Anomalous diffusion in random media of any dimensionality

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Anomalous diffusion in random media of any

dimensionality

J.P. Bouchaud, A. Comtet, A. Georges, P. Le Doussal

To cite this version:

Anomalous

diffusion

in random media

of

_any

_{dimensionality}

J. P. Bouchaud

_(1),

A. Comtet

_(2),

A.

_Georges

₍₃₎

and P. Le Doussal

₍₃₎

(1)

Laboratoire de

_{Spectroscopie}

Hertzienne de l’Ecole Normale

_Supérieure

_(*),

24, rue Lhomond,

75231 Paris Cedex

05,

France

(2)

Division de

_{Physique Théorique (*),}

Institut de

_Physique

Nucléaire, Université de _Paris-Sud, 91406

_Orsay

Cedex,

France ,

(3)

Laboratoire de

_{Physique Théorique}

de l’Ecole Normale

_{Supérieure (**),}

24, rue Lhomond,

75231 Paris Cedex _05, France

(Requ

le 22 mai 1987,

accept6

le 27 mai

₁₉₈₇

Résumé. 2014 Nous montrons

que la diffusion est anormale en toute dimension dans les milieux aléatoires où les forces

_présentent

des corrélations à

_longue

_portée.

Ce résultat est _{obtenu à la fois par des arguments}

_physiques

et _par une

_analyse

_{de groupe de renormalisation. Les comportements} obtenus sont en

_général

surdiffusifs sauf

lorsque

la force aléatoire dérive d’un

_potentiel.

Dans cette situation on obtient un _comportement

sous-diffusif. Dans le cas

_critique

_supérieur

(D

= 2 pour des corrélations à courte

portée),

celui-ci est caractérisé par

un _exposant

_dépendant

continûment du désordre. La raison en est l’annulation de la fonction 03B2

qui

est

démontrée à tous les ordres de la théorie des

_{perturbations.}

Dans le cas

_général,

des _arguments

_simples

suggèrent

qu’une

force

_potentielle

conduit à des diffusions

_{logarithmiques (c’est-à-dire}

à du bruit en

1/f).

Abstract. 2014 We

show,

through

physical

arguments and a renormalization group

_analysis,

that in the presence

of

long-range

correlated random

_forces,

_{diffusions is anomalous in any dimension. We obtain in}

_general

surdiffusive behaviours, except when the random force is the

_gradient

of a

_potential.

In this last

_situation,

with either short or

_long-range

_{correlations,} a subdiffusive behaviour with a disorder

_dependent

_exponent is found in the upper critical case

_(D

= 2 for

short-range

correlations).

This is because the

03B2-function

vanishes, which is

explicitly

proven at all orders of the

_{perturbation theory. Apart}

from this case, a

_potential

force is

_expected

to

lead to

_logarithmic

diffusion

_(1/f

_noise),

as

_{suggested by}

_simple

_arguments.

Classification

Physics

Abstracts 05.40 - 05.60 -

47.55M

1. Introduction.

Random walks in random media

_(RWRM)

are idealized models of a considerable number of

physi-cal situations.

_Examples

are

provided

by

the

hopping

conductivity

of disordered materials

_[1],

the diffusion

of a test

_particle

in a _{porous medium}

[2]

or in turbulent flows. More

_abstractly,

perhaps

also more

loosely, they

could

_capture

the essential features of the time evolution of a disordered

_system

in its

complicated

configuration

space, for

example

a

spin-glass.

In its one-dimensional

_version,

the model is

_by

now well understood

[3-7]

and exhibits

_quite

a rich

variety

of behaviours.

_Imagine

a

particle

on a

_line,

submitted to a

_Langevin

thermal noise and to a

quenched

random force

_{independently}

chosen at

each

_point.

If this force is of zero mean it has been shown

₍₁₎

that the

_particle

diffuses

_{extremely slowly :}

its mean

_squared

_position

increases

_{logarithmically}

in time instead of

_linearly.

This can be understood

through

an Arrhenius

_{argument :}

in one

_dimension,

a force is

_always

the

_gradient

of a

_potential,

which in our case increases

_typically

like the square root of the distance

_(as

it is the sum of

_independent

random

(1)

This result is due to Sinai

_[5]

in the case of a walk on a lattice.

