Fractal traps and fractional dynamics

(1)

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Fractal traps and fractional dynamics

Pierre Inizan

To cite this version:

(2)

PIERREINIZAN

Abstra t. Anomalousdiusionmayariseintypi al haoti Hamiltoniansystems. A ord-ing toG.M.Zaslavsky'sanalysis,ades ription anbedone bymeansoffra tionalkineti s equations. However,thedynami alorigin ofthosefra tionalderivativesisstillun lear. In thistalkwestudyageneralHamiltoniandynami srestri tedtoasubsetofthephasespa e. Starting fromR.Hilfer'swork,anexpressionfor theaverageinnitesimalevolution of tra-je toriessetsisgivenbyusingPoin arére urren etimes. Thefra taltrapswithinthephase spa e whi haredes ribedby G.M.Zaslavskyarethentakenintoa ount, andit isshown that inthis ase, the derivative asso iated to this evolution may be ome fra tional, with orderequaltothetransportexponentofthediusionpro ess.

1. Introdu tion

Fra tional al ulus [26 , 21 , 22 ℄ is e iently used in several elds of physi s [12, 24℄. For example,itmaybeusedtotakeintoa ountmemoryee tsandanomaloustransport. Several equations of physi s have hen e been generalized to non-integer orders soas to provide new models. Among them,one nd theEuler-Lagrange equation [23 ,1, 2,5 ,7℄ and thediusion equation[20 , 4,9 ,18,27 ,29 , 6℄.

However,reasonsforemergen eofsu hoperatorsarestillun learandtheusethisformalism oftenremains heuristi . R.Hilfer [10 , 14, 11℄ and G.M.Zaslavsky [31, 34 , 25, 33℄ have tried through dierent ways to understand the physi al origin of fra tional derivatives. Both of theirmodelsrelyon the re urren e time notion.

The rst of those authors studies the evolution operator of a subsystem and shows that afteratemporalrenormalization, the asso iatedinnitesimalgenerator isafra tional deriv a-tive. However, the interpretationof this operator mayseem di ult and therenormalization pro edure isambivalent.

Zaslavsky is interested in haoti hamiltonian systems. He makes fra tional derivatives appear in the diusion equation related to the kineti (i.e. probabilisti ) des ription of the system. Withouta truejusti ationfor the introdu tion ofthose derivatives, he nevertheless onne tsthetransportexponent

µ

withthe fra tionalordersofderivationandthe oe ients of theself-similarstru tureswhi h appear inthe phasespa earound resonan e areas.

Inthepresentarti le,westudythedynami sofanHamiltoniansystem,presentedinse tion 2. With ideas taken from Hilfer, we fo us in se tion 3 on the evolution of a phase spa e subset under the hamiltonian ow. More pre isely, the asso iated innitesimal generator is onsidered. In several exampleswe showthat itis proportional to theusual derivative

d/dt

. Then we pre ise our model by taking into a ount the phase spa e stru ture des ribed by Zaslavskyandsumedupinse tion4. Inthat ase,weproveinse tion5thattheinnitesimal generator may turn into a fra tional derivative of order

µ

. We dis uss the relevan e of this exponent inse tion6,before on ludinginse tion 7.

(3)

Let

H

bean Hamiltonian dened ona ompa t manifold

M

. The indu edowis denoted

φ

t

. Let

m

beameasuredened on

M

. Let

G

beameasurable subsetof

M

and

m

′

ameasure adapted to

G

(su h that

m

′

_{(G) > 0}

). Two ases may happen: if

m(G) > 0

, then we an hoose

m

′

_{= m}

. Conversely if

m(G) = 0

(important ase inthis arti le),

m

annot measure subsets of

G

,so

m

′

mustdierfrom

m

.

Let us suppose that we only have a ess to

G

. Thus we are interested in the dynami s restri ted to

G

,andwe onsiderthe measurable spa e

(G, T

′

_{, m}

′

₎

,where

T

′

isa

σ

-algebra of

G

overwhi h

m

′

is dened. We introdu e

G

inv

,theattra tive subset of

G

:

G

inv

=

x ∈ G | ∃t

0 > 0, ∀ t > t

0 , φ

t

x ∈ G

.

If

x ∈ G

inv

, after some time it be omespossible to ompletely follow thetraje tory starting from

x

. The assumption that we only have a ess to

G

is thus invisible on erning the dynami s on

G

inv

. Conversely, traje tories starting from

G\G

inv

leave

G

and then annot be tra ked. Fortunately, from Poin aré re urren e theorem, if

m(G) > 0

, then almost all traje tories omeba kinto

G

. Morepre isely,we maydenethePoin aré re urren e timeas

∀x ∈ G\G

inv

, τ

G

(x) = inf

t > 0 | φ

t

x ∈ G, ∃t

0 ∈ (0, t), φ

t

0 x /

∈ G

.

We remark that if

τ

G

(x) < ∞

, then

φ

τ

G

(x)

_{x ∈ ∂G ∩ (G\G}

inv

)

,where

∂G

is theboundary of

G

. Let

G

ext

be the set of the starting points of traje tories whi h never ome ba k into

G

, i.e. points

x

su h that

τ

G

(x) = ∞

:

G

ext

=

x ∈ G | ∃t

0 > 0, ∀ t > t

0 , φ

t

x /

∈ G

.

Theorem 1 (Poin aré re urren e theorem). The set

G

ext

is negligible:

m(G

ext

) = 0

. Inthat ase,

G

is said

m

-re urrent. Until the end,if

m(G) = 0

, we supposethat

G

is

m

′

-re urrent. We note

G

rec

thesetof traje tories whi halternatively wanderinside andoutside

G

:

G

rec

= G\ (G

inv

∪ G

ext

) .

We may remark that if

x ∈ G

rec

, then

φ

τ

G

(x)

_{x ∈ G}

rec

. Although it is impossible to have a ontinuous des ription of the dynami s within

G

rec

, we may then tra k by intermitten e the traje tories stemmingfrom thisset, thanksto re urren e times. Following Hilfer[11 ℄,we introdu e the mapping

S :

G

rec

−→

G

rec

x

7−→ φ

τ

G

(x)

_x.

