Entropy ultrametric for dynamical and disordered systems

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Entropy ultrametric for dynamical and disordered

systems

Sergio Caracciolo, Luigi A. Radicati

To cite this version:

Entropy

ultrametric

for

_dynamical

and

disordered

_systems

Sergio

Caracciolo and

_Luigi

A. Radicati

Scuola Normale

_Superiore,

Piazza dei Cavalieri, Pisa 56100,

Italy

(Reçu

le 6 avril 1989,

accepté

le 7

_juin

₁₉₈₉₎

Résumé. - Le but de cet article est de montrer :

1)

que l’ensemble de toutes les histoires d’un

système dynamique

à un _temps donné a une structure

_{ultramétrique ;}

₂₎

_qu’on

_{peut définir pour}

un

_système

dynamique (X, F,

m,

T)

une distance

_Dm

sur l’ensemble des histoires, en terme de

l’entropie

de von _{Neumann-Shannon ;}

₃₎

_qu’on

_peut définir un ensemble à un

_paramètre

de

distances

_{ultramétriques}

_D(03B1)m

en utilisant les mesures d’information de

degré

a, a ~

_{[0, 1];}

4)

que le

paramètre

Y utilisé dans la théorie des

systèmes

désordonnés est une fonction de la

mesure d’information de

_degré

2 ;

5)

qu’une

construction similaire

_s’applique

aussi aux

_systèmes

désordonnés dans

lesquels

l’ultramétricité n’est _{satisfaite que dans} un sens

_{probabiliste.}

Abstract. 2014

The purpose of this paper is to show :

₁₎

that the set of all histories of a

_dynamical

system at any

given

time has an ultrametric structure ;

2)

that for a

_dynamical

_system

(X,

F, m,

T)

a distance

_Dm

on the set of histories can be defined in terms of the von Neumann-Shannon _{entropy ;}

₃₎

that a one _parameter set of ultrametric distances

_D(03B1)m

can be defined

_by

using

instead of the von Neumann-Shannon _entropy the information measures of

degree

03B1 for

03B1 ~

_{[0, 1] ; 4)}

that the parameter Y used in the

_theory

of disordered _systems is a function of the information measure of

degree

2 ;

5)

that a similar construction holds also for disordered systems where

_{ultrametricity}

is satisfied

_only

in a

_{probabilistic}

sense.

Classification

Physics

Abstracts 02.50 - 05.20

1. Introduction.

Ultrametric _{spaces, known since}

_long

to mathematicians

_(1),

have

_recently

found

_applications

in

_physics

in connection with Parisi’s solution of the

_{Sherrington-Kirkpatrick}

model for

_spin

glasses

[2].

A recent review on the

_importance

of

_{ultrametricity}

for

_physics

and other sciences

is

_given

in

_[3].

The purpose of this paper is to show that a similar ultrametric structure can be defined on

the set of all histories of a

_general

_dynamical

_system

at a

_given

time. Such a set is a

_partition

of the

_phase

_{space that becomes} more and more refined as time goes on. Therefore the

_entropy

of the

_partition

increases with time and we shall show that it is

_possible

to define an

ultrametric distance between histories from the

_entropy

distance between

_partitions.

Such a

definition is

_quite

natural because the

_complexity

of a set of histories is

_naturally

related to its

entropy.

(1)

For a recent discussion of

_{ultrametricity}

in mathematics see for

example

[1].

The paper is

organized

as follows. In section 2 we shall define a distance between the

elements of a

_partition,

a distance that will turn out to be ultrametric. In section 3 we will introduce a new ultrametric distance between two histories at a

_given

time that

_depends

_upon

the

_entropy

_{produced during}

the evolution from the common ancestor

_(to

be

_defined)

of the

two histories. Thus this distance

_depends

_{upon the choice} of the

_probability

measure. In

section 4 we shall

_generalize

the distance defined in section 3

_{by considering}

instead of the usual

_entropy

the set of

_a-entropies

_[4]

for a E

_[0,1].

In section 5 we shall show that in the

case of disordered

_systems

the

_analogue

_{of the time evolution is the resolution process of}

clusters of

_ergodic

states into their

_components.

The order

_parameter

of these

_systems

is related to the

_2-entropy,

thus

_showing

its true connection with the notion of

_complexity.

Finally

we shall

_give

the relation between the

_geometric

distance on

_phase

_{space and the}

probabilistic

distance

_{provided by}

the von Neumann-Shannon

_entropy.

