Anomalous diffusion in presence of boundary conditions

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Anomalous diffusion in presence of boundary conditions

A. Giacometti, A. Maritan

To cite this version:

Anomalous

diffusion

in

_{presence of}

_boundary

conditions

A. Giacometti

₍₁₎

and A. Maritan

_(2,

*)

(1)

Dipartimento

di Scienze Ambientali, Universitá di _Venezia, Calle

_Larga

S. _Marta, 30124

Venezia,

Italy

(2)

Dipartimento

di Fisica dell’Universitá di Bari e _I.N.F.N., Sezione di Bari,

Italy

(Reçu

le 10

_juillet

1989,

accepté

sous

_{forme définitive}

le 14 mars

1990)

Abstract. 2014

Anomalous diffusion in presence of a

_{(fractal) boundary}

is

_{investigated. Asymptotic}

behaviour of the survival

_probability

and current

_through

an

_{absorbing boundary}

are

_exactly

calculated in

_{specific examples. They}

agree with recent

_findings

related with anomalous

_Warburg

impedance experiments.

The autocorrelation function for sites near the

_boundary

is

_carefully

discussed ; in _presence of

_{reflecting boundary}

conditions it can be different from the

(spatial)

average of the autocorrelation at variance with the normal diffusion case.

_Interesting

issues from a

methodological point

of view are discussed in the renormalization groups

analysis. Scaling

arguments are

_{given relating}

various _exponents. Classification

Physics

Abstracts 05.40 - 82.65

1. Introduction.

Very

recently

the old

_problem

of diffusion

_{impedance (Warburg impedance) [1] has regained}

much attention in the framework of fractal

_geometry

_[2-5].

There exists both numerical and

_{experimental [2]}

evidence that the

_long

time behaviour of the Faradaic current,

J(t),

across a fractal

_surface,

scales like

where ds

is the fractal dimension of the surface. Some heuristic considerations

_[2-4]

and rather

simple exactly

solvable models

[2, 3],

have been

_suggested

in order to

_{justify equation (1).}

In reference

_[4]

it has been shown that the survival

_{probability, PS(t),}

of a

_{particle starting}

at random near the

_boundary

has the same

_asymptotic

behaviour of

_{J( t )}

and that the

a - _exponent is related to the surface and bulk

_magnetic

_exponents

of the Gaussian model

associated to the diffusion

_equation.

One

_might

ask what

_happens

when a fractal bulk is substituted to the euclidean one.

Two differences are to be

expected :

an anomalous diffusion instead of the normal one and

the _presence of the fractal

_{dimension dB}

of the

bulk,

instead euclidean dimension d.

By

means of

_general

and exact

_arguments,

the authors of reference 4 have been able to

generalize equation (1)

to the case when besides fractal

_boundaries,

a self-similar

_waiting

time

distribution is

_present

on an Euclidean d-dimensional

_bulk,

thus

_leading

to anomalous diffusion.

In this case

is found instead of

_{equation (1). dw}

in

_{equation (2)}

is the

_exponent

_governing

the

long

time behaviour of the _{average square}

_distance,

_r2

travelled

by

a random walker after a time

t, i.e.

2 "" t 2/ dw

.

Concerning

the behaviour of the survival

_{probability Ps(t),}

some heuristic

_arguments

_[4]

suggest

that the power law

decay

is now different from the one

_{of J(t),}

and that is should be

characterized

_by

the new

_exponent

Our aim is three-fold : to

_present

_exactly

solvable models

_(two-

and three-dimensional

Sierpinski gaskets)

where the

_scaling

relations obtained in reference

_[4]

are verified. We think

it is

always

welcome to have

_{examples confirming scaling}

laws and

_indirectly

the

_arguments

leading

to them. Indeed as mentioned in reference

_[4]

a

straightforward generalization

of

preexistig

arguments

leading

to

_{equation (1)}

_[2]

does not lead to

_equation

_(2).

If the

_waiting-time

is an effect due to the presence of a self-similar distribution of

_symmetric

transition rates between

neighbouring

sites then the

arguments

of reference

[4]

would

suggest

in this case

_{P s(t) ’" J(t) ’" t- a}

with a

_{given by equation}

_(2).