(2.1)

Iterationsof

S

permittofollowthetemporalevolutionofa point of

G

rec

. Letusremarkthat for all

k ≥ 1

,

S

k

_{x ∈ ∂G}

. Sin e

m

′

_{(G) = m}

′

_(G

inv

) + m

′

(G

rec

)

, it is now possible to tra k almost all traje tories starting from

G

,at leastbyintermitten e.

The following hara teristi timeswill alsobe useful:

∀x ∈ G

rec

,

τ

r

(x) = inf

t > 0 | φ

t

x /

∈ G

,

τ

e

(x) = inf

n

t > 0 | φ

τ

r

(x)+t

x ∈ G

o

.

Thetime

τ

r

(x)

isthetime thatthetraje torystarting from

x

staysin

G

beforeleaving,while

τ

e

(x)

isthetime this traje torythenspendsoutside

G

(see gure1). Those timesverify

τ

G

(x) = τ

r

(x) + τ

e

(x).

(4)

Inorder to obtain global informations on the dynami s inside

G

, sets of traje tories - i.e. evolution of subsets of

G

- should be studied, for instan e through the evolution of their measures.

Thisproblem is studied indetails in[11, 10℄. The operator

S

is redened at pre ision

∆t

onthesetofmeasureson

G

andappearsasa onvolutionprodu t. Forameasure

ρ

on

G

and asubset

A ⊂ G

,Hilferobtains

S(∆t)ρ(A)(t) = p

∆t

∗ ρ(A)(t).

Then hefo usesonthe indu eddynami safter arenormalizationintimes ale ( ontinuous time-limit in [11℄ and ultra-long time limit in [10℄) and obtains a new operator asso iated

to a new time step,

S(f

˜

∆t)

. In that ase, he shows that the hara teristi derivative of this operator, more pre iselytheinnitesimal generator[8, p.356℄

G

asso iated to thesemi-group

˜

S(f

∆t)

f

∆t≥0

,dened by

Gρ(A)(t) = lim

f

∆t→0

˜

S(f

∆t)ρ(A)(t) − ρ(A)(t)

f

∆t

,

may be equal (up to the sign) to the Mar haud fra tional derivative of order

α

[26, p.109℄, with

α ∈ (0, 1)

. A tually,this approa h ispartof adeeperquestioningontimestru ture and irreversibility[14 , 13℄.

However some points may still seem un lear, su h as the signi ation of

S(∆t)

and the renormalization pro edures. Furthermore,theexponent

α

remains unspe ied.

While keeping a similar approa h, we propose here a simple dynami al model for whi h we study the innitesimal generator. In several examples, it is proportional to the ordinary derivative. Then we useZaslavsky analysisonHamiltonian haoti systems: itthat ase, the generatormaybe omea fra tional derivative.

3. Constru tion of a simple model

Letusre allour obje tive: we wouldliketodes ribethedynami s ofour systemrestri ted to

G

,ina global way,i.e. by onsidering setsof traje tories.

(5)

Todo so,weintrodu e the mapping

N

given by

N :

T

′

_{−→ F(R, R)}

A

7−→

N

A

,

where

N

A

is areal-valued fun tion dened by

N

A

:

R

−→

R

+

t

7−→ m

′

_((φ

t

_{A) ∩ G).}

Let

A ∈ T

′

and

t

0 ∈ R

. We want to know the innitesimal evolution of

N

A

(t

0 )

. Several su essive steps willlead us to ageneral formula.

3.1. Model with one trap. Let

∆t > 0

. We onsider thefollowing binarydynami s: all of thetraje tories whi h leave

G

aretrapped within

P ⊂ Γ

,with

P ∩ G = ∅

. Thenthey ome ba kafter

2∆t

,andstay in

G

duringa multiple of

∆t

,until possiblyleaving again.

Wemay thensplit

G

withthetwo following sets:

G

0 (∆t) = {x ∈ G | τ

r

(x) ≥ ∆t} ,

(3.1)

G

1 (∆t) = {x ∈ G | τ

r

(x) < ∆t, τ

e

(x) = 2∆t} .

(3.2) Weremark that

G

1 (∆t)

mayalso be written as

G

1 (∆t) = {x ∈ G | τ

r

(x) = 0, τ

e

(x) = 2∆t} ,

= {x ∈ G | τ

r

(x) < ∆t} ,

= {x ∈ G | τ

r

(x) = 0} .

This setisdire tly linkedto trap

P

.

Hen e we have

G

0 (∆t) ∩ G

1 (∆t) = ∅

and

G = G

0 (∆t) ∪ G

1 (∆t)

. Asin [10℄,we dene the numbers

p

k

(∆t) =

m

′

_(G

k

(∆t))

m

′

_(G)

,

k ∈ {0, 1}.

Those twoquantitiesprovideaprobabilitydensityasso iatedtore urren etimes(

p

0 (∆t) +

p

1 (∆t) = 1

).

Wesuppose thatthese setsarewell mixed:

∀B ∈ T

′

_{, m}

′

_{(B ∩ G}

0 (∆t)) = p

0 (∆t)m

′

(B), m

′

(B ∩ G

1 (∆t)) = p

1 (∆t)m

′

(B).

Starting from

N

A

(t

0 ) = m

′

_(A)

, we determine the following states. The shifts will o ur every

∆t

, sowe may just onsider

N

A

(t

0 + n∆t)

, with

n ∈ N

. Those su essive instants are nowdetailed.

(1) At

t

+

0

, traje tories starting from

A

are splitting: some of them stay in

G

while the others leave

G

during

2∆t

. We note

A

0

the set of initial onditions of the rst ones and

A

1

the set of the se ond ones. Consequently,

m

′

_(A

0 ) = p

0 m

′

(A)

and

m

′

_(A

1 ) =

p

1 m

′

(A)

(weomit the dependan e of

p

0

and

p

1

in

∆t

). (2) At

t

0 + ∆t

,only

A

0

is in

G

:

N

A

(t

0 + ∆t) = m

′

(A

0 ) = p

0 N

A

(t

0 ).

