Finitely Generated Multifractals Can Display Phase Transitions

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Finitely Generated Multifractals Can Display Phase Transitions

Thierrey Huillet, Bernard Jeannet

To cite this version:

Thierrey Huillet, Bernard Jeannet. Finitely Generated Multifractals Can Display Phase Transitions.

Journal de Physique I, EDP Sciences, 1996, 6 (2), pp.245-256. �10.1051/jp1:1996146�. �jpa-00247181�

Finitely Generated Multilfactals Can Display Phase Transitions Thierry Huillet (*)

and Bernard

Jeannet

Laboratoire

d'Ingénierie

des Matériaux et des Hautes Pressions

(**)

Institut Galilée,

Université Paris 13, Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

(Received

11

September

1995, received in final form 10 October 1995 and

accepted

25 October

1995)

PACS.64.60.Ak

Renormahzation-group, fractal,

and

percolation

studies of

phase

transitions

PACS.05.50.+q

Lattice

theory

and

statistics; Ising problems

PACS.75.10.-b General

theory

and mortels of

magnetic ordering

Abstract. A _new Mass of multifractal

objects ("skewed" multifractals)

_is introduced, ^the

multiplicative

generator of which has _a finite number of branches of different real-valued

depths.

Both

microscopic

and macroscopic ^scales _are represented by such

objects,

each of these _corre-

sponding

to _a

specific thermodynamical regime.

In the "diluted"

regime,

the

partition

function Zt is

exactly

renormalizable which _means in the

sequel,

_{as is} the _{case in} trie

general

multifractal

theory,

that t~~

log

Zt _{as a non} trivial limit _as t tends ta infinity. In trie "condensed" _one the

partition

function _converges. Details about the transition between these two regimes are

given.

Résumé. Une nouvelle classe de "multifractales" est

introduite,

_pour

laquelle

le

généra-

teur

présente

_un nombre fini de branches de

longueur

variable à valeurs réelles. Les échelles

macroscopiques

et

microscopiques

sont

représentables

_par de tels

objets,

chacune d'elles _cor-

respondant

à

un

régime

thermodynamique

spécifique.

Dans la phase "diluée", la fonction de

partition

Zt est exactement

renormalisable,

_{en ce sens}

(classique)

_que la limite

quand

t _{- oo} de t~~

log

Zt est _non triviale. Dans la

phase

"condensée" la fonction de partition _converge. Les

détails

thermodynamiques

concernant cette transition de

phase

sont fournis.

1. Introduction

Inhomogeneous

natural

objects

_can

generally

be considered as assemblies of

fragments

_or

"atoms",

the

sizes, weights

and

spatial organization

of which _{cover an}

extremely

wide _range of orders of

magnitude,

_rangmg from the

microscopic

scale to the

macroscopic

_one. A _num-

ber of _common

examples

of such

composite objects,

such _as

beaches, forests, galaxies

etc...,

have

already

been

widely

discussed in the hterature about fractals. When

only microscopic arbitrarily

small

fragments

_are

considered, then,

_a

complete description

of the

assembly

_is not

accessible and _a statistical

description

is the

key

point of the multifractal

theory.

In most _cases however both

microscopic

and

macroscopic fragments

coexist in the

object.

Therefore the first

question

that _arises is: what kind of model would be

capable

of

describing (*)

Author for correspondence

(e-mail: huillet@limhp.univ-paris13.fr)

(**)

UPR 1311 CNRS

©

Les

Éditions

_de

_Physique

₁₉₉₆

such _a situation? The second

question

is: how does _one handle _or understand the transition between these t~vo

phases,

for instance in _a way similar to that used to understand the initial process of

condensation,

1-e- the

clustering

of molecules into

droplets

when the temperature of _a gas is lowered. One of the

goals

of this _paper is to ~vork out this

analogy

and _see its limitations.

According

to an

hypothesis commonly

used in natural sciences the structures

investigated

here will be assumed to present a character of

self-similanty.

The _purpose of this paper is ₁₎ to construct _a

simple

mathematical model able to descnbe

composite objects

_as those described

previously, ii)

to _use the statistical

properties

of this model in order to

reproduce

the main features of these

objects, namely:

coexistence of both diluted and condensed

phases

and

phase

transition.