We will

_give

an

_example

that this

indeed occurs while other « bulk

_{properties »}

like the average over sites of the autocorrelation

function and the mean end-to-end distance after a time t behave like in the related

waiting-time

_problem.

Last but not least the renormalization _group

_(R.G.)

_analysis

of diffusion in _presence of

boundary

conditions is not a

_simple

modification to be done in the

already existing

cases

_[7].

A first

_example

of it was

_given

some time _ago in reference

_[6].

In section 2 the one dimensional case will be discussed in order to

_exemplify

how the

implementation

of a R.G.

analysis

must be done when

_boundary

conditions are

properly

treated. An

_application

to the Huberman and

_{Kerszberg [8]}

model of ultradiffusion is done and some

_asymptotic

behaviours are

_compared

with those found for the related

_waiting-time

problem already

discussed in reference

_[6].

In section 3 the two- and three-dimensional

Sierpinski gasket

is studied where besides an

absorbing boundary

also a hierarchical

distribution of

_waiting

time is

_present.

In a sense our results could be considered

complementary

to those of reference

[7]

where

_boundary

conditions were not taken into

account. Section 4 contains

_scaling

arguments

and heuristic considerations

_{supported by}

the

exact results of

_previous

sections.

2. 1-D

_examples.

2.1 UNIFORM CASE. - Let _us start with the

very

simple

and

exactly

solvable

example

of

diffusion on a semi-infinite one-dimensional lattice with an

_absorbing

wall at x = 0. It will be shown that of two

_possible

choices for the sites to be decimated in order to

_implement

a

Let

_PXo,x(t)

be the

probability

of the

_particle

to be at site x after a

_t-step

random walk

beginning

in x = _Xo. P’s

satisfy

the discrete time master

equation

where

_{P,,,@,, = 0 (t)}

=

0,

for _{any t,} is the

boundary

condition and

P Xo, x(O)

=

5 x, Xo

is the initial condition.

We assume in the

_following

that _.xo =

1,

that is the

particle

starts its walk close to the

absorbing

wall.

By

introducing

the

_generating

function

_{(discrete Laplace transform)}

equation (4)

can be rewritten as :

where

In

_{equation (7)}

_{a x = 1 (lù )}

=

â ( w )

has been written in order _{to start} with the minimal subset of

_parameters

_entering

the master

_{equation (6)}

which remains invariant under R.G. recursions.

_Pathological

role

_{played by x}

= 1 in

fact,

could make

a (lù ) =F a (.w )

under R.G. transformations even if

_actually

the initial condition for the flow

_{is lX (lù )}

=

a ( w ).

It is

easy to

see that the choice

_{a (lù) = 1 + lù}

_corresponds

to a

_reflecting

barrier at x = 0. The Gaussian model related to this

_problem

is described

_by

the reduced Hamiltonian

Indeed

_using

the

_identity

one sees

_{that pxo ’Q x>}

satisfies the same

_equation

as

_{G Xo,}

_{x (lù)}

for

_{h = h}

_{1 = 0.}

A R.G. transformation can be obtained

_{by decimating}

the field variables in a subset E of sites and

_recasting

the effective

Hamiltonian,

for the

_surviving

field variables

_’P’,

in the form

(8a)

with renormalized

_parameters

_{a

_{a’, h’,}

h

_Í}.

If the set of even sites

_Eeven

is chosen as subset

_E,

the

following

recursions for

a and a are found :

together

the field variable renormalization _{cp2 x} =

NFa

ço’.

R.G. flow in the

{a, a}

plane

is shown in

_figure

1 a ;

the

_only

fixed

_point

A =

_{(a *}

_{= 2,}

1ii * =

1 )

turns out to be

repulsive

and it cannot be reached

_by

_starting

with 1ii = a. For úJ -+

_0,

the « fixed

point » (a

* =

2,

1ii* =

oo)

is reached which does not describe the

problem

of a walk

starting

at .X{) = 1 since

P Xo = I,x(úJ)

= 2( ’PI ’Px)

= 0 for a = oo!