2. The space of histories and its ultrametric structure.

Let us consider a

_dynamical

_system

_(X,

_J7,

m,

T )

where

(X, 57, m )

is a

_probability

_{space and}

T is a measurable

_automorphism

of

_{(X, J7).}

Here X denotes the Cartesian

_product

of the

phase

spaces of a

_physical

_system

at times

_{t = 1, 2, ... ; :F is a}

_{Q algebra}

of measurable subsets

(events)

of X and m a

_probability

measure

(a state)

not

_{necessarily preserved by}

T. An

_experiment

with a finite number of outcomes

_performed

at time t = 1 on

(X,:F, m)

is

_{represented by}

a measurable

_partition

y =

{Ao,

...,Ak-l}

of X. The _same

experiment

performed

at t = n

_(n

is an

_integer)

is

_{represented by}

the

_partition

T- (n - 1) y

=

{T- (n - 1) A °,

...,

T- (n -1 ) A k -1 }

and the sequence of

experiments repeated n

times from t = 0 to t = n is

_{represented by}

the

_partition

{ y

_{(j)j =,}

_{1 is}

a _{sequence of}

_increasingly

refined

_partitions,

i. e. _y

(j) >

y (j - 1>,

indexed

_bey

time.

The number of elements of

_{y (’)}

is kn. Each element of y (n)

(ij

e {0,

...,

k -1 }

for any

j)

represents

a sequence of outcomes of the

experiments

y,

T -1

_{y ,}

..., T - n y. We call

A (i 1,

...,

i n )

a

n-history

or

history

of

length n originating

at time

t = 1 from

A (i 1 ).

Similarly

T- i A (i 1, - - -,

i n )

is _a

n-history originating

at time t

= j

+ 1 from

A (i 1 ). Equation (2.4)

shows that each

_n-history

can be

_regarded

as the intersection of a

f

-history

(l

n )

starting

at time t = 1 with a

(n -

f)-history

_starting

at t

= i

+ 1. We shall call

A (i 1,

...,

if)

the common

f-prehistory

of all the

kn - f

histories that branch off from it. The

(n -

f

_)-histories

are the elements of

T- f

y (n - l) so that :

Thus the collection of all the histories can be

arranged

in a tree, indexed

by

time,

as shown in

figure

1. Each

_point

_represents

a

_{history ;}

those that

belong

to the same

_partition

_{occupy the}

same horizontal line. Each

_point

of

_{l’ (n)}

is connected downwards to a

_{single point}

of

l’ (n - 1)

and

_upwards

to k

_points

of

_{l’ (n + 1)}

that

_represent

the k

_possible

extensions of a

A

Fig. 1.

- A tree of histories

corresponding

to a

_partition

with k = 2. The common closest ancestor of E and _F, i.e. P

_{(E, G ),}

_belongs

to the

_partition

_{l’ (2)}

and has a

_prehistory

P

_{(E, F),}

which

_belongs

to y and

is the common closest ancestor of E and F.

It _{is easy} to see that the collection of all

_points

on a horizontal

_line,

i.e. the elements of a

partition,

y (n)

say, is an ultrametric set.

_Indeed,

let us consider two events E and F of

y (n)

and let

y (t)

with

Q , n

be the most refined

_partition

in the

_{collection { y}

_{(j)} oo= 1}

such that

one of its

_elements,

denoted

_by

P

(E, F),

contains both E and F. We call P

(E, F)

the common

closest ancestor

_of

E and F. One can

_{easily verify}

that the non

_negative

number.

is an

_acceptable

definition of distance between E and F.

In addition to the

_properties

of a distance D satisfies the more restrictive ultrametric

condition

where

_{E, F,}

G are

_generic

elements of

_{’Y (n).}

The maximum distance between two elements of

_{y (n)}

is n which obtains when

_they

have no common

_prehistory.

The distance

_{(n -}

i ) kt

is the same for all the

n-histories, E,

F e y

(n) with

common closest

ancestor

_{P (E, F)}

E

y (l).

Hence

where

E,

F are any

pair

in

_{y (n)}

such that P

_{(E, F)}

= G. D

(E, F)

is the _same for all such

pairs.

Thus the sum is the time difference between the

_lengths

of the histories and that of their

prehistory.

3.

_Entropy

ultrametric.

The distance

_(2.6)

is

_{purely dynamical}

and does not take into account the

_{probabilistic}

nature

probabilities

and on that of their common closest ancestor. We shall remove from the tree

histories with zero measure

_and,

in this way, we include the

possibility

of

obtaining irregular

trees.