In order to

exploit

the

analogy

with _a

condensating

fluid mentioned

previously,

the

language adopted

here will be

naturally

that of

thermodynamics.

A

purely geometrical

^model has been chosen. It is based _{upon a}

special partition

of the unit-interval1 ^~~~

(0

< _x <

1)

into _a set of

sub-intervals,

the

lengths

of which _are either "observable" _or not. This set is derived from _a

special

kind of "skewed" non-random

finitely-generated multiplicative

cascade. In this

cascade,

the

partitiomng

of _a finite number of the initial sub-intervals _is

stopped,

whereas it is

continued for the other sub-intervals. Under these conditions the

limiting

distribution of the mtervals presents at the _sanie time both

microscopic

and

macroscopic

scales. Some critical value of _an external

parameter (separation point

_or

line) separates

^these two

phases.

We then sho~v that two different

thermodynamical

limits exist for these two

phases.

In the diluted

phase, fragmentation

is

complete, entropy diverges

but remains

renormalizable,

_or

self-similar,

as it is _m a standard multifi.actal

theory;

this

phase

is dominated

by

chaos in this _sense. In the condensed

phase,

the limit entropy

exists;

we shall

conversely

_say that order dominates.

The statistical

physics

of such hierarchical

systems

is then

developed.

Particular attention _is

paid

to the

divergence

mode of the

thermodynamical

functions _near the critical

point

_or line.

From this view

point

the

present

model exhibits

special

structures which _are worth exammmg in _some detail.

Similar

problems

_were first

investigated

ⁱⁿ

iii by using

a reductible transfer matrix. As it will be _seen later ouf conclusions _are also in agreement with other recent

investigations [2j.

^{The loss} of

analyticity

of the

thermody.namical

functions has also been introduced in multifractal models

descnbing

diffusion-limited

aggregates [3,4j.

It is shown

by

these Authors that the anomalous behaviour of

multiphcatively generated

_{measures is}

compatible

with the

self-similanty

of these

measures which is _a direct consequence of the fact that the multifractals considered there

are not

finitely generated.

This leads the Authors to derme left-sided multifractals. Such _a situation is known to be _a

signature

^of a

phase transition,

which requires some

generalization

of classical fractals

(see

for instance the multinomial _{measures m}

là-?] ), keeping analytical

the

thermodynamical

^{functions. Simflar} treatments and conclusions _cari be found in the field of

dynamical _systems

_[8].

It is worth

mentiomng

however that these

approaches

should not be confused with the present one, where it is shown that

finitely generated

multifractals _may. fail to be renormalizable.

2. The Model

Now _{we come} to _a brief

description

of the

phenomena

_{we are}

deahng

with

here,

_{I.e. a} hier- archical

multiphcatively generated system

^of intervals _on: 1

=

(0

< _z <

1).

At time _zero, this initial

interval, _generates offsprings

in

finite

number

(say AI).

There _are two

categories

of

"sons" _{in our}

model, namely

the

"productive"

_ones that will repeat their ancestors'

"growth

program",

and the "sterfle" _ones that do _trot

spht

_anymore.

~ l

t xi x2 x3 x~ ^{X> ~2 x3 x> x2}

~ 0

t2

Fig.

l.

First-generation

tree and

first-generation

of intervals _on 1.

Each of the

productive first-generation

mtervals _possesses its _own real-valued lifetime before it

generates

a

second-generation

in the _same ratio of sizes.

Thus,

the

length

^{of the} intervals is

multiplicatively propagated

ⁱⁿ the cascade. As

multiplicative phenomena

_{are qmte} diflicult to

handle,

_{one can m an}

equivalent

_manner work with _an additive structure, rather

considering

the

logarithm

of the

length

_process.

At _a

given generation

the lifetimes _are trot all the _same. Trie different lifetimes of the first

generation

^intervals are assumed to take their values into _a finite set of I( real distinct values:

(tk)Î=1

^It is convenient to

represent

this first

generation

of intervals

by

_a

generator tree,

as m

Figure 1,

^{for which} each branch

symbolizes

_an mterval. The

length

of each branch indicates the time at which the interval is cut into sub-intervals,

The different

lengths

of the first

generation

intervals _are assumed to take their values into _a finite set of L real distinct values:

(xi _)Î=1

The "sterile"

fragments

remain forever.