On the

_contrary

if the set of odd sites

_{E.dd (which}

includes x =

1)

is

chosen,

the same

recursion

(9b)

is

_recovered,

and

instead of

_{equation (9a).}

R.G. flow in

_{the {a, a}}

_plane

is shown in

_figure

_{1 b ;}

there are two fixed

_points

A =

_{( a}

*

=

2, «

* =

1 ), B = (a * = 2, a * = 2) in

this _case, but B is the

only

one that can be

reached

_{by starting}

with « _ « i.e. in the case of

_{absorbing boundary}

condition. The fixed

point

A is related to _{the presence of} a

reflecting boundary

at x = 1 since

(a ( w ) = 2 + w,

The recursions for the

magnetic

fields

_h

1 and h are :

whose

_eigenvalues

_{A H}

and

_{A HS}

are related to the bulk

_magnetic

_exponent

_YH and surface

magnetic

exponent

YHS

through

the relation

2yi‘

=

_A

i

(2).

We

get

yH

= 3 ,

YHS

= 1

at

2 2

A _{and YH}

= 3/2 ,

yH - - 1

at B which are the exact ones.

2 S 2

Fig.

1. - Flow

_diagram

for the recursions

_{(9a, b) (a)}

and

_{(9b, c) (b).}

The use of recursion

_equations

like the ones obtained to derive the

_asymptotic

_scaling

will be done in the next subsection in a more

_general

model. It is well known that different R.G.

schemes can lead to different fixed

_points

but to

_equal

_exponents,

that is the same

_physics

must be described. However in the first of the two cases

_presented

_above,

due to the R.G.

flow,

one is not allowed to derive

_exponents

by

standard

_scaling

_arguments.

We must notice however that as far as the bulk

_properties

are concerned

only

the recursion

_(9b)

is

important

and it is

_independent

of the R.G. schemes we use ! It is worth

_noting

that,

in the

_analysis

of reference

_[6],

the

_right

scheme has been chosen.

2.2 HIERARCHICAL TRANSITION PROBABILITIES DISTRIBUTION. - Let

us

_apply

what we

have

_just

learned to the non-trivial case of the Huberman and

Kerszberg

model

[8] :

the

(discrete)

Laplace

transform

_{P Xo,}

_{x( w)}

satisfies the

_{following equation}

with x -- 1 and

The transition

_{probabilities}

_{Wx, x +}

₁ to

_jump

from site x + 1 to site

_{x = (2 m + 1 ) 2n =F 1}

(m, n

=

0, 1, 2,

...)

are

representing

a

_{particle hopping}

across barriers

_{(low Wx -}

_{l, x} means

_high

barriers between.

x

_{and x - 1).}

In order to take into account

_boundary

conditions we choose

indicating

that x = 0 is

partially absorbing

if W 0 or

_reflecting

if W = 0.

By

a

_simple

redefinition

of P’s

one can fix _eo = 1. The model

(11),

without

paying

attention _to

boundary

conditions,

was studied

_by

_approximate

renormalization group

techniques

in references

[8,

9]

and solved

_by

an exact R.G. in reference

_[10].

It exibits a «

_{dynamical phase}

transition » at R

_{= Rc}

=

1 /2

between an anomalous diffusion

_regime

_(R

_{Rc )}

and a normal diffusion one

(R > Rc )

[9, 10].

Let us see how the

_things

_go with

_boundary

conditions. Also in this case it is more

_practical

to work with the related Gaussian model whose Hamiltonian is

The minimal subset of « external field »

_parameters

we

need,

which remain invariant under

R.G.

_recursions,

is

_hx

=

h ( 1-

5 x, 1)

₊

h

1d X,

I.

The

_right

subset of sites to be decimated is E -

_{x

= 4 m +

_1,

4 m + 2 : m =

0, 1, 2,

...

}

and _{apart from}

_boundary

effects was the same used in

_[10].

where H’ has the same form as in

_equation

₍₁₄₎

with the

_following

renormalized fields and

parameters

and in

_{equation (15)}

is constant and the

relevant terms for our

_{applications,}

are

Now let us discuss what these recursions

_imply

as far as the

_asymptotic

behaviour

w -> 0

_{(i.e. t}

-->

ao)

is concerned.

