ξ

⎫

⎬

⎭

(34)

This expression reflects the fact that the field u

(ξ )

at position

ξ

induces a flux at x, independently of the distance between

ξ

andx, evenif, asexpected, theweight affectingthe contributionof u

(ξ )

onthe flux atx decreaseswiththe mutualdistance,inparticularwith

|

x

− ξ|

1+_{α .Thisresultisnottotallysurprising,becauseitseemsinagreementwiththe}

Einsteinrationaleexposed intheIntroduction orits extendedCTRWformulation[4]. Inthatapproach,thereisanon-zero probabilitythataparticlelocatedfarfromx reachesx inasingletimestep.

Ifweconsider,inEq. (34),bothintegralsforthelimitcase

α

≈

1 andassumethatthemaincontributiontobothintegrals occursintheinterval

[

x

−



/2,

x

+



/2

]

,itresults

x



0 1

(

x

− ξ)

2u

(ξ )

d

ξ

−

L



x 1

(ξ

−

x

)

2u

(ξ )

d

ξ

≈

x



x−/2 1

(

x

− ξ)

2u

(ξ )

d

ξ

−

x+

_

/2 x 1

(ξ

−

x

)

2u

(ξ )

d

ξ

≈

1

(

₂

)

2u



x

−



2





2

−

1

(

₂

)

2u



x

+



2





2

=

2





u



x

−



2



−

u



x

+



2



(35) Now,byexpressing u



x

−



2



=

u

(

x

−



)

+

u

(

x

)

2 (36) and u



x

+



2



=

u

(

x

+



)

+

u

(

x

)

2 (37)

introducingbothintoEq. (35) leadsto

2





u



x

−



2



−

u



x

+



2



=

u

(

x

−



)

−

u

(

x

+



)



≈ −

2 du

(

x

)

dx (38)

which,asexpected, isrelatedto theusual gradientwiththe opposite sign,which combinedwithEq. (34), resultsinthe standard flux.The main differencewith fractionaldiffusion isthat, when

α

differs fromone, theintegral requires richer approximationinvolvinglargerintervalsaroundx.

An immediate consequence is that, if the field u

(

x

)

is symmetric with respect to x, the net flux vanishes. Another important consequence is that the physics induces a characteristic length, the one characterizing the domain around x, whose contributionto the integral can not be ignored. When

α

decreases, this characteristiclength increases, andthen transportpropertieswilldepend onthedomainsize L,afact whichcouldexplain thefindingsreportedin[15,16].When

α

≈

0,theintegralsinvolvedinEq. (34) diverge.

Toapplynowthespacederivativeinvolvedinthebalanceequation(31),itsufficestoconsiderthederivativeofEq. (34) usingagaintheLeibnizrule,whichimpliesthesecondderivativeofthekernelfunction.

Thus,themainconsequenceofusingtheRiemann–Liouvillederivativeistheobtentionofanon-localintegralform im-plyingthatthefarfield(farwithrespecttox)affectsthefluxatpositionx,inagreementwiththemicroscopicinterpretation ofdiffusionfromcontinuoustimerandomwalk,andmoreparticularlyitsLevyvariant.Thecaseofstandarddiffusionisnot anexception;itcanbeformulatedandinterpretedwithinthesameframework,evenifitsfinaldescriptionbecomeslocal. 4. Caputofractionaldiffusion

Now,weperformasimilaranalysis,butnowconsideringtheCaputofractionalderivative.Thefluxinthiscasewrites

Q

α_∗

= −

1 2

K



x 0

D

α∗u

(

x

)

+

Lx

D

α∗u

(

x

)



(39) with x 0

D

α∗u

(

x

)

=

1

(

1

−

α

)

⎛

⎝

x



0 1

(

x

− ξ)

α du

(ξ )

d

ξ

d

ξ

⎞

⎠

(40)

Itisimportanttonotethat,byusingthepreviousresults,integral(40) resultsin

x

0

D

α→∗ 1u

(

x

)

=

du

(

x

)

dx (41)

Thus,thenetflux

Q

α_∗

(

x

)

reads

Q

α_∗

(

x

)

= −

1 2

K

1

(

1

−

α

)

⎧

⎨

⎩

x



0 1

(

x

− ξ)

α du

(ξ )

d

ξ

d

ξ

+

L



x 1

(ξ

−

x

)

α du

(ξ )

d

ξ

d

ξ

⎫

⎬

⎭

(42)

Thisexpression statesthattheflux atpositionx dependsontheexisting fieldfirst-orderderivative atposition

ξ

with anundoubtablephysicalcontent,expressingthefluxfromthederivativesatdifferentlengthscales,asdiscussedlater.