They

have infinite lifetimes. What

only

matters is to

keep

track of their distinct

length, taking

their different

possible

values in _a finite set of

Î

real distinct values:

(11))~~.

Let

(ai,k)Î(~i

denote the number of

first-generation

sons of

length

_{xi among} those that will

split

at té,

ànd _{(âi ~)~~~}

L denote the number of sterile

first-generation

_sons of

length fi.

L

There _are thus: a_,k ^~~~

~

ai _k

Productive

mdividuals whose lifetime is

tk,k

= 1:.I( and

i=1 K

ai, ^~~~

~ai,k

_whose

_length

_is

_xi,1

=

1, ,L.

We shall

speak

of

degeneracy

to describe this

k=1

K Î

fact. Let also A ^~~~

~

_a,k be the total number of

productive

_sons, and D ^~~~

~âi,co

the

k=1 ~=i

total number of sterile _sons. ive

get

^of _course, A ₊ D

= ilf.

"Length

conservation" is assumed,

L £

which _means:

~

_{ai xi +}

~

_{ài coi}

= 1.

Figure

1 is _an

example

of such _a generator ^for

which,

i=i i=i

L =

3,

K

=

2,

_{ai,1 " a2,1 " a3,1 #}

1, îli,co

_{# î12,co}

#

1,

_ai,.

" a2, # a3,. #

1,

_a

1 "

2,

_{a 2 #} 1,

A = 3 and D

= 2.

No

special ordering

_is chosen in the

arrangement

^of

lengths

of intervals. It is however

useful,

for

notftional

convenience

only,

to introduce _some

particular

values of these

numbers, namely:

~°~~

~~~

max

~°~~~

Let

(1*, k~)

^~~~

Arg

max

~°~~~

ive also define

t~ i,k=1 .L,K tk i k=i...L,K tk

0

~~ ~

~~~ +' % _~' ~~

t

~ ~ ~~ ~~

t ti

t

iii

₁₂

'((((( l Il Il 1Il1 t2

,

2

x2x3 x2

ii

_x212

,~2~( ^~~ ' ' ,

Fig.

2. Generation and

fragmentation

of trie unit~interval at any time t.

an associated

value, ~°)/~

^~~~

^°~~~'~

k i~,k~

In _a similar

fashion,

if:

logl*

^~~~ _max~

(- log t ),

and 1* ^~~~

Arg

_max~

(- log A ),

we define

i=i...L i=i...L

log

à~ the associated

log-number

value of such sterile branches:

logà*

^~~~

logài,co

i*

At any time

t, (see Fig. 2),

_we thus have constructed _a

system

of intervals embedded in the unit interval. We shall call

X,

the limit _set of

points

^obtained

by considering

the

endpoints

^of

these intervals when t _- oo. Observe

that,

due to _our construction

including

stenle

individuals,

there _{is an} infinite number of

adjacent

sites

separated by

_a measurable

distance,

and of _course

a

large

amount of

"infinitely"

^close _pairs.

We have

adopted

here the

"population theory" language

because the

presentation

of _our model _appears to be easier this _way. We

might

have descnbed _our model in ternis of "contact

processes"

_{m a very} similar fashion.

So,

let

X~

^~~~ U

B~(x)

stand for the

"coarse-gramed"

~ex set obtained from X

by centering

balls

,

of radius _{e on} each

point

x of this

set,

and

considenng

the reunion of these halls. When trie resolution is very

low,

_say when _e exceeds _some value 0 _{< go <}

1, X~

is limited to _a

single

^connected comportent.

By increasing resolution,

that

is to say

by positioning

e below _go, this component ^is

split

into M

sub-components

within which each

point

^is e~indiscernible. Part of these M

sub-components (A

^of

them)

will then

split

into

sub-sub-components

when _{e crosses}

respectively

the value

(eh )k=1..K, (not including degeneracy).

Part of these

(D

of

them)

will _not

split,

which _{means ci}

#

0,

₌

1...1,

_in

this _case.