Withinthe trap,

A

1

be omes

A

11

. At

t

+

0 + ∆t

,itisnow

A

0

whi hsplitssimilarlyto

A

,andgivesbirthto

A

00

and

A

01

:

m

′

_(A

00 ) = p

0 m

′

(A

0 )

and

m

′

_(A

(6)

(3) Traje tories whi h es aped from

G

at

t

+

0

ome ba k at

t

+

0 + 2∆t

. Consequently, at

t

0 + 2∆t

,only

A

00

is present in

G

:

N

A

(t

0 + 2∆t) = m

′

(A

00 ) = p

0 N

A

(t

0 + ∆t).

At

t

+

0 + 2∆t

,

A

00

splits into

A

000

and

A

001

,

A

11

omes ba k (it turns into

A

110

), and

A

01

stays outside

G

whilebe oming

A

011

.

(4) At

t

0 + 3∆t

,

G

ontains

A

000

and

A

110

. Hen e wehave

N

A

(t

0 + 3∆t) = m

′

(A

000 ) + m

′

(A

110 ),

= p

0 m

′

(A

00 ) + m

′

(A

1 ),

= p

0 N

A

(t

0 + 2∆t) + p

1 N

A

(t

0 ).

At

t

+

0 +3∆t

,

A

000

splitsinto

A

0000

and

A

0001

,

A

110

into

A

1100

and

A

1101

,

A

011

omes ba kand be omes

A

0110

. Con erning

A

001

,it staysoutside

G

andturns into

A

0011

. (5) At

t

0 + 4∆t

,we nd in

G

the sets

A

0000

,

A

1100

and

A

0110

:

N

A

(t

0 + 4∆t) = m

′

(A

0000

) + m

′

(A

1100

) + m

′

(A

0110

),

= p

0 m

′

(A

000 ) + m

′

(A

110 )

+ m

′

(A

01 ),

= p

0 N

A

(t

0 + 3∆t) + p

1 N

A

(t

0 + ∆t).

A sket hof the dynami s isgiven ingure2. An immediate generalization leadsto

∀ n ∈ Z, N

A

(t

0 + n∆t) = p

0 N

A

(t

0 + (n − 1)∆t) + p

1 N

A

(t

0 + (n − 3)∆t).

In parti ular,

N

A

(t

0 + ∆t) = p

0 N

A

(t

0 ) + p

1 N

A

(t

0 − 2∆t).

(3.3) Keeping in mind denition (2.1) , we note

S(∆t)

the operator of innitesimal temporal evolution, whi h leads roughly speakingto the next temporal step. In this example, time is dis ete and takesits valuesin

t

0 + ∆tZ + 2∆tZ = t

0 + ∆tZ

.

Sowe have

S(∆t)N

A

(t

0 ) = N

A

(t

0 + ∆t).

(3.4) Giventhat

lim

∆t→0

+

p

0 N

A

(t

0 ) + p

1 N

A

(t

0 − 2∆t) = N

A

(t

0 )

,

S(∆t)

veries

S(0) =

id

.

(3.5)

Moreover,from (3.4)

S(∆t)

also veries

∀∆t

1 , ∆t

2 > 0, S(∆t

1 ) S(∆t

2 ) = S(∆t

1 + ∆t

2 ).

(3.6) Letus remarkthat(3.3)maynot usedto he kthisproperty. Byverifying (3.5) and(3.6) ,

(S(∆t))

_∆t≥0

denes a one-parameter semi-group. If

N

A

possessesa left derivative at

t

0

,the asso iated innitesimalgenerator

G

isgiven by

GN

A

(t

0 ) = lim

∆t→0

+

S(∆t)N

A

(t

0 ) − N

A

(t

0 )

∆t

,

(3.7)

= lim

∆t→0

+

p

1 (∆t) (N

A

(t

0 − 2∆t) − N

A

(t

0 ))

∆t

,

(3.8)

= −2p

1 (0

+

)

d

dt

−

N

A

(t

0 ),

(3.9)

(7)

(8)

where

d

dt

−

N

A

(t

0 )

isthe left derivative of

N

A

at

t

0

and

p

1 (0

+

₎

istherightlimit of

p

1

at

0

. Remark1. Inthisexample,thefun tion

N

A

annotbedierentiableat

t

0

,unless

N

′

A

(t

0 ) = 0

. Indeed,

S(∆t)N

A

(t

0 ) = N

A

(t

0 + ∆t)

in that ase, so we also have

GN

A

(t

0 ) =

d

dt

+

N

A

(t

0 )

. 3.2. Model with two traps. We generalize theprevious example bysupposing that there arenowtwosets

P

1

et

P

2

outside

G

,whi htrap traje toriesduring

2∆t

and

3∆t

respe tively. Trapped traje tories thenstayin

G

duringa multiple of

∆t

.

Aspreviously,weintrodu e thefollowingsets:

G

0 (∆t) = {x ∈ G | τ

r

(x) ≥ ∆t} ,

G

1 (∆t) = {x ∈ G | τ

r

(x) < ∆t, τ

e

(x) = 2∆t} ,

G

2 (∆t) = {x ∈ G | τ

r

(x) < ∆t, τ

e

(x) = 3∆t} .

On eagain,thosesetsformapartitionof

G

. For

k ∈ {0, 2}

,wenote

p

k

(∆t) =

m

′

_(G

k

(∆t))

m

′

_(G)

. We stillhave

p

0 (∆t) + p

1 (∆t) + p

2 (∆t) = 1

.

By pro eeding similaralyto theprevious model, we nd:

N

A

(t

0 + ∆t) = p

0 N

A

(t

0 ) + p

1 N

A

(t

0 − 2∆t) + p

2 N

A

(t

0 − 3∆t).