_Explicit

results for the continuous case mentioned here can be found in

_[7].

1446

variables).

Thus the

_typical

time needed to travel a distance x is

_{given by t}

~

exp ( J x),

which leads _to

J x2 -

In 2 t.

If the mean of the force is _non-zero, one obtains a

_{large variety}

of

_behaviours,

like for

example X - t 1,

where g

is an

_exponent

_directly

related to the value of the mean force.

It was soon realized that such a

_logarithmic

diffusion was tantamount to a

1 / f

_spectrum

of the fluctuations and thus that

_{i f}

_logarithmic

diffusion

could occur in

_higher

dimensions,

this mechanism

would be a

_general

and rather

_convincing

source for the

_ubiquitous

_{1/ f}

noise

_{[8]. Unfortunately,}

it has been shown

_[9,10]

that if the random force is chosen

independently

from site to

_site,

the diffusion is normal

_(i.e.

X2 -

_t)

for any dimension

_larger

than

two. The same remark holds when the force is

divergence

free

_{(hydrodynamical}

flow,

magnetic

field)

or is the

_gradient

of a

_potential

_(Hamiltonian

motion) [2,

11,

12],

provided

correlations remain

short-ranged

(see

however the last

_paragraph

of this

paper).

Renormalization group

(RG) analysis

can be

performed

(as

first shown in

_{[10, 13])}

to obtain the

dynamical

behaviours in dimension D = 2 - E.

This paper has two _{purposes. The first} one is to

emphasize

that RWRM can

_display

anomalous dif-fusion laws in any dimension

_provided

the disorder has

_long-range

_{correlations,}

and to

_present

the results of a RG

_analysis

of this case. We shall

_point

out, in the next

_section,

the

_physical

relevance of such

correlations,

which are essential to _the argu-ments of reference

_[8]

but have not been

_analysed

so far

_through

RG

_{techniques (except,}

as a

_byproduct

of the

_study

of the « TSAW _», in the

special

case of an unconstrained force

[13]).

The second purpose concerns the case where the random force is the

gradient

of a

_potential,

with either short or

long-range correlations. It has been remarked

[11, 12]

that in this case the RG fails

_(up

to two

_loops)

to

predict

the

_dynamical

behaviour. We will make a

step

towards the

_complete

solution of this

_problem

by

an all-orders

_analysis

of

_perturbation

_theory.

We are motivated in

_{particular by}

the recent mathemati-cal results of Durrett

_[14]

who finds

_logarithmic

diffusions in a

_long-range

correlated

_potential

case.

We will show

that,

contrarily

to his

claim,

this result can _{very well be} understood

_through

a RG

_argument.

2. The relevance of disorder.

We consider a

_particle

submitted to a thermal noise

’I) (t)

and to a

_quenched

random force

_{V (x ).}

Its

motion is described

_by

a viscous limit of the

_Langevin

equation

(such

that the

_velocity

is

_{instantaneously}

adjusted

to the local

_{force) :}

with

The constraints on

_{V (x )}

determine the

_{(i j )}

structure

of

_{Gij ;}

this is more

_easily

_expressed

in Fourier

transform

_by

_{defining :}

we will

_mainly

consider three models :

(model I)

unconstrained force :

_{G L}

=

GT

(model II) divergenceless

flow :

_{G L}

= 0

(model III) potential

case V = -

grad

(H) : GT

= 0 .

We define an

_{exponent a}

_{characterizing}

the

_spatial

decay

of the correlations :

Power law correlations can be of

_{great physical}

relevance,

for

_example

to the diffusion of a test

particle

in a

_steady

flow accross a _{porous medium}

(model II).

Indeed,

the flow can be

_highly

in-homogeneous

and nevertheless

_display

_preferential

paths

along

which correlations remain

_strong.

Another

_{interesting application}

lies in

_spin-glass

dynamics

(to

which model III could be

_relevant).