Thus,

each

sub-component splits

at different levels of

resolution,

if it

splits

at all.

Increasing

the resolution ad infinitum

(that

is to say,

taking

the limit _{e -}

0+) subcomponents

shrink therefore _into the continuum of points ^X.

Thus, considering

the

change

of variables: _e

=

eoe~~, according

to which the sequence

(ek)k=i:.K

_con be obtained from

(tk)k=i...K by:

_eh

#

eoe~~L,

these two models _appear very much alike.

Thus,

_more and _more details _are discovered from X _as resolution _goes to

mfinity (e

_-

0+

_{or t}

-

oo). Reversing

the _arrow of

time,

^details

amalgamate

until _we

get

a final

blur,

just

_as

epidemics eventually

contaminates _a given

population.

3. Evidence of _a Phase Transition in the

Thermodynamic

Limit

3.1. STATISTICAL PHYSICS _OF THE CASCADE. From the statistical

physics point

of

view,

all information _on inter-distance distribution is enclosed in the

partition

^function:

Zt(fl)

^~~~

~

N,

_xt(1)~, fl

_E

lit,

t > 0 where

Nt

stands for the total number of individuals available at time

i=i

t,

^{both sterile and}

productive

_ones, and _xi (1) the distance

separating

the _two consecutive sites

1- 1 and 1, ₌

1,

...,

Ni,

at time t. We define the local

partition

^{functions of the} sterile and

L _L

productive offsprings, a~(fl)

^~~~

~îli,colt, ak(fl)

^~~~

~

_ai

kxf, respectively.

One _can then

i=i i=i

easily

check that

Zt(fl)

meets the renewal

equation:

K

Z~(fl)

₌

aco(fl)

+

~ iak(fl)lt~>t

₊

ak(fl)Zi-i~(fl)li~çii,

_t

2 °, Zo(fl)

=

i, (1)

which

basically

describes the "self-similar" character of _our model

(see Fig. 2).

In this

equation,

"1"

represents

the indicator which is 1 if its

argument

is

true,

^and 0 otherwise.

K

We observe that

Zi(0)

=

Ni

is accurate with:

Ni

" D +

~[a,k.11~>1

_{+ a_}

k.Ni-i~.li~<iÎ,

k=i

t >

0, giving

the number of intervals

Ni,

in the

cascade,

at time _t.

Hence, Zi(1)

=

1,

which

means that it is _a

fragmentation

of the interval

[0, ii,

at any time.

Letting

_{now, :}

Î(fl, _s)

^~~~

Î

^~

e~~~Zi(fl)dt,

_one

gets, taking

the

Laplace

transform of

il),

and after

elementary computations:

K

acojfl)

+

~

_ah

_(fl) Il

e~~~L

~~~'~~

^~

Î

' ~~~

s

~ ak(fl)e~~~k

k=1

provided

_s >

max(0, log~(fl))

^~~~

logz(fl), considering

the two

singularities

of

Î(fl, _s).

Those

singularities

_{are s}

= 0 and the

largest

solution _s

=

log~(fl)

to the

"generating

L,K L,K

equation": ~j

a~

kxfe~"~

₌

~j

ai

kxf~(fl)~~~

₌ 1.

Moreover, i(fl)

is

analytic

_on IR. It

i,k=i i k=i

should be observed

that,

on the

contrary,

^function

log z(fl),

as

defined,

is not itself

analytic.

Indeed, -logz(fl)

= 0 for

fl

>

flc,

with 0 <

flc

<

1,

the _unique cntical value solution to

L

i(fl)

= 1; ^hence:

~j

ai

xf~

= 1.

Moreover, -logz(fl)

_is

finite, non-decreasing

and _concave

i=i

(see Fig. 3).

It should be noted that:

log z(0)

₌

~°)~,

^{where T is solution of the}

^equation:

~j a,kAÎ

K

= 1.

(Note

that if all

productive

branches have the _same

length

_7, solution of the

k=i

last

equation

is T

=

7).

-é~yzl>)

à

r

fl~£

Fig.

3. Function

logz(fl):

the "renormalized

partition

function".

The existence of _a

phase

transition in _Dur

mortel, crossing

the critical

point,

^has thus been

proved.

Note that if

ao~(p)

=

0, 1(p, _s)

_has

_only

one

singularity.

In this _case there _{is no}

phase

transition.

It therefore follows from the

study

of

singularities

that the

"thermodynamic

limit" halas:

lim

log Zt(fl)

_"

log z(p), (3)

~vhere function

log z(fl)

is the "renormalized

partition

function".

At

p

=

0,

formula

(3)

reads:

hm

Zt(0)1 ^Î~

lim

(Nt)Î

=

Al Ù z(0)

_> 1

(4)

Thus,

the total number of intervals of the cascade _grows like:

Ai.

Let _now

Nt(e)

^{~ ~~} i

=

1,.

,

Nt/ logzt(1)

_{> e} denote tl~e

number,

_among these

Nt

in-

tervals at time t;

sharing

the property ^{within the bracket:} zt

Ii)

<

exp(-et).

It follows from the

large

deviation theorem _on

general

random _processes [9j

that, provided

_{e >}

~~°~~~~~

~Ù