The pointinwhichtheCaputorouteseems advantageousfromaconceptualviewpointconcerns itsapplicationonthe balanceequationthat,makinguseofthefractionalderivativescompositionrule,reads

d dx

Q

α ∗

(

x

)

= −

1₂

K



x 0

D

α+∗ 1u

(

x

)

+

Lx

D

α+∗ 1u

(

x

)



(43) or d dx

Q

α ∗

(

x

)

∝

⎧

⎨

⎩

x



0 1

(

x

− ξ)

α d2u

(ξ )

d

ξ

2 d

ξ

+

L



x 1

(ξ

−

x

)

α d2u

(ξ )

d

ξ

2 d

ξ

⎫

⎬

⎭

(44)

When

α

→

1,byusingthepreviousresults,

Lim_α→1 1

(

1

−

α

)

⎧

⎨

⎩

x



0 1

(

x

− ξ)

α d2u

(ξ )

d

ξ

2 d

ξ

+

L



x 1

(ξ

−

x

)

α d2u

(ξ )

d

ξ

2 d

ξ

⎫

⎬

⎭ =

2 d2u

(

x

)

dx2 (45)

whichprovesthatthecurvatureatx canbeobtainedfromtheintegraloftheweightedcurvaturesatpositions

ξ.

Again,when

α

differsfromone,theintegralinvolveslargercharacteristicintegrationdomainsandthefinalintegraldoes notreducetoalocalexpression.

Ifwecomebacktoexpression(42),andapplythederivativeusingLeibniz’srule

d dx

Q

α ∗

(

x

)

=

1₂

K

α

(

1

−

α

)

⎧

⎨

⎩

x



0 1

(

x

− ξ)

α+1 du

(ξ )

d

ξ

d

ξ

−

L



x 1

(ξ

−

x

)

α+1 du

(ξ )

d

ξ

d

ξ

⎫

⎬

⎭

(46)

Asbefore,ifweconsiderbothintegralsinthelimitcase

α

≈

1 andassumethatthemaincontributiontobothintegrals occursintheinterval

[

x

−



/2,

x

+



/2

]

,itfollows

x



0 1

(

x

− ξ)

2 du

(ξ )

d

ξ

d

ξ

−

L



x 1

(ξ

−

x

)

2 du

(ξ )

d

ξ

d

ξ

≈

x



x−/2 1

(

x

− ξ)

2 du

(ξ )

d

ξ

d

ξ

−

x

_

+/2 x 1

(ξ

−

x

)

2 du

(ξ )

d

ξ

d

ξ

≈

2



du

(ξ )

d

ξ

x−₂

−

du

(ξ )

d

ξ

x+₂



(47)

Now,byexpressingbothderivativesfromtheonesexistingatx

−



,x andx

+



,itresults

2



du

(ξ )

d

ξ

x− 2

−

du

(ξ )

d

ξ

x+ 2



=

1





du

(ξ )

d

ξ

x−

−

du

(ξ )

d

ξ

x+



≈ −

2d 2_u

₍

_x

₎

dx2 (48)

When

α

differsfromone,thepreviousdiscreteapproximationisnotvalidanymore;however,thediscreteexpressionof bothintegralsin(46) reads

d dx

Q

α ∗

(

x

)

=

1 2

K

α

(

1

−

α

)





i 1

(

x

−

i



)

α+1 du

(ξ )

d

ξ

ξ=x−i



−



i 1

(

x

+

i



)

α+1 du

(ξ )

d

ξ

ξ=x+i





(49) orgrouping d dx

Q

α ∗

(

x

)

=

1 2

K

α

(

1

−

α

)



i

β

i 1 2i





du

(ξ )

d

ξ

ξ=x−i

−

du

(ξ )

d

ξ

ξ=x+i



≈ −

1 2

K

α

(

1

−

α

)



i

β

i

C

i

(

x

)

(50) where

C

_i

(

x

)

=

du(ξ ) dξ

_ξ=x+i

−

du(ξ ) dξ

_ξ=x−i 2i



(51)

whichresultsinasortofmulti-scalecurvatures

C

iallthemevaluatedatpositionx andallthemcontributingtothesolution

atpositionx.

5. Discussionandconclusions

Afterthe previous discussion onthe physicalcontent ofstandard andanomalous diffusion, thelast considered under twodifferentperspectives,theonesrelatedtotheRiemann–LiouvilleandCaputofractionalderivatives,theintegralnature ofdiffusionbasedonthesedescriptionsseemsestablished.Whenthederivativeconsideredinthefluxdefinitionisinteger, the integralcollapses intoa localstandard gradient (definingstandard flux)ora localstandard curvaturewhenapplying thedivergenceofthefluximposedbythebalanceequation.Theremainingquestion concernsthephysics thatimpliesthe existenceofanon-integerderivative.

A natural explanation consists in assuming that it emerges from larger particle flights than the ones concerned by standarddiffusion.Forexample,whenconsideringLevyflights,thejumplengthvariancediverges.

ByusingtheCaputoformulation,whichconsidersthefluxatpositionx fromthecontributionofasequenceofgradients involving differentcharacteristic lengths(discrete orcontinuously distributed), one could expect that, in the presenceof multi-scaleorfractalmedia,thefluxatpositionx iscomposed bythecontributionofthedifferent-scalefluxes,justifying theintegralexpression.

Inthecasewhenconsideringalmostone-dimensionalmedia,particlesatpositionx exhibitaspectrumofcorrelated be-havior,andconsequentlyagainthegradientsinvolvedinthefluxesshouldconcerndifferentcharacteristiclengths,explaining once moretheintegral expression ofthe fluxes.When dimensionalityincreases, long-distancecorrelationsdisappear, be-causeoftheinteractionscomingfromthenewdimension(s)andconsequentlyphysicsregainsitslocality.

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