The fixed

_points

of recursion

_{equations (16b, c)}

are 0 =

(w*, e1-l) =

_{(o, 0 )}

and

and the

_eigenvalues

of the linearized recursions near them are

_{A1 = 4,}

_À2=1/(2R)

and

_{A1’ = 2/R,}

_{A2 = 2 R}

_{respectively.}

Thus for

R

_{1/2(> 1/2) 0’ (0)}

is the stable fixed

_{point implying}

an

_asymptotic

behaviour for the

average end-to-end distance after a

_large

time t

with

_[10]

leading

to the above mentioned

_{« dynamical phase}

transition » at R =

Rc

=

1/2 [9, 10].

From

_{equation (15)}

_{the free energy}

_density

scales like

where the bar means average over sites.

_{Using equation (19) at h = hl = 0}

and at the stable

fixed

_point,

one has

implying

as one should

_expect

from standard

scaling

_arguments.

The fixed

_points

of

_(16d)

are W * = 0 and W * = 1

corresponding

_to the

reflecting

and

absorbing

wall at x = 0

respectively.

It is

easy to see that W* = 1 is

always

attractive and thus

as far as

_{W =1=}

0 the

_asymptotic

behaviour is the one

_{corresponding}

to the

_absorbing

wall.

The

Laplace

transform of the

_probability

to be at the

_{boundary starting}

from the

_boundary,

Pl,1

1

= (cp 1 cP 1 >,

can be obtained

differentiating

the

partition

function twice with

respect

to

h1. Recalling

the recursion

₍₁₅₎

and

_{( 16f)}

and G from

_{equation (17)}

we

_get,

at the fixed

_point,

for R

_1/2.

If R >

_{1/2 equation (23)}

still holds if R is substituted with

_1/2.

From

₍₂₃₎

one

derives the

_{following asymptotic}

behaviour

One can also derives the

_{asymptotic expression}

for the

_probability,

_FI,

_(t),

to be for the first time

_again

at the

_origin

x = 1 at time _t. Since

dW >.

2,

using

a

_simple

result

_reported

_e.g. in

reference

[ 11 ]

and

_{equation (24a)}

or

(24b)

one has

From recursions

_(16e)

and

_(16f)

one

_gets

_{(the procedure}

is the same as in the

_previous

subsection)

It _{is easy} to see that for

_{large t}

Pl,I (t) ~ t - (1 -

2 yHS/ dW) with the _YHs

_appropriate

to the chosen

_boundary

condition.

It was shown in

_[4]

that the current at an

_absorbing

_boundary

of an initial uniform

which is

_proportional

to the survival

_{probability PS ( t)=}

_.Ix

_{P 1, x ( t)}

of a

particle starting

near

the

boundary

since in the present case

_{(symmetric hopping probabilities)}

_{Pxo, X}

=

Px, xo.

Using

equations (26)

and

₍₂₇₎

it is immediate to show that

where _yHS

is,

of course,

given by equation (26c). Equations (28a, b)

confirm what we had

already

announced in the introduction

_concerning

diffusion in presence of

symmetric hopping

probabilities.

2.3 HIERARCHICAL WAITING TIME DISTRIBUTION. - Let

us now _{compare the}

_exponents

obtained above with the ones of a related model where the bulk transition

_{probabilities}

are

This means that at a site

x = (2 m + 1 ) 2n

there is an _average

_waiting

time ~ R - n and that this

distribution is hierarchical in the same sense as the Huberman and

_Kerszberg

model

previously

studied. At variance with the model of the

_previous

subsection

_{(Eq. (13a))}

now

sites

_represent

wells

_(low

_{Wx + l, x}

means

_deep

well at

_x)

and the transition rates are not

symmetric.

This model with

_boundary

conditions taken into account was studied in full detail

in reference

[6] (R

in

_Eq.

₍₂₉₎

is

R-1

in

Ref.

_[6])

so that we

_report

here

_only

the results.