~~~

lim

Nt(e)~/~

= exp

s(e) (5)

t-m

Formula

(5),

thus grues the

asymptotic shape

of the distribution of interval

length.

In _case

e <

~

~°~~~~~

,

change

> into < in the formula

defining Nt(e).

dp

~~o

In tl~e last

formula,

_s

je)

^~~~ inf

je p+log z(fl)

_is the _concave

Legendre

transform of

log z(fl),

peur

i e. "renormalized

entropy".

We shall call

e(fl)

^~~~

-~~°~~~~~,

_the "renormalized internai

dfl energy". Thus,

formula

(5)

is valid

provided

_{e >}

e(0).

3.2. THE ItENORMALIzED ENTROPY SHAPE. What follows is _a brief

description

of the

shape

of

s(e),

which

essentially

diifers front the left-sided _ones of

[3,4j (see Fig. 4). Taking

5(e)

@

---~-

T ^'

l '

.

'

~°8~

t* ~~~~~<~~~~~~~~~~~~

°

e(flc)

e(oj

e(-oeJ

_e

Fig.

4. The renormalized entropy

shape.

the

Legendre

transform of

logz(fl),

_as defined

by (3),

_one

gets: s(e)

is

defined, continuous, nonnegative

and concave when 0 < _e <

e(-ce)

< ce. On the "renormalized internai

energy"

interval: 0 < e <

e(-ce), entropy

is

strictly

_concave, with _a linear

part s(e)

=

fl~e

for ail

e E

[0,e(fl~)]. Here, e(-ce)

=

~°~~~, s(e(-ce))

=

~°~~~

(definitions

of these numbers _are

t* t*

given in Section

2)

and

s'(e(-ce))

= ce. s attains its

maximum, logA,

at

e(0).

Remarks

1)

^One can show that

pc

has _a natural

interpretation,

_in terms of box

counting dimension, namely:

dim X ^~~~

limsup ~°~~j~~,

wl~ere

N(e)

is the minimal number of intervals

~_o+

log

j necessary ^to caver X.

2) Up

to now, we have

only

been

dealing

with the statistical

properties

of _set

X, through

the

Laplace parameter fl.

At this

step, indeed, fl

^has

only

been used _{as a} distortion

parameter, allowing

to

study

the

family

of distributions:

(zt(1)P,1

=

1,...,Nt).

For

example, negative

^values of

fl

underhne small intervals in this

distribution,

whereas at

fl

=

0,

it is net

possible

to discriminate the small _ones from the

large

ones at ail. The statistical

physics language

_can be

adopted, provided

_{one gives a} richer

meaning

to this central

parameter fl.

TO do _so, observe that _Dur formulation _amounts to consider two

repelling partiales

whose

potential

_energy is E

=

logz,

that is to say, the

logarithm

of the distance between them

[2]). Suppose

that the

only

reachable _energy

levels,

at time t, are

precisely: Et Ii)

^~~~

log~t Ii),1

=

1,..., Nt.

This

pair

of

partiales,

at

"temperature"

T

=

(related

to its _own vibrational

energy),

_may then be used to observe the diiferent

fl

energy levels.