In this context, one has _{the intuitive idea that energy} barriers between two

_{configurations}

should be an

increasing

function of their distance. In the

Sherring-ton-Kirkpatrick

[15]

infinite-range

model for

example,

a

_simpler

_quantity,

such that the

corre-lation (H( {Si} ) - H( {Sf} ))2)

is

_easily

shown to

be an

_increasing

linear function of the « distance »

between the two

_{configurations}

_{Si}

and

_{Sf},

in a wide range of distances. This

corresponds

to a correlation between forces with a 2. We wish to

emphasize

that,

physically,

(4)

needs not hold on very

large

scales

_{(x - y )}

when diffusion is

ultra-slow,

but

_only

in the restricted

_region

of _{space that}

the

_{diffusing particle}

can

_probe

within a

_given

time. In the case of a

_logarithmic

diffusion for

_example

this defines a

frequency

cut-off

exponentially

small in the

_length

and in

_{temperature :}

_{fc - exp (-}

_LIT)

down to which

_{1 / f}

noise will be observed. Let us mention

_finally

that

_explicit

construction of fractal energy

landscapes

can be

_{given [16],}

for which

(H(x)H(y) = Ix-yI2-a

at all

_scales,

with

a - 2 = 2 (D - DF) 0, DF

being

the fractal

dimen-sion of the

_landscape.

,

We now show that the

region

in the

_plane

(a, D )

where disorder is relevant

_(in

the sense that it leads to an anormal diffusion law and not

_only

correlations. In the pure case

(V

=

0)

_one has

x2(t ) - t2 vo

with

_{VO = 1/2.}

The number of sites which are visited

_by

the walker in a time t is thus

inf (t, t v° D),

equal

to t if

_Dvo::>

1,

that is for D > 2.

_{Reasoning perturbatively,}

it means that when a small amount of disorder is

_present

in D :::.

2,

the walker

essentially

probes

of the order of t

different

values of

V,

which

_according

to the central limit

_theorem,

results in an

_apparent

mean value of the

_{velocity given}

_{by :}

The extra

_displacement

due to the disorder is thus

given by :

leading

to

which

_implies

_that,

for an

ordinary

random walk in

D :> 2,

short-range

correlated disorder

_simply

changes

the value of the diffusion coefficient

₍₂₎

(5 DO

= (T).

Below this

_{dimensionality,}

the walker is

extremely

sensitive to local fluctuations of the force

V,

since in the pure case it visits

_infinitely

often a

given

site. This can lead either to subdiffusive behaviours

_(if

_{trapping regions}

_dominate)

or to surdiffusion

_(if

_locally

ballistic motion is

_preserved,

like in model

II,

see

_[11]).

One can discuss

_along

the same lines the case where there are

_long

_{range correlations of}

_type

_(4).

(4)

means that at a distance R from a

_point

0 one

R

roughly

finds

f

rD -1

_dr/ra

sites

_carrying

the same value of V as 0. For

_large

_R,

this

_integral

behaves as a constant if a :::. D and as

R D - a

if a D. In a

_sphere

of size

_RD,

one can thus

_organize

sites in sets of size

R D-a

inside which V takes

_roughly

the same value. The number of

_independent

values of the disorder is therefore

_{RDIR D - a - R’.}

In other

words,

the

_long

_range correlations have reduced the

dimensionality

of space to an « effective » _{one, a,} to

which the above

_argument

on residual average

velocity

can be

_applied.

Thus,

for a >

2,

the disorder

simply

modifies the diffusion

coefficient,

while for a -- 2 one

_expects

anomalous diffusions

independent-ly

of the dimension D : the

_resulting

domain in the

(a, D )

plane

where disorder is relevant is shown in

figure

1. For model

III,

the

_analysis

has to be

_slightly

modified,

and leads to the conclusion that disorder is

only

relevant if a -

2,

independently

of

D,

i.e. in the

case where correlations of the

_potential

increase with

(2)

Note that this also means that for

_Levy

_flights

(V,:::. 1/2),

Gaussian disorder is irrelevant in D >

_{1/ Vo.}

Fig. 1.