Concerning

« bulk

_{properties »}

like

_{x2( t) >}

and

_{P Xo, xo (t)}

they

behave like the

_previous

case

with the same

_exponent

_{dw given by (18).}

The «

_magnetic

_{exponent »}

_{yH and yHS} are the same as in the uniform case of subsection

1 and thus do not coincide with those

_given

in

_{equation (26)}

in the anomalous diffusion

_regime

R

_{1/2. Concerning}

the

_asymptotic

behaviour of the correlations introduced above

_they

are

Equations (32a)

and

_(32b)

lead to the

_exponents

₍₂₎

and

_{(3) respectively}

for the 1-D case.

We believe these two last

_examples

are instructive at least for two reasons :

_i)

in

_spite

of

having

the same « bulk

_{properties »}

that could also be obtainable

_by

heuristic _arguments of references

_[7]

and

_{[6] they}

have

_completely

different « surface

_properties

». We would like to

remark that such a difference was

_{already partially guessed}

on the basis of numerical

simulations

_by

Havlin and Ben-Avrahan

_[12]

but never

_{explained. ii)}

Renormalization _group

techniques

we have

_{applied, explicitly}

exhibits the care that must be

_payed

to the

_boundary

conditions. This is uncommon

_expecially

because most of the subtleties related to an exact

3.

_{Sierpinski gaskets}

with

_boundary

conditions.

Let us consider now an

_example

of diffusion on a fractal structure. With reference to

_figure

_2,

consider a 2-dimensional

_{Sierpinski gasket}

with a side

aA

acting

like a

perfecting absorbing

surface. We assume that the

particle

could start the walk in each of the site

_belonging

to

aA =

{.xo: Iy-xol = 1, ye aA}

with

_{equal probability.}

If A is the

_bulk,

define

_Em

as the set of sites x

_belonging

to

_A,

such that x is a vertex of a

m-th order

_triangle

_(m

=

0, 1, 2,... ) ;

_a hierarchical distribution

of waiting

times is obtained

by assigning

a

_probability

_qx =

T m

to jump

when the

particle

is one a site x

_belonging

to

Em [7].

Thus _Tm is the

_waiting

time on sites of

_Em.

It can be further assumed

To = 1

by

_{a proper}

rescaling

of w.

The fractal dimension of the bulk is

_dB

= ln 3

_/In

2 and the master

_equation

is such a case

is :

where,

as

_{usual, E}

means sum over all

_{nearest-neighbour}

of x whose number is

y(X>

In our model :

Fig.

2. -

Subset of an infinite

Sierpinski gasket.

The

_waiting

times related to some

_{representative}

site

where z =

4,

2 before the first decimation is

performed, corresponding

to

absorbing

and

reflecting boundary

conditions

_{respectively.}

The

_probability

of

_{being trapped}

in 3A,

after a

_t-step

_walk,

_{Q (t),}

satisfies :

as it can be inferred

_{by using}

the constraint of conservation of the total

_{probability. By}

means

of

₍₅₎

and our définition of the initial

conditions,

we

_get

from

_{(33) :}

where

1 aA

1

is the number of sites on the surface and

_{8 x. aA}

is 1 if x e 6A and zero otherwise.

As we learnt from the one dimensional case the decimation

_procedure

consists of the

elimination of the whole set

_{Eo, including}

_atl,

in favour of

_{Em (m}

>

0 ) ;

the whole

_system

is

thereby

scaled

_by

a factor 2. This leads to new

_{expression G x,, ax,}

«’ in terms of

After some

_{manipulations}

the recursion

_equations,

in the limit w -->

_0,

are :

where

The recursions for w and

_Tn’s

are obtained

by imposing Ti

= 1 while the ones for

T and z are derived from the recursion for à

by requiring

j’ =

z’ (1

₊ w’

T’ ).

It is

_important

to stress that the G’s renormalization is determined

_by

the

_{inhomogeneous}

term in

_{equation (36)}

which,

for the renormalized

_problem,

must

be dx, aA’

_{( x}

in

_{Eq. (38a)}

1 6A

x

has the same functional

dependence

on

_{{ T’ }}

and w’ as

_{G x}

has in terms of the unrenormalized

counterparts).

The fixed

points

of the recursions

_(38b-e)

are :

The fixed

_point

reached

_by

f under recursions is

where

We have used

that,

since

_{(az’ /az)z*}

= 2 =

10/3

and

(8z’Iaz)z.