Indeed,

_one of the _axioms of statistical

physics

tells _us that the

probability

of

configuration

is:

Zt(fl)~~ ^e~P~'fi)

=

Zt (fl)~~

_zt

(1)P.

The statistical

physics

of this pair

can thus be derived from the

study

of the

partition

function

Zt(fl),

where the central parameter

fl

is _now

interpreted

_as the _inverse of the

temperature.

This

analogy

allows _us to

skip

from the statistical

language

to the _one of statistical

physics. Any

allusion to statistical

physics

in the

sequel

should be understood in this

physical

_context,

only.

3)

It should be

emphasized

that the presence of _a

phase

transition is _a direct consequence

of _our structure

presenting macroscopic scales,

due to the presence of sterile individuals.

Thus,

_fl~

separates

two

phases:

the "cola" _one

Ii.e.

when

fl

_>

pc)

for which macroscopic order

prevails,

and the "hot" _one for which disorder is the rule and which is identified with the _exact "renormalization" of the

partition

function.

3.3. BEHAVIOUR OF THE THERMODYNAMIC FUNCTIONS NEAR THE CRITICAL POINT

Going deeper

into the

analogy

with

thermodynamics,

mentioned

above,

it is

possible

to intrc- duce the

following quantities: fl

_can thus be

interpreted

_as the inverse of _some

"temperature".

Occurrence of _a

phase

transition appears in the limit: t _{~ oe.}

At

given (finite)

t, the statistical

properties

of this mortel _are enclosed into the "free

energy"

function=

Ft (4)

^~~~

j _{iog zt (4)}

Entropy St(E),

the

Legendre

transform of

log Zt(fl),

in the variable

fl,

satisfies:

~-pE,(i) ~pE,(i)

St(E)[~

~~~ ^{d<o~z jp) =}

flEt(fl)

₊

logZt(fl)

"

~ _log

,

where

Et(fl)

is the

' d~

~~~

~t(~) Zt(~)

average of the

"energy

levels" _:

log

_zt

(1),

as a function of

fl.

Looking

at t _as the resolution _power, its dual

variable,

in the _sense of

Legendre, P,

will represent

"thermodynamical pressure" [loi.

This allows to

defipe

the

thermodynamical

"free

enthalpy" G(fl, P),

the

Legendre

transform of

which,

ⁱⁿ the variable

P,

is

nothing

but

Ft(fl).

Pressure _{as a} function of

fl

and t is:

Pt(fl)

^~~~

~~~~~

ôt As t _~ oe, hm

St (E)[E,(p)

^~~~

^s(e)[

~~~ ^{diogzjp) =}

fie(fl) +log z(fl),

where the "renormal-

t-m t e "- do

ized internai

energy" e(fl)

^~~~ lim

~Et(fl)

"

-~~°~~~~~

is the hmit

(t

_~

oe)

average of the

t-m t

à~

"normalized energy levels":

logzt(1).

One _can show that pressure:

P(fl)

^~~~ lim

Pt(fl)

t t-m =

~°~~~~~

(> 0). Moreover,

the

"specific

heat _per unit"

~~~~

=

-p~~~~ ^°(~~~~~

can be

fl d(p- dp

defined.

We shall _now

briefly

discuss the behaviour of these functions _as

p

_~

pl.

From the definition of

log z(p)

and

P(fl),

it follows that

"pressure"

vanishes for

fl

>

pc,

_is

regular

and

decreasing

as

fl

<

pc.P(flc)

= 0. Thus

"pressure"

_{is a}

non-increasing,

continuous function of

fl.

On the

contrary,

"internai

energy" e(fl)

is _a

non-increasing

function of

fl, vanishing

when

fl

_>

pc

but with _a

jump

at

pc,

whose

amplitude

can

easily

be

computed.

Occurrence of _a

jump

at

p

=

pc

_is shared

by

the

"specific

heat per unit".

3.4. INCLUDING _THE GENERATION NUMBER. Additional information _on the hierarchical

structure under

study

is enclosed in the

bi-partition (grand canonical)

function:

Et là, p)

^~~~

P~(t)

N<(P)

~j e~~P ~j zt(1)°,

_À,

p

E

IR,

t >

0,

where

Nt(p)

stands for the number of individuals at t

p=i i=1

In the Cascaàe WÎtÎI

eXactly

_p ancestors

(file generatlon nUmber).

p+(t)

^~~~

Integer part

indicates the maximal number of such ancestors at t.

min

(tk)

=1...K

P+(n)

Indeed,

bath

marginais: Bt(0, fl)

₌

Zt(fl),

and

Bt(À, 0)

₌

~j _e~~PNt(p)

are of

interest,

in p=i

this _case. This is _one of the main interest of the "skewed" character of _our multifractal

family.

One _can

easily

check that

Bt(À, fl)

meets the extended renewal

equation:

~~~~ ~~ ~~~°~i~~

₊

e> ia~i~)it~>~

+

a~i~~~~-~~ j~, ~~i~~<~j,

> ~

Bo(À,fl)

= 1

(6)

This follows from the

straightforward

recurrence formula

defining Nt(p)1

K

Nt(p)

= D

ip~i

+

~ _la

~

ip=i it~>t

+ a,~

Nt-t~ (p i) it~~tj

,

t

2 o,

_p

=

i,.

,

p+(t).

k=1

Letting

_now,:

É(À, fl, s)

^~~~

Î~ e~~~Bt(À, fl)dt,

_one

_gets, taking

the

Laplace

transform of

(6):

e~~ lK

are

(~)

+

~

ak

(~)(Î

e~~~~

j

É(À>4> s)

=

~)l

(71

S Î e-~

~

_ak

_(~)e~~~~

k=1

provided

_s >

max(0, _log~y(À, fl))

^~Î~

logf(À, fl).

Here _s

= ~y(À,

fl)

is

(for

each

(À, fl))

the

largest

K L,K

solution to the

"generating equation": e~~ ~jak(fl)e~~~L

=

e~~ ~j ai,kzfe~~~L

₌ that

is,

k=i 1k=i

L,K

satisfying: ~j

ai

k~f~y(À, fl)~~L

₌

e~.

l,k=i

Function

log f(À, fl)

thus faits _to be

analytic,

which is the

signature

of _a

phase

transition.

This function _is

finite, non-decreasing

and _concave. Note that

f(0,fl)

=

z(fl).

It satisfies

logf(0,0)

=

log Al,

_with

Al ~ÎÎ f(0, 0)

>

1,

and T defined _{as in} Section 3.1.

Moreover,

logf(À, fl)

=

0,

when >

T(fl),

with

T(fl)

the

strictly

_convex critical hne of

equation:

L,K

T(fl)

₌

log ~j

ai

kzf, (8)

k=1

for which:

T(0)

=

logA, T(flc)

₌ 0. One has thus

proved

the existence of _a

phase transition, crossing

the critical fine this

time,

the

equation

of which is

=

T(fl),

in the

(fl,

_À)

plane.

It aise follows that the

following thermodynamic

limit halas: hm

-~

logBt(À,fl)

=

t-m t

logf(À, fl),

and that the behaviour of ail

thermodynamic

^variables can of _course be derived from this _more

general

situation.

The behaviour of the

thermodynamic functions,

_as

fl

_~

pi

_can

easily

be obtained in _a

similar way.

&àÇ*

ôc

/

r' à

-logâ*1'

'

Fig.

5. Function

log Z(fl).

4.

Thernlodynamics

of trie Condensed Phase

4.1. CONVERGENCE OF THE PARTITION FUNCTION. Iii _case

fl

>

pc,

it should _now be clear

that the limit of the

partition

function

Z(p)

^~~~ hm

Zt(fl)

itself exists and is:

t-m

~ al,mÎÎ

Z(fl)

" lilll _S

Z(S, fl)

_#

~)