Time evolves here in

t

0 + ∆tZ + 2∆tZ + 3∆tZ = t

0 + ∆tZ

. The innitesimal evolution operator

S(∆t)

on eagainveries

S(∆t)N

A

(t

0 ) = N

A

(t

0 + ∆t),

thus semi-groupproperties (3.5) and (3.6)are stillfullled. Con erning the innitesimal generator,wehave

GN

A

(t

0 ) = lim

∆t→0

+

p

1 (N

A

(t

0 − 2∆t) − N

A

(t

0 ))

∆t

+

p

2 (N

A

(t

0 − 3∆t) − N

A

(t

0 ))

∆t

,

= −(2p

1 (0

+

) + 3p

2 (0

+

))

d

dt

−

N

A

(t

0 ).

3.3. Generalizations. Let

{P

k

}

k∈N

∗

be a setof traps withrespe tive trapping times

n

k

∆t

,

n

k

∈ N

∗

. We assume thatea h timea traje toryleaves

G

,it istrapped byexa tly one trap. Hen e, ifwenote

G

0 (∆t) = {x ∈ G | τ

r

(x) ≥ ∆t}

and for all

k ∈ N

∗

,

G

k

(∆t) = {x ∈ G | τ

r

(x) < ∆t, τ

e

(x) = n

k

∆t} ,

westill obtain apartition of

G

. For all

k ∈ N

,ifwe set

p

k

(∆t) =

m

′

_(G

k

(∆t))

m

′

_(G)

,

(3.10)

the evolutionof

N

A

veries

N

A

(t

0 + ∆t) =

X

k≥0

(9)

On e again,the su essiveinstantsbelongto

t

0 + ∆tZ

. Then

S(∆t)N

A

(t

0 ) = N

A

(t

0 + ∆t)

, and

S(∆t)N

A

(t

0 ) =

X

k≥0

p

k

(∆t)N

A

(t

0 − n

k

∆t).

(3.12)

Now we onsider any trapping times, denoted

T

k

(∆t)

with

k ∈ N

∗

, and we suppose they are well-ordered:

0 < T

1 (∆t) < · · · < T

k

(∆t) < · · · .

Inthat ase,the group

P

k≥1

T

k

(∆t)Z

annotanymorebewrittenas

τ

0 Z

butisdensein

R.

Inparti ular,a minimaltime step annotanymorebedened. Butthegroup

P

k≥1

T

k

(∆t)Z

remains ountable, so it is still possible to move to the next step: the operator

S(∆t)

still makessense,but isno longerequal to

N

A

(t

0 + ∆t)

. Consequently,ageneralization annotbe done with(3.11) , but with(3.12) :

S(∆t)N

A

(t

0 ) =

X

k≥0

p

k

(∆t)N

A

(t

0 − T

k

(∆t)),

(3.13)

where

p

k

(∆t)

isgiven by (3.10) ,with

G

0 (∆t) = {x ∈ G | τ

r

(x) ≥ T

1 (∆t)} ,

(3.14) and,for all

k ∈ N

∗

,

G

k

(∆t) = {x ∈ G | τ

r

(x) < T

1 (∆t), τ

e

(x) = T

k

(∆t)} .

(3.15) This formulais to be linked withexpression(8)in[10 ℄.

We assume by now that

∆t 7→ p

0 (∆t)

has a right limit at

0

, denoted

p

0 (0

+

₎

. Now we spe ifyvalues of

p

k

(∆t)

and

T

k

(∆t)

in the aseof haoti Hamiltonian systems.

4. Dynami al traps and anomalous diffusion

Zaslavskystudiesin[33℄ thegeneral shapeof haoti Hamiltonianphasespa es. In hapter 12, he introdu es the notion of dynami al trap so as to des ribe thebehavior of traje tories near KAM tori. This area possesses a self-similar stru ture: it is omposed of imbri ated subsets

P

k

whi h verify

m(P

k+1

) = λ

M

m(P

k

),

with

λ

M

< 1.

Moreover,the trapping times

T

k

asso iated also have a self-similarproperty:

T

k+1

= λ

T

k

,

with

λ

T

> 1.

Trappingtimes arehen eallthe longer astraps aresmall. Thisanalysis analso be found in[32 , 34 ℄.

The kind of stru ture has ma ros opi onsequen es: when one studies diusion of par-ti ules through a probabilisti des ritpion of the system, the moment of order 2 is ruled by the following law:

hx

2 i ∝ t

µ

.

The lassi al ase (normal diusion) orresponds to

µ = 1

. The terms subdiusion and superdiusionare respe tively usedfor

µ < 1

and

µ > 1

. See [33, part.3℄ and [28 ℄ for more details.

Those anomalousdiusionphenomena anbedes ribed withtheintrodu tionof fra tional derivativesinto some spe i partialderivativesequations [20 , 31 ,25℄, [33, hap. 16℄.

(10)

Plan k-Kolmogorov equation, isgiven by

∂

β

∂t

β

P (x, t) =

∂

α

∂x

α

(A(x)P (x, t)) , 0 < β ≤ 1, 0 < α ≤ 2,

(4.1) where

P (x, t)

istheprobabilityto nd theparti ule at position

x

at time

t

.

Ifweassume

A

onstant,thisequationleadstothefollowingtransportequation[33 ,p.251℄:

hx

α

i ∝ t

β

.

The lassi al ase orresponds to

β = 1

and

α = 2

. The transport exponent [33, p.192℄ is denedby

µ =

2β

α

.

(4.2)

A ording to Zaslavsky [33, p.251, p.263℄, the inuen e of the dynami al traps appears through thefollowing relation:

µ =

| ln(λ

M

)|

ln(λ

T

)

.

(4.3)

Equality between (4.2) and (4.3) provides a onne tion between the fra tal stru ture of the phase spa e and the fra tional derivatives of (4.1) . However, the justi ation for the introdu tionofthosederivativesinequations(16.3)and(16.4)of[33℄isnot lear. Anapproa h basedonContinuousTimeRandomWalks(CTRW)[18,20,30 ℄leadstosu hderivatives, but thoseprobabilisti models do not relyonthemi ros opi dynami s of thetraje tories.

Weproposeheretolinktheemergen eoffra tionaloperatorswiththeself-similarstru ture ofthephasespa e des ribed above.