- The

region

of the

plane

(a, D )

where disorder is relevant is shown

_by

hatches. In the

_potential

case,

only

the

_(dotted)

line a = 2 remains. Previous studies of the

short range correlated case _{span the}

_{(dashed-dotted)}

line a = D.

the distance. The

_{power-counting analysis developed}

below will confirm these conclusions

_{rigorously :}

in the field-theoretical

framework,

our

_qualitative

argu-ment can be seen as a « Harris criterion »

(whose

modification in

_long-range

correlated cases has been studied in

_[17]).

3. Renormalization group

analysis.

The field-theoretical

_analysis

starts from the Fokker-Planck

_equation

for the

_{probability density}

_{P (x, t )}

associated with

_(1).

_{The average of P} over the random force

_(taken

to be

_Gaussianly

distributed

according

to

₍₂₎₎

can be

expressed,

using

the

replica

trick,

as the

_2-point

function of a field

_theory

involving

2 N scalars

_{fields 0 a,}

_4>2:

where a

_Laplace

transform in time has been taken. The action S reads :

In the static limit

_{(w = 0 ),}

this field

_theory

has a critical

_point

at V = 0. This is related to

long-distance

_{singularities}

_arising

when

_{(Vo) -}

0 in the weak-disorder

_expansion

of lattice

_hopping

models

with

_asymmetric

_hopping

rates.

_Indeed,

it can be shown

_[10]

that

_{(7) reproduces}

the dominant

I . R

_{singularities}

of a

_large

class of

_hopping

models

(see,

however,

last

_paragraph

of this

_paper).

In one

1448

(7),

the critical

_point

at

VO

= 0 is

replaced

by

a whole domain of

Vo

where the

_velocity

_It

vanishes at t - oo

_{(critical phase) [6].}

The

_{exponent v}

characterizing

the

_long-time

be-haviour of

_{x2(t )}

can in fact be related to the

exponent

q of model

(7)

in the

purely

static limit

w = 0.

Indeed,

probability

conservation

implies

that

the

_{susceptibility}

_exponent

y is

exactly

equal

to 1

(since

w P

_(q

=

0,

1).

The

scaling

relation

2

y = v

_{(2 - q )}

thus leads to

_{X2(t) - t 2 -,,}

where q is

the

_usually

defined

_exponent

_{characterizing}

the

algebraic

decay

of the

_2-point

function at

_criticality

in the static

_theory

_{( w}

=

0,

Vo

=

0).

The

_superficial

I . R

_degree

of convergence of a

diagram

with

_{E/2 0}

_(and

_E/2

external

_legs

at

the order n of

_perturbation

theory

is

_easily

shown to

be :

W V is the momentum

dependence

of the vertex,

arising

from the derivative

_couplings

and from the

momentum

_dependence

of the non-local interaction.

Owing

to

_(4),

we see that

_{GT, L (k2)}

behaves

_(when

k2-+ 0)

as a constant,

_{0- T, L, if}

a ::. D and as

uT, L (k2)(D - a)/2 if

a. D. This leads to: _c

2 if a :> D and úJ y = 2 + a - D if a - D.

We deduce from

₍₈₎

that the _{upper critical « dimension » is}

D = 2 if a :::. D and a = 2 if a -- D : this confirms the

analysis

of the

_preceeding

_section,

summarized in

figure

1.

_{Renormalizability}

is

_{likely [10]}

in _{the upper} critical case

(for w

=

0),

despite

the non-local character of the model and the fact that an infinite number of

_{dimensionnally}

admissible counterterms

a

_priori

exists. This is due to

_{global symmetries}

(U(N), a _ oa

+

_Cst.)

and to the fact that

non-locality

arises

_only

from the « non-canonical » kinetic

term of the force

_Vi,

seen as an

_auxiliary

field. A non zero w could nevertheless

spoil

these

renor-malizability

properties.

The

_one-loop

RG

_analysis

can be

_{performed along}

the lines of reference

_[18]

_{by working}

in the framework of the renormalized

_{theory. Defining}

dimensionless

_coupling

constants _gT

_{and gL}

_{through :}

0- L =

gL JL 2 - a and

0"T = _9T

U 2 - aat

the tree

level,

we obtain at one

_loop

order :

where e denotes

Models

_I,

II and III thus appear as remarkable cases

whose structure is

_{preserved by}

renormalization. If

one starts with a bare

_Gij

which is neither of

_type

II nor of

_type

_III,

the

_one-loop

_{RG flow goes} to the fixed

_point

_{g* = g*}

=

E/4 CD(I - 11D):

thus a

«

generic

» RWRM

(without

constraints II or

_III)

is in the

_universality

class of model I.