^~~~

"

~~~

~

(9)

s-o+

~j _ak(flj ~j

_ai,

xl

k=i 1=1

from the initial value theorem of

Laplace. Figure

5

gives

the

shape

of the function

logZ(fl).

Thus,

in the "cola"

phase (fl

>

fl~),

order

prevails

and the

partition

function has itself

a

thermodynamic

limit. This

phase

thus deserves its _own statistical

physics.

For

example, defining

the

Legendre

^transform:

S(E)

= inf

(Efl

₊

log Z(fl)),

the

entropy,

it should be clear P>P~

that the whole

"machinery"

of statistical

physics

_can be defined. In

particular,

the "internai

energy" E(fl) ^~ÎÎ -@

_{the "free}

energy" F(fl)

^~~~

-) logZ(fl)

and the

"specific

heat"

C(fl) ^~ÎÎ -fl~~

can be of interest, in the condensed

phase.

4.2. THE SHAPE _{OF THE} ENTROPY. We first locus _on the

shape

of

entropy,

as a function of the internai energy

(see Fig. 6).

For obvions _reasons, this function _{is concave,}

non-negative,

defined for E >

-logl*,

where Î" is the

largest

value among the

(fli)i=i

_z.

Moreover,

S(-logl~)

=

logé'

~vith à* the number of individuals attached to this

vllul. _S(E(1))

=

E(1)

_> 0 where

E(1)

is the bounded derivative at

fl

₌ 1 of

logZ(fl).

The

slope

of

S(E)

is infinite at E

=

logfl*, (fl

_~

+oe),

_one at E ₌

E(1).

The behaviour of

S(E)

_near E

= ce is:

S(E)

_m

fl~E+log

E+ cste,

traducing

the fact that

E(fl) diverges

ai

fl

E-m

=

fl~. Thus,

this

entropy hnearly diverges,

_{as a} function of E. TÎ~is result

is in accordance with the linear

part

of the renormalized

entropy s(e),

defined in Section 3.2.

Divergence

of E is due to the

divergence

of the _average value of the

logzt Ii),

wl~en

fl

> fl~, In tl~is _case

Et(fl)

"

~~°~~~~~

"

-t~~°~~~~~

=

te(fl) provided e(fl)

<

e(fl~). TÎ~us,

E

àfl

_t-m

à~

SIC

,

, ,

Iogi* _{'_}

,

E@J £

Fig.

6. The entropy

shape.

diverges

like

te,

where t _ce ,vith _e <

e(p~)

and

s(e)

= lim ~~~~~

=

p~e,

under

hypothesis

t-m t

~ <

~(flc).

4.3. ASYMPTOTIC BEHAVIOUR NEAR THE CRITICAL REGION. We

briefly

stress here how

it can be obtained _as

fl

_~

fl/,

Observe from

(9)

that the denominator of

Z(fl)

_is aise:

L L

~j ai.(z/~ zf), referring

to the

equation defining

the critical value. Thus:

~j ai.(zf~

i=i i=1

zf)

L

m

-(fl pc) ~j

_ai,

_{z/~ log}

_{xi, so that:}

Z(fl)

_m

I(.(fl fl~)~~,

^{with K} as a

positive

4=4t

~~~ 4=4t

known constant. In _a similar ~vay, one con show that the "internai

energy" behavej

^like:

E(fl)

_m

K'+ (fl fl~)~~,

that the

"specific

heat" behaves hke:

c(fl)

m 1 ~~

,

and

o=ot o=Pf 4

that the "free

energy"

grows like:

F(fl)

_m

(K"(fl _fl~)),

where

K',

K" _are

positive P=Pt flc

known _constants.

Finally,

the entropy

S(fl) ~# S(E)j

⁼

^flE(fl)

⁺

^logZ(fl),

^{as a function}

E(P)

of temperature _grows ^like:

S(fl)

_m

Ki(fl flc)~~ K2 log(fl pc ),

where

Ki

and

K2

_are

P=àf positive

known constants.

5.

Concluding

Remarks

In this

article,

we

emphasize

_a basic

property

^{of nonrandom}

multifractals,

under which

phase

transition _occurs at finite

temperature, namely

the coexistence of _micro and

macroscopic

fea-

tures. It is indeed the presence of

macroscopic

characteristics within them which induces loss

of

analyticity

of the

thermodynamic

functions. We illustrate these ideas _{on a} continuous-time

(resolution) family

of skewed

multifractals,

which in itself deserves interest. Part of the "ther-

modynamics"

of such structures _is studied in _some

detail, including

behaviour _near trie critical

point (or hne).

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B.,

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