= 13/3 =

3/10,

z* =

13/3

is the

stable fixed

_point

and it is reached

_starting

with the initial condition z = 4. z * = 2

corresponds

to

_{reflecting boundary}

conditions and it is unstable as usual.

If

_Rn

= R recursion

equations (38b-d) imply

that the stable fixed

point describing

the

asymptotic

behaviour w - 0 is

_T1*

= Max

( l, R - 2 ),

where the maximum

eigenvalue

of the

matrix associated to the linearized recursion

_gives

the diffusion

exponent

defined

_through

the

_scaling

law

rl(t) - t’l dw

of the _{average square} distance travelled

_by

the

particle

after a

_long

time t.

As

_expected

a «

_{dynamical phase}

transition » at

_Rc

= 3

[9,

10,

7]

arises,

separating

the normal and anomalous

_regime.

In the

_following

we will call

_dwO)

the diffusion _exponent in

equation

(42)

when

R

_Rc.

The result

₍₄₂₎

has

_already

been

_{given by}

Robillard and

_{Tremblay [7].}

We

_point

_out, however that neither

_boundary

conditions nor

_magnetic

_exponents

have been discussed in

their work.

In view of the fixed

_{points quoted}

_above,

it is

_{straightforward}

to derive the

_long

time

where

_Q

is the discrete

Laplace

transform of

_Q

and

which

satisfies,

by using

the recursion

_{equations (38a, b)}

at the fixed

_{point :}

Equations (43), (45), (35)

and the recursion

_(38b)

at _{T =}

_{71 *}

lead to the

_{following asymptotic}

behaviour of the survival

_probability

and autocorrelation function

where

which agrees with the

_{prediction (3),}

since in this

_{case ds}

= 1 and the normal diffusion

exponent

in the

_present

case is

_dw(°)

and not 2 as was the case

_{equation (3)}

refers to

_(in

order to

obtain

_{Eq. (46b), Eq. (35)}

has been

_used).

In order to

_compute

_magnetic

_exponents,

we write down the Gaussian model associated to

the

_{equation (36),}

i.e.

A decimation of

_Eo

similar to the one

_performed

in the one-dimensional case,

yields

the

previous

recursions and a new

_couple

of recursion

_{equations involving}

h and

h 1.

In the limit

w - 0 and at the fixed

_point,

these latter turn out to be :

independent

of T * ! !. Thus bulk and surface

_magnetic

_exponents

are

which can be

_thought

as

_{generalization}

of the standard Euclidean case

_[4].

As shown in reference

_[4]

the current

_arriving

at the

_absorbing

wall for

_initially

uniform

and

_{using equation}

(50a, b)

which

_{generalizes equation (1)}

to the

_present

case.

In the

_{reflecting boundary}

condition case one finds

confirming

a naive

_scaling

_argument

_according

to which the sum

_{of yHS’s corresponding}

to the

reflecting

and

_{absorbing boundary}

conditions is

_equal

to the fractal dimension of the

boundary (see

next

_section).

At least we want to check our result in the three dimensional

Sierpinski gasket.

This time

the surface is fractal and it is the two dimensional

_{Sierpinski gasket}

of the

_{previous example,}

i.e.

_ds

=

ln 3 .

Rather tedious calculations follow the same

_procedure

as the two dimensional

In 2

case.

Instead of

₍₄₂₎

we have

(see

also Ref.

[7]).

and the

_{« dynamical phase}

transition » occurs at

_Rc

= 4.

Equation (47)

is substituted

by

while

_{equations (50)}

and

₍₅₁₎

retain their form.

In this case it is found z* =

13/2.

4.

_Scaling

_arguments

and conclusions.

The exact

_asymptotic

behaviours obtained in the

_previous

sections

_together

the results contained in references

_[4,

_6],

_help

us to formulate naive

_scaling

_arguments

which

_might

hold in

_general.

The

_singular

_part

_{of surface free energy}

_density

of the Gaussian

models,

associated to the diffusion

_problems

studied

_above,

scales like

under a

_spatial

_rescaling

l.