_Starting

from

model II leads to the fixed

_point

g*

=

E/8 CD(l - 11D),

g*

= 0

(for

which gL

= 0 is _an

attractive

direction,

but which is

_repulsive

other-wise).

For these two cases, the

_long

time behaviour of

_X2(t)

_resulting

of this one

_{loop-analysis}

is

given

in table I for both

_short-range

and

_long-range

cor-relations.

_(In

the upper critical cases, a direct

integration

of the one

_loop

RG

_equations

has been

performed).

In _{the short range} case, we recover the

results of reference

_{[2, 9-12] (a}

_two-loop

_computation

of q

is then necessary for model

I.)

The behaviour V =

(V 0) "

of the renormalized

velocity

as a function of

V 0

is also

_given.

The

_exponent

is related to the RG function _yv associated with insertions of the

operator 0’

aio’

through cp

=

II (1 -

yy).

_yv is

related to

_vertex-type

corrections of the

_4-point

function,

and reads at

_one-loop

order :

Table I.

The fact that model II

_always

leads to

_surdiffusion

can be understood

_(as

noticed in

_[11])

from the

existence of closed lines of force

_along

which ballistic

motion is

_preserved.

On the other

_hand,

_introducing

long-range

correlations in model I turns the

subdiffu-sive behaviour into a surdiffusive one.

Model III is

_{extremely peculiar}

from the RG

_point

of

view,

since no contribution to the

_j8

function

_(and

thus no fixed

_point)

is found at

_one-loop

order.

Explicit

calculations in _{the short range} case

_[11]

reveal that this

_property

is maintained at two

_loops.

The rest of _{this paper} is devoted to a discussion of this model.

4. The

_potential

case

_{(Model III).}

We will first show that for this model one has

f3 L

=

EgL without corrections at _{any order} of

pertur-bation

_{theory. Arguments}

in favour of this

_point

in the short range case have been

_given

in

_[11].

_Setting

V = -

grad

(H),

_we choose _{not to}

perform explicitly

the,

Gaussian _average

_leading

to

₍₇₎

and to

_keep

instead H as an

_auxiliary

field. In _{the short range}

case, where

GL(k2)

is a _constant, the action S is thus

replaced by

a local field

_theory

whose

_Lagrangian

reads :

_{(0" --+ i 0" )}

We

_recognize

a non-linear a-model

[19]

for the

2 N + 1 fields

_{(0 ’, ’, H).}

_Computing

the Ricci and curvature tensor leads to

_Rab

oc -

_Ngab

and

Rabcd OC

(gac

gbd - gad

gbc),

with det

(gab) = 1 -

Sur-prisingly

enough,

we discover that the model has a

hidden non-linear

_{0 (N + 1, N + 1 )}

_non-compact

symmetry.

(We

have found in

_explicit

form the

transformation on the fields which allows to _express

(11)

in the standard form

_{of a 0 (N + 1, N + 1 )}

field

theory.)

When the N - 0 limit of the

_replica

trick is

performed,

one is thus left with a free

_theory

for which

_{f3 L}

= - _EgL at all orders. In the

long

_range

correlated case, the u-model becomes non local in space, but the same conclusion holds

(since

it comes

only

from an « internal

symmetry »

property).

In _{the upper} critical case

_(short

or

_long-ranged,

E =

0),

the function

i3 L is

_{zero ;} the

running coupling

constant g

(,k )

thus remains

_equal

to the bare one at

any scale k. This leads to a continuous

_dependence

of the

_exponents

on the variance of the disorder

(in

this case, the results of Tab. I are valid to all

_orders).

When E :> 0, 9 (À ) is

driven to

_infinity

in the infra-red limit A - 0 :

_{g (A ) -}

_{go À -}

E.