Coe ients

λ

M

and

λ

T

apriori dependon

∆t

. Be auseofthedynami aldenitionoftraps

P

k

,thesubsets

G

k

(∆t)

also verify,for

k ≥ 1

,

m

′

(G

k+1

(∆t)) = λ

M

(∆t) m

′

(G

k

(∆t)).

Con erning the hara teristi times, wehave,for all

k ≥ 1

,

T

k

(∆t) = T

1 (∆t) λ

T

(∆t)

k−1

.

Wewouldlikethatthestru tureofthetrapsbe omesthinerwhen

∆t → 0

,whileremaining self-similar. Thisleads us toassume

λ

M

(∆t) = (λ

M

)

∆t

and

λ

T

(∆t) = (λ

T

)

∆t

_,

(4.4)

where by sake of lisibility,

λ

M

and

λ

T

are now two real numbers su h that

0 < λ

M

< 1

and

λ

T

> 1

.

Remark 2. The transportexponent remains un hanged with denition (4.4) :

∀∆t > 0, µ =

| ln(λ

M

(∆t))|

ln(λ

T

(∆t))

=

| ln(λ

M

)|

ln(λ

T

)

.

Consequently, forall

k ≥ 1

,

m

′

_(G

k

(∆t)) = (λ

M

)

(k−1)∆t

m

′

(G

1 (∆t)),

and

(11)

In orderto obtain smaller hara teristi timeswhen

∆t → 0

,we supposethat

lim

∆t→0

+

T

1 (∆t) = 0.

(4.6) Usingrelation

X

k≥0

p

k

(∆t) = 1

,we nd thatfor all

k ≥ 1

,

p

k

(∆t) = (1 − p

0 (∆t))(1 − λ

∆t

M

)λ

(k−1)∆t

M

.

(4.7)

The innitesimalevolution (3.13) ofthesystemthus be omes

S(∆t)N

A

(t

0 ) = p

0 (∆t)N

A

(t

0 ) + (1 − p

0 (∆t)) (1 − λ

∆t

M

)

X

k≥0

λ

k∆t

_M

N

A

t

0 − T

1 (∆t)λ

k∆t

T

.

(4.8)

The innitesimalgenerators relatedto (4.8) an now be determined.

5. Fra tional infinitesimalgenerator

Hölder onditions on

N

A

appear in this part, so we need the following denitions. Let

Ω ⊂ R

,

f : Ω → R

and

α ∈ (0, 1]

.

Denition 1. Let

x ∈ Ω

. The fun tion

f

satises the Hölder ondition of order

α

at

x

if

∃c > 0, ∃η > 0, ∀ y ∈ Ω, |x − y| ≤ η ⇒ |f (x) − f (y)| ≤ c|x − y|

α

.

Denition 2. The fun tion

f

lo ally satisestheHölder onditionoforder

α

ifforall

x ∈ Ω

,

f

α

at

x

.

Denition 3. The fun tion

f

satises the Hölder onditionof order

α

if

∃c > 0, ∀ x, y ∈ Ω, |f (x) − f (y)| ≤ c|x − y|

α

.

If

α = 1

,

f

is alled Lips hitz ontinuous.

Nowwe goba kto our problemand we begin to showthat

S(∆t)

stillfullls (3.5). Lemma 1. If

N

A

β

, with

β < µ

,then

lim

∆t→0

+

S(∆t)N

A

(t

0 ) = N

A

(t

0 ).

Proof. Thedieren e

S(∆t)N

A

(t

0 ) − N

A

(t

0 )

veries

S(∆t)N

A

(t

0 ) − N

A

(t

0 ) = (1 − p

0 (∆t)) (1 − λ

∆t

M

)

X

k≥0

λ

k∆t

_M

h

N

A

t

0 − T

1 (∆t)λ

k∆t

T

− N

A

(t

0 )

i

.

We remark that

λ

M

λ

β

T

< 1

if and only if

β < µ

. Given that

N

A

satises the Hölder onditionoforder

β

,we obtain

|S(∆t)N

A

(t

0 ) − N

A

(t

0 )| ≤ (1 − p

0 (∆t)) (1 − λ

∆t

M

)

X

k≥0

λ

k∆t

_M

T

1 (∆t)λ

k∆t

T

β

≤ (1 − p

0 (∆t)) T

1 (∆t)

β

1 − λ

∆t

M

1 −

λ

M

λ

β

T

∆t

Onthe onehand,

lim

∆t→0

+

(1 − p

0 (∆t))

1 − λ

∆t

_M

1 −

λ

M

λ

β

_T

∆t

= (1 − p

0 (0

+

₎₎

ln(λ

M

)

ln(λ

M

λ

β

_T

)

,

(12)

and on the otherhand,

lim

∆t→0

+

T

1 (∆t) = 0

fromassumption (4.6) . Consequently,

lim

∆t→0

+

(S(∆t)N

A

(t

0 ) − N

A

(t

0 )) = 0

.

As it has already be seen, denition (3.3) annot be used to he k property (3.6) . So we just assumethat(3.6) isfullled.

Were all thatthe innitesimalgenerator

G

asso iatedto this semi-groupveries

GN

A

(t

0 ) = lim

∆t→0

+

S(∆t)N

A

(t

0 ) − N

A

(t

0 )

∆t

.

In [14 , 10, 13, 11 ℄, Hilfer shows that fra tional derivatives may appear as innitesimal generators of renormalized evolution operators. A similar result will now be obtained, but withoutusing anyrenormalization.

For all

∆t > 0

,we note

G(∆t)N

A

(t

0 ) =

S(∆t)N

A

(t

0 ) − N

A

(t

0 )

∆t

.

Wealso introdu e the fun tion

f

dened by

f :

R

+

_{× R}

+

_−→

R

(∆t, y)

7−→ λ

y

_M

N

A

t

0 − T

1 (∆t)λ

y

_T

− N

A

(t

0 )

.

Consequently,

G(∆t)N

A

(t

0 ) = (1 − p

0 (∆t))

1 − λ

∆t

_M

∆t

X

k≥0

f (∆t, k∆t).