In all cases we have

_{studied (Qx Q Xo)}

=

A x fi Xo,

x(w)

where

Ax

does not

_depend

on x for the barrier case

_(see,

_{e.g., Sect.}

_{2.2) ;}

for the wells case,

Ax depends

on x

_except

when x E âA. From

_{equation (53)}

we thus have that the

_Laplace

for

_{small w,}

_implying

the

_{following large t}

behaviour

yHS at variance with

_respect

to _yH, will

_depend

on the

_boundary

conditions.

Given the

asymptotic

law

_(54b)

a

_simple

theorem

_(see

_e.g.

_[11])

states that the

_asymptotic

behaviour of the first retum

_{probability, F6A,}

will be

It is crucial to observe

_that,

_by

_definition,

_F6A

cannot

_depend

on the

_boundary

conditions

and thus the

_scaling

laws in

_{equation (55)}

must coincide for the

_absorbing

and

_reflecting

cases

(we

will

_distinguish

the two cases with the

_superscripts

a and r,

respectively).

This

_implies

that

Since

x (r)

x(a)

we have

x (r)

_1,

x(a)

> 1 and

p (a) ~

_F6A

For hierarchical distributions of

_symmetric

transition rates

_(see

_e.g.

_{subsect. 2.2)}

with a

simple

redefinition of the time scale one has that the

_particle

moves at each time

_step.

If after

a

_t-step

walk the

_particle

visits

_{(almost) uniformly}

a

_region

of linear

size e (t) - t’l dw,

then we

should have

implying

i.e. the same behaviour we would

_get

if aA were a

_(fractal)

subset in the bulk.

The

_exponent

_yH can be determined

_by

_observing

that the zero field bulk

_{susceptibility}

must

behave like

Now it is easy to see that

_equation

_(28a)

holds with

to be

_compared

with

_equation

_(2).

The case of hierarchical distribution of

waiting

times

(equal probability

to

_jump

from a site

to _any of its

_neighbours

like in Subsect. 2c and Sect.

₃₎

was

_already

considered in reference

_[4]

for diffusion on a

_regular

d-dimensional lattice. The

_arguments

_given

there are

_easily

generalized

to a fractal bulk. Indeed if renormalization transformations are

_analytic

in a

neighbour

of w = 0 then the

_exponents

YH and yHS are the same as in the case without

_waiting

time. In the

_example

of section 3 if w = 0 the Hamiltonian

(48)

becomes

independent

of

{qx}

(see Eq. (37b))

and thus the recursions for the bulk and surface field h and

Aj

are the same as in absence of

_{waiting times !}

Thus

_using

the results of the

_previous

case we

_get

where

_dw(o)

is the diffusion

_exponent

in absence of

_waiting

times.

_Proceeding

as in reference

[4]

we

_easily

_get

the

_decay

exponents

for current and survival

_{probability, respectively.}

In the

_examples

considered here the

_waiting

times on the

_boundary

do not

_depend

on the

site and

_{equation (54b)}

holds

_implying

All the _exponents found in the

_previous

sections

_by

an exact R.G.

_analysis

agree with the

corresponding

ones

_{heuristically}

derived here. In

_particular

we wish to stress the

independ-ence of

_yHS(a)

from the surface fractal dimension as

_{predicted already}

in reference

_[4]

and the

relations

₍₆₀₎

and

₍₆₃₎

_relating

the

_asymptotic

_{behaviours of the average autocorrelation} function and of the survival

_probability.

We should mention that

_{recently adsorption}

of

_{self-avoiding}

walks have been studied in

reference

_[13]

on the same

_{Sierpinski gasket}

used here.

It would be

_{extremely interesting}

to

_study

the case of the ideal chain

_adsorption,

i.e. the

problem

of reference

_[13]

without the self-avoidance

_constraint,

on self-similar structures with

coordination is

uniform,

and in

_general

can

_present

rather

unexpected surprises,

at least on

deterministic self-similar structures

_[14].

Acknowledgment.

We wish to thank Attilio Stella for discussions and for a critical

_reading

of the

manuscript.

We

thank also the referee whose criticism stimulated the

_scaling

_arguments

of section 4.

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