We

_{begin by}

_noticing

that the distance between two

_partitions

y(n )

and

y (l),

with

l

n, can be measured either

_by

their time

separation n -

f

or

_by

their

entropy

(information)

distance which is

_{given by}

_{(2) :}

where

_{H(y(j) ,}

_{m )}

is the

_entropy

of the

_partition

_{y (j)}

with

respect

to the

_probability

measure m. If we

_keep

f

fixed

dm(y

(n),

y

(f»)

increases with n, not

linearly

as n - f

but in a _{way that}

depends

upon the rate of information

_{produced by}

the

_mapping

T on the

partition

y.

Let E and F be two histories of

_y(n)

with common closest ancestor P

_{(E, F)}

e

y

(l).

We can

define their distance

_by

where for any G E F

(mP

is the measure conditioned

_by

_P).

_{Obviously only}

those elements of

y (’)

that

_belong

to

the subtree

_arising

from P

_{(E, F)}

contribute to

H(y (n)

_mp).

Since the

_entropy

of the

_partition

y (l)

in the measure _mp,

P r= y (f),

vanishes we can write the distance

(3.2)

in the form

where

_dmp

is defined in

_{(3.1 ).}

The distance

_{Dm (E, F)}

_represents

the

_uncertainty

about which element of

_{y (")}

the transformation T

_generates

from the element P

_during

the time interval At = n -

f.

In

particular,

if all the histories

arising

from P

(i.e.

all the n-histories

having

a common

f-prehistory)

have the same

_probability,

_{1 / v}

_{say, then}

The distance increases with the number of descendants of P. We shall now _{prove the}

_{following :}

Proposition

1. Let

_{y (n)}

= _y v

T- 1

y v ... v

T- (n - 1)

y be a

finite

_{partition o f}

the

_dynamical

system

(X,

[F,

m,

T ).

The distance

_{Dm defined by}

(3.2)

between elements

o f

y (n) with non zero

probability

is ultrametric.

Proof :

Dm (E, F)

is

_{obviously symmetric}

in E and F and non

negative.

It vanishes if and

only

if E = F = P (E, F )

because E and F have non zero measure. Let

E,

F and G be three

histories in the

_partition

y(n)

and _{suppose that}

_{P (E, G),}

the common closest ancestor of E and

G,

be contained in

_{P (E, F ).}

Then

_{P (G, F ) = P (E, F ).}

If we set

L =

P (E,

F)BP(E,

G),

then the

_probability

measure

_mp(E,

F) can be

decomposed

as the

convex combination

Owing

to the

_concavity

of the

_entropy

we have

and therefore

From this it follows

If instead

_{P (E, G) P P (E, F)}

we obtain

_by

a similar

_argument

From these two

_inequalities

it follows

which proves the statement.

If 1£

is the uniform measure,

i. e. J.L

(G) =

k - j

for any

history

G E y

(j >,

then

where

_{D (E, F)}

is defined in

_(2.6).

The sum over G of the distances

_{Dm (E, F )}

such that

E,

F _{is any}

_pair

in

_-Y(")

with ancestor

G=P(E,F)E

l’ (1) ils

This relation for the sum of the

_entropy

distances

_{Dm generalises equation}

_(2.8).

Remarks :

(i)

the above results are valid also for

_partitions

y with a countable

_infinity

of

_elements,

provided

that for each n the

_entropy

of the

_partition

y (n) be

finite,

because this condition ensures that all the distances are well defined. In this case the number of branches at each

event in the tree is no

_longer

limited ;

(ii)

the maximum distance between two histories of y (n) is

H( y

(n),

m ),

which

is the mean

information _necessary to reconstruct the tree. The difference

(iii)

if m

_{preserved by}

T the limit lim

_{s (n )}

exists

_(it

is called in Ref.

_[9]

the tree’s

n -,m

silhouette)

and coincides with the rate of

_entropy

_production

_[5-7]

_by

T on _y defined

_by

If m is not

_{preserved by}

T neither the existence of the rate of

_entropy

_production,

nor, a

fortiori,

that of the tree’s silhouette can be

_proved.

4. a-distances and

_a-overlaps.

In the definition

_(3.2)

of the ultrametric distance

_{Dm (E, F)}

we have used the von

Neumann-Shannon

_entropy

of the

_partition

_{y (n)}

to which E and F

_belong

relative to the measure m,

namely :

As is well known

_{H( y}

_(n),

_{m )}

is the limit for au 1 of the information measures of

_degree

a defined

_by

(see

for

_example

_{[4] ;}

the case a = 2 was first considered in

[10]).

For any

positive

real a

Ha

is a

_positive

function whose value increases with the refinement of the

partition,

and which

takes values in the interval

_[0,

n

_log

k ].

It can be verified that for a E

_{[0,1 ],}

_{H" is}

a concave

function on the space of

_probability

measures m.