For all

k ∈ N

,we note

I

k

(∆t) = f (∆t, k∆t) =

Z

k+1

k

f (∆t, k∆t) dx,

J

k

(∆t) =

Z

k+1

k

f (∆t, x∆t) dx.

Hen e

G(∆t)N

A

(t

0 )

an bewritten as

G(∆t)N

A

(t

0 ) = (1 − p

0 (∆t))

1 − λ

∆t

_M

∆t

X

k≥0

I

k

(∆t).

5.1. Case

µ > 1

. Inthat ase,

λ

M

λ

T

< 1

.

Theorem 2. If

N

A

isdierentiable andLips hitz ontinuouson

R,

and if

T

1

isdierentiable at

0

, then

GN

A

(t

0 ) = −γ

d

dt

N

A

(t

0 ),

where

γ = (1 − p

0 (0

+

₎₎

µ

µ − 1

T

′

1 (0)

. Proof. Firstwe prove that

lim

∆t→0

+

X

k≥0

(I

k

(∆t) − J

k

(∆t)) = 0

.

The fun tion

N

A

isdierentiable, sois

y 7→ f (∆t, y)

for all

∆t ≥ 0

,and

∂

2 f (∆t, y) = ln(λ

M

)f (∆t, y) − T

1 (∆t) (λ

M

λ

T

)

y

N

A

′

(t

0 − T

1 (∆t)λ

y

_T

).

(13)

If we note

c

the Lips hitz onstant, we have

|∂

2 f (∆t, y)| ≤ ln(λ

M

)cT

1 (∆t) (λ

M

λ

T

)

y

+ cT

1 (∆t) (λ

M

λ

T

)

y

.

By setting

c

′

_{= c(1 + ln(λ}

T

))

,we obtain

|∂

2 f (∆t, y)| ≤ c

′

T

1 (∆t) (λ

M

λ

T

)

y

.

Let

k ∈ N

. Then

|I

k

(∆t) − J

k

(∆t)| ≤

Z

k+1

k

|f (∆t, x∆t) − f (∆t, k∆t)| dx,

≤ ∆t sup

[k,k+1]

|∂

2 f (∆t, x∆t)|,

≤ c

′

_∆tT

1 (∆t) (λ

M

λ

T

)

k∆t

.

Giventhat

lim

∆t→0

+

∆t

X

k≥0

(λ

M

λ

T

)

k∆t

=

1 | ln(λ

M

λ

T

)|

and

lim

∆t→0

+

T

1 (∆t) = 0

,we infer that

lim

∆t→0

+

X

k≥0

(I

k

(∆t) − J

k

(∆t)) = 0.

Consequently,

G(∆t)N

A

(t

0 )

∼

∆t→0

+

− 1 − p

0 (0

+

₎

_ln(λ

M

)

X

k≥0

J

k

(∆t).

(5.1)

Nowwe anevaluate

lim

∆t→0

+

X

k≥0

J

k

(∆t)

. Firstly,

X

k≥0

J

k

(∆t) =

Z

∞

0 λ

x∆t

_M

N

A

t

0 − T

1 (∆t)λ

x∆t

T

− N

A

(t

0 )

dx.

Withsubstitution

t = λ

x∆t

T

,weobtain

X

k≥0

J

k

(∆t) =

1 ∆t ln(λ

T

)

Z

∞

1 t

−(1+µ)

[N

A

(t

0 − tT

1 (∆t)) − N

A

(t

0 )] dt,

(5.2)

=

1 ln(λ

T

)

T

1 (∆t)

∆t

Z

∞

1 t

−µ

N

A

(t

0 − tT

1 (∆t)) − N

A

(t

0 )

tT

1 (∆t)

dt.

For all

t ≥ 1

,

t

−µ

N

A

(t

0 − tT

1 (∆t)) − N

A

(t

0 )

tT

1 (∆t)

≤ ct

−µ

, and

t 7→ ct

−µ

is integrable on

[1, +∞)

. Moreover,

lim

∆t→0

+

N

A

(t

0 − tT

1 (∆t)) − N

A

(t

0 )

tT

1 (∆t)

= −N

′

A

(t

0 ).

By dominated onvergen e,

lim

∆t→0

+

X

k≥0

J

k

(∆t) =

1 ln(λ

T

)

T

′

1 (0)(−N

A

′

(t

0 ))

Z

∞

1 t

−µ

dt,

=

1 (1 − µ) ln(λ

T

)

T

₁

′

(0)N

_A

′

(t

0 ).

(14)

lim

∆t→0

+

G(∆t)N

A

(t

0 ) = − 1 − p

0 (0

+

₎

ln(λ

M

)

(1 − µ) ln(λ

T

)

T

₁

′

(0)N

_A

′

(t

0 ),

= − 1 − p

0 (0

+

)

µ

µ − 1

T

′

1 (0)N

A

′

(t

0 ).

Remark 3. If

N

A

veries assumptions of theorem 2, then

N

A

1

and onsequently fullls onditions of lemma 1.

5.2. Case

µ < 1

. We antrytoestimate

lim

∆t→0

+

P

k≥0

J

k

(∆t)

,assumingthat

N

A

issmooth enough and rapidly de reasing in

−∞

, in order that all the following quantities are well-dened. We integrate byparts(5.2) :

X

k≥0

J

k

(∆t) =

−T

1 (∆t)

µ∆t ln(λ

T

)

Z

∞

1 t

−µ

N

′

A

(t

0 − tT

1 (∆t)) dt +

N

A

(t

0 − T

1 (∆t)) − N

A

(t

0 )

µ∆t ln(λ

T

)

dt.

Substitution

u = T

1 (∆t)t

leads to

X

k≥0

J

k

(∆t) =

−T

1 (∆t)

µ

µ∆t ln(λ

T

)

Z

∞

T

1 (∆t)

u

−µ

N

′

A

(t

0 − u) du +

N

A

(t

0 − T

1 (∆t)) − N

A

(t

0 )

µ∆t ln(λ

T

)

du.

The integralisnot problemati :

lim

∆t→0

+

Z

∞

T

1 (∆t)

u

−µ

N

′

A

(t

0 − u) du =

Z

∞

0 u

−µ

N

′

A

(t

0 − u) du.