This means

that it is the

_strong

disorder

regime

which controls the

physics,

regardless

of the value of _{go. In this}

limit,

particles

are

_trapped

for

_{extremely long}

times in the bottom of the

_potential

wells,

where we

_expect

the

probability

distribution

_{P (x, t )}

to be well

approxi-mated

_by

its

_equilibrium

value which is the Boltzmann

_{weight exp (- H (x)).}

This distribution

behaves

_typically

as

_{exp (-}

gl/2)

at

_large

_{g, and}

inserting

9 (À ) - À - E

this leads to a

_long

distance

behaviour

_{exp (-}

_XE/2).

This

_suggests

_(requiring

that

tP (x )

should be of order

₁₎

a

_logarithmic

diffusion

(see

Tab.

_I).

This result is the one that can be

_guessed

from an

Arrhenius

_argument

similar to the one

_given

in the

introduction. In D

_{= 1,}

this

_argument

is

_certainly

correct, since there is

_only

one

_path

_going

from one

point

to another. In

_higher

dimensions,

diffusion

between a

_point

0 and any

_point

A at a distance R will be dominated

_by

the smallest _energy barrier

encountered

_by

_paths

_going

from 0 to A and

restricted to a volume

R D.

As we are interested in

most efficient

_paths,

_they

can of course be taken as

self-avoiding. Now,

on _any

_path

of

_{internal length}

N,

the typical

energy barrier one encounters scales like

-,IN

(we

deal here with

_short-range

correlations

such that a =

D) ;

nevertheless the minimal energy

barrier

_{between p}

_independent

_paths

is

_obviously

-,IN lp.

In a

_sphere

of radius

R,

one has

_roughly

R D/N

independent paths

of size

N,

and as the

paths

are

_{self-avoiding}

one also has R = N

VSAW.

Therefore,

we

_predict

that the minimal energy barrier scales in this

_system

as R

[3/2 VSAW - D]

Diffusion is thus

logarith-mic if _{V SAW} --

3/2 D,

i. e. if D = a :

2,

as

v SAW (D

=

2 )

= 3/4.

Inserting

the

Flory

value

(exact

in D =

1, 2, 4),

_one obtains t

= to

exp(R(2-D)/2),

and thus R =

(In tlto Y/ (2 - D)

which

is,

quite

sur-prisingly,

exactly

the result obtained

_through

the RG

argument

presented

above. In D >

2,

the

_great

number of

_{paths always}

allow the walker to _go around « hills ». In the

general

case of

_long-range

correlations,

the condition a 2 means that the

potential

grows at

_large

distances like

_{H(bx) =}

b(2 -

a)/2

H(x)

and the Arrhenius

_argument

allows to

recover the behaviour of table I.

Lastly,

we would like to

_point

out that our model is a continuous limit of the discrete one for which Durrett

_[14]

has proven that this

_logarithmic

diffusion takes

_place

_{in any}

dimension.

_Indeed,

if one identifies

where

_W,,,,

is the

_hopping

rate from site x to site

x’,

one

_gets

_exactly

with Durret’s

definition :

V

_{= - grad}

_(H),

with

_{H(bx )}

=

b"H(x),

his

ex-ponent

a

_being

(2 - a )/2.

Let us

_finally

mention that not all RWRM fall in

the

_universality

classes of table I. For

_example,

random walks among

traps

with a broad

distribution

of release time T of the

_type

p(T)2013T’" (with

a

2)

reveals subdiffusive behaviour in all dimen-sions with an

_{exponent v}

=

a 2 1

for D :> 2

[20].

1450

[21]

falls in this

_universality

class rather than in those studied in this paper : the

potential

constructed in

[21]

has

_short-range

correlations with a local distri-bution of

_{potential depths}

p

(H) -

e- aH. Upon

res-caling

of space, this induces

traps

with a release time

given by

Arrhenius law

_{T - exp (f3 H).}

Its distri-button is thus

_{given by}

r

_,

_leading

us to

conjecture

that the diffusion law for this model is

X2 -

_talp,

for D > 2.

Acknowledgments.

We thank E.

_Brezin,

D. Fisher and D. Ruelle for their interest in the

_problem

and useful discussions.

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