If

T

1

isdierentiable at

0

,then

lim

∆t→0

+

N

A

(t

0 − T

1 (∆t)) − N

A

(t

0 )

µ∆t ln(λ

T

)

= −

T

′

1 (0)

µ ln(λ

T

)

N

_A

′

(t

0 ).

Conversely,

lim

∆t→0

+

T

1 (∆t)

µ

∆t

= +∞

. So we annot ndanyinnitesimal generator. The assumption onthe dierentiability of

T

1

at

0

should hen eberepla ed.

In orderthat

T

1 (∆t)

µ

∆t

hasanite limit,wesuppose thatthereexists

b > 0

su hthat

T

1 (∆t)

∼

∆t→0

+

b (∆t)

1/µ

_.

Fromaphysi al point of view,

T

1 (∆t)

and

∆t

arehomogeneous totime, sowe introdu e a onstant of time

τ

su h that

b = τ

1−1/µ

:

T

1 (∆t)

∼

∆t→0

+

τ

1−1/µ

_(∆t)

1/µ

_.

(5.3)

Under thisassumption on

T

1

,afra tional derivative dened asfollows will appear.

Denition 4. Let

f : R → R

and

α ∈ (0, 1)

. The Mar haud fra tional derivative of order

α

is dened as

D

α

+

f (t) =

α

Γ(1 − α)

Z

∞

0 u

−(1+α)

[f (t) − f (t − u)] du,

where

Γ

is the Gammafun tion.

(15)

This derivative is well-dened if

f

is bounded and lo ally satisesthe Hölder ondition of order

δ

,with

δ > α

. See [26,p.109℄ formore details.

Nowwe anenun iate themain resultof the arti le.

Theorem 3. If

N

A

β

and lo ally satises the Hölder onditionof order

ν

,with

β < µ < ν

, then

GN

A

(t

0 ) = −˜

γ τ

µ−1

D

+

µ

N

A

(t

0 ),

(5.4) where

γ = Γ(1 − µ)(1 − p

˜

0 (0

+

₎₎

.

Proof. Aspreviously,we rstlyprove that

lim

∆t→0

+

X

k≥0

(I

k

(∆t) − J

k

(∆t)) = 0

. Let

k ∈ N

. For all

x ∈ [k, k + 1]

,

f (∆t, x∆t) − f (∆t, k∆t) = λ

x∆t

_M

N

A

(t

0 − T

1 (∆t)λ

x∆t

T

) − N

A

(t

0 )

−

N

A

(t

0 − T

1 (∆t)λ

k∆t

T

) − N

A

(t

0 )

i

+

h

λ

x∆t

_M

− λ

k∆t

_M

i

N

A

(t

0 − T

1 (∆t)λ

k∆t

T

) − N

A

(t

0 )

,

= λ

x

_M

N

A

(t

0 − T

1 (∆t)λ

x∆t

T

) − N

A

(t

0 − T

1 (∆t)λ

k∆t

T

)

+

h

λ

x∆t

_M

− λ

k∆t

_M

i

N

A

(t

0 − T

1 (∆t)λ

k∆t

T

) − N

A

(t

0 )

.

Con erning the rst right-hand member, we obtainthefollowing inequality:

Z

k+1

k

λ

x∆t

_M

N

A

(t

0 − T

1 (∆t)λ

x∆t

T

) − N

A

(t

0 − T

1 (∆t)λ

k∆t

T

)

dx

≤ λ

k∆t

_M

Z

k+1

k

N

A

(t

0 − T

1 (∆t)λ

x∆t

T

) − N

A

(t

0 − T

1 (∆t)λ

k∆t

T

)

dx,

≤ λ

k∆t

M

Z

k+1

k

T

1 (∆t)

λ

x∆t

_T

− λ

k∆t

T

β

dx,

≤ λ

k∆t

_M

T

1 (∆t)

β

λ

(k+1)∆t

_T

− λ

k∆t

_T

β

,

≤ T

1 (∆t)

β

λ

∆t

T

− 1

λ

M

λ

β

T

k∆t

.

For these ondone,wehave

Z

k+1

k

h

λ

x∆t

_M

− λ

k∆t

M

i

N

A

(t

0 − T

1 (∆t)λ

k∆t

T

) − N

A

(t

0 )

dx

≤ T

1 (∆t)

β

λ

kβ∆t

_T

Z

k+1

k

λ

x∆t

M

− λ

k∆t

M

dx,

≤ T

1 (∆t)

β

λ

kβ∆t

_T

λ

(k+1)∆t

_M

− λ

k∆t

_M

,

≤ T

1 (∆t)

β

λ

∆t

M

− 1

λ

M

λ

β

_T

k∆t

.

Consequently,

|I

k

(∆t) − J

k

(∆t)| ≤ T

1 (∆t)

β

λ

∆t

T

− 1 + λ

∆t

M

− 1

λ

M

λ

β

_T

k∆t

.

(16)

Sin e

β < µ

,

λ

M

λ

β

T

< 1

. Thus,

X

k≥0

|I

k

(∆t) − J

k

(∆t)| ≤ T

1 (∆t)

β

λ

∆t

_T

− 1 + λ

∆t

M

− 1

1 − λ

M

λ

β

_T

.

Giventhat

lim

∆t→0

+

λ

∆t

_T

− 1 + λ

∆t

_M

− 1

1 − λ

M

λ

β

_T

= −

ln(λ

T

) + ln(λ

M

)

ln

λ

M

λ

β

_T

and

lim

∆t→0

+

T

1 (∆t)

β

_{= 0}

,weinfer that

lim

∆t→0

+

X

k≥0

(I

k

(∆t) − J

k

(∆t)) = 0.

Relation (5.1) is hen e still valid here. Furthermore, (5.2) holds for

µ < 1

. Substitution

u = tT

1 (∆t)

leads to

X

k≥0

J

k

(∆t) =

T

1 (∆t)

µ

∆t ln(λ

T

)

Z

∞

T

1 (∆t)

u

−(1+µ)

[N

A

(t

0 − u) − N

A

(t

0 )] du.

Bydenition,

0 ≤ N

A

(t) ≤ m

′

_(G)

forall

t ∈ R

. Sin ewehavealsoassumedthat

N

A

lo ally satisestheHölder onditionof order

ν > µ

,its Mar haud fra tional derivative of order

µ

is well-dened. Asa onsequen e,

lim

∆t→0

+

Z

∞

T

1 (∆t)

u

−(1+µ)

[N

A

(t

0 − u) − N

A

(t

0 )] du = −

Γ(1 − µ)

µ

D

µ

+

N

A

(t

0 ).

Withrelation (5.3), we obtain

lim

∆t→0

+

X

k≥0

J

k

(∆t) = −

τ

µ−1

ln(λ

T

)

Γ(1 − µ)

µ

D

µ

+

N

A

(t

0 ).

Finally,

GN

A

(t

0 ) = (1 − p

0 (0

+

))

τ

µ−1

ln(λ

T

)

ln(λ

M

)Γ(1 − µ)

µ

D

µ

+

N

A

(t

0 )

= −(1 − p

0 (0

+

))τ

µ−1

Γ(1 − µ) D

µ

+

N

A

(t

0 ).

We have deliberately let the onstant

τ

appear in(5.4) for reasons of dimensional homo-geneity [16℄: the relevant derivative isnot

D

µ

+

,but

τ

µ−1

_D

µ

+

,inorder to be homogeneous to the inverseof atime.

Remark 4. So astorespe t dimensionalhomogeneity,a onstant of time

τ

′

shouldhave been introdu ed for the traps onstants:

λ

T

(∆t) = (λ

T

)

∆t/τ

′

,

λ

M

(∆t) = (λ

M

)

∆t/τ

′

.

(17)

6.1. Chara terization of

µ

. From distribution

(p

k

(∆t))

k≥0

and hara teristi times

(T

k

(∆t))

k≥0

(with

T

0 (∆t) = 0

), we an evaluate moments

hT

α

_i

∆t

with

α > 0

,dened by

hT

α

i

∆t

=

X

k≥0

p

k

(∆t)T

k

(∆t)

α

.

In the aseof dynami altraps des ribed by(4.5) and (4.7) ,we obtain:

hT

α

i

∆t

=











(1 − p

0 (∆t))

1 − λ

∆t

M

1 − λ

M

λ

α

T

∆t

T

1 (∆t)

α

if

α < µ,

+∞

if

α ≥ µ.

Consequently, ifwenote

hT

α

_{i = lim}

∆t→0

+

hT

α

_i

∆t

,parameter

µ

appears asa riti alpoint:

hT

α

i =

0

if

α < µ,

+∞

if

α ≥ µ.

However,if

m(G) > 0

and

µ ≤ 1

,

hT i = ∞

,whi hdoesnot respe ttheKa lemma[17 ,19 ℄. Then we shouldassume

m(G) = 0

. Thisremarkis loselyakinto theapproa h ofHilfer[11 ℄, wherethe fra tionalinnitesimal generatoronly appears for sets ofmeasure

0

.

Remark 5. If

G

is a se tion transverse to the Hamiltonian ow (a Poin aré se tion for instan e) then

m(G) = 0

, and sin e all the traje tories ross

G

,

τ

r

(x) = 0

for all

x ∈ G

. Consequently,

G

0 (∆t) = ∅

and

p

0 (∆t) = 0

, for all

∆t > 0

. Hen e

p

0 (0

+

_{) = 0}

.

6.2. Fra tional kineti equation. The modelpresentedheredonot explaintheemergen e of fra tional derivativesinequations su h as(4.1). However, thefra tional exponent we have obtained isexa tlythetransport oe ient(4.3) . Thisresultis ompatible withrelation(4.2) whi h involvesthe fra tional exponentsof Zaslavsky.

Indeed, letus assumethat

S(∆t)

ouldbeapplied to

P (x, t)

inorderto des ribe a gener-alized shift of

P (x, t)

along

t

by

∆t

[33, p.246℄. Then thetemporal derivative asso iated to thetemporal evolution of

P (x, t)

is the innitesimal generator

G

. In the ase of anomalous diusion,exponents

α

and

β

in(4.1) be ome ompletely determined.

•

If

µ > 1

(superdiusion),thenthetemporalderivativeis lassi :

β = 1

. Superdiusion is ex lusively taken into a ount by the spatial derivative of order

α =

2 µ

. Equation (4.1) thus be ome

∂P (x, t)

∂t

=

∂

2/µ

∂x

2/µ

(A(x)P (x, t)) .

•

If

µ < 1

(subdiusion),

β = µ

so

α = 2

: the temporal derivativeis theonly oneto be fra tional. Consequently,(4.1) turnsinto

∂

µ

∂t

µ

P (x, t) =

∂

2 ∂x

2 (A(x)P (x, t)) .

Inparti ular,ifourmodelappliesto

P (x, t)

,fra tionalderivativesinspa eandtime annot oexist.

(18)

The model whi h has been des ribed in this arti le attempts to explain, from a dynam-i al view point, the emergen e of fra tional derivatives in haoti Hamiltonian systems. It seemssimplierthantheformalismofHilfer,inparti ularbe ausenorenormalizationappears. Moreover, it strongly relies on fra tal properties of the phase spa e. Our approa h is obvi-ously perfe tible on several aspe ts. It does not explain why

T

1 (∆t)

should fulll (5.3) , and ondition

m(G) = 0

imposedbythe Ka lemmashould be laried. Soasto testthevalidity of themodel,othersystems shouldalsobe onsidered, inparti ular strongly haoti systems, wherethe distributionof re urren etimesissimilarto anexponential law[3, 15℄. Finally,we believethatthere arestillenough freedom degrees inour modelfor allowing us to enhan eit inforth oming studies.

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Institut de Mé anique Céleste et de Cal ul des Éphémérides, Observatoire de Paris, 77 avenue Denfert-Ro hereau,75014Paris,Fran e

Fractal traps and fractional dynamics

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