_Therefore,

_by

the same

_argument

used in the

proof

of

_Proposition

1 we can _prove.

Proposition

2. Let y (n) = _{y v}

T- 1 V ...

v T - (n - 1) y be a

finite

_partition

_{o f}

the

_dynamical

system

(X, F ,

m,

T ).

The distance

D (a )

between the elements

o f

y (n) with nonzero

_probability

defined by

is ultrametric

_for

_every choice

_of

a in the interval

_{[0, 1].}

We shall call

_Dm(a)

a a-distance.

To each ultrametric distance we can associate a

_{a-overlap by}

the relation

Clearly

Q(a)(E,

E )

= 1. Moreover from the

ultrametricity

of

_{D (a)}

it follows

Since the maximum a-distance from two histories of

_{y (n)}

is

_{Ha (1’ (n),}

_{m )}

it follows that the minimum value of the

_a-overlap

on the

_partition

_{y (n)}

is

In terms of

_{Yn" a,}

the a-silhouette

_slope

at time t = _n, defined

(see (3.14))

_as the difference of

the

_a-entropies

of

_{l’ (n + 1)}

and

l’ (n),

reads

The

limit,

if it

_exists,

_{of s,,,,}

_{(n )}

for n - oo, is the a-silhouette of the tree

_(the

usual

_proof

of the convergence of

sl (n )

fails when a # 1 because

a-entropies

are not in

_general

_{subadditive).}

If

m is a measure for which T is a Bernoulli shift we have

In this case the a-silhouette is

The

_quantity

_Y,,(")

is defined

_by

_(4.6)

_{for any}

_positive

value of a, also for « > 1 when no

entropic

distance is defined. As

Ha

increases with refinement

_(and

therefore with

_time)

yn(a)

decreases

with n,

i.e.

In the case a = 2

represents

the

_probability

that two histories at time t > n have a common

_{n-prehistory.}

5. Disordered

systems.

The ultrametric structure of the

_partition

_{y (n)}

_depends

_{upon the existence of} a _{sequence of}

partitions {y (i)}

with i =

0,

..., n, ... such that

y()

is more refined than

y (i - 1).

In the

_theory

of disordered

_systems

_{there also appears} an ultrametric structure

_[11]

which however has its

_origin

in the metric

_properties

of the

_phase

_{space and which} can

_only

be

defined in a

_{probabilistic}

sense.

Consider a

_probability

_space

_{(X, 57, m )}

where the

_phase

_{space X is assumed} to be

_equipped

with a metric

_{g (x, y),}

x, y E

_X,

and where m is an

_equilibrium

measure, that is a convex

combination of

_ergodic

measures

_[12]

_{pi :}

Let

E’

be the non

_empty

subset invariant for the

_ergodic

state _{p i , i. e. such that} pi

(Ei)

= 1. Then for any

ergodic

state

_{p j}

we have

because for i

_{# j}

Therefore

If Z is the

_subspace

of X with zero measure for all the

_equilibrium

states _m, the set q =

f E‘ } i E I is

a

_partition

of the space

XBZ

of

_equilibrium

_{configurations.}

The

_partition

is non trivial

_only

when there is more than one

_ergodic

state, that is when the

_system

_undergoes

a

_phase

transition. In the mean field solution of

_{spin glasses}

there is an

infinity

of

_ergodic

states not related to each other

_by

a

_symmetry

_(this

is a _{consequence of}

what is called

_frustration

_[13]).

The

_entropy

of such a

_partition

has been considered

_{previously by}

Palmer

_[14].

From the

metric g

(x, y)

in X we can define the Hausdorff distance between the elements of

Q 1

Let (do > dl > ... > dn

>... >

0 }

be a _{sequence of different}

_decreasing

distances between

the elements

of Ti

and let us consider a _{sequence of}

partitions

{ {X} , 0 }

= 77o - 7?1 - - - . , 71n ":-- - - - « ’n

ordered

_according

to

_refinement,

whose elements are subsets

off

with

_decreasing

_diameters,

respectively

do, dl,

...,

dn,

...,

0,

i.e.

where

In reference

_[2]

the elements

_En

are referred to as clusters.

The

_{sequence {11 n}}

is not

_uniquely

determined

_by

_{the sequence}

_{t dn }}

as several subsets

En

with the same diameter can in

_general

exist. Moreover it should be remarked that the

distance

_{d (Ei , Ej)}

between two elements

E’

E

_En

and

Ej E

_En

_{may well be smaller than}

dn.

For

_example,

consider four

_points

at the corners of a _{square. If d is}

equal

to the side of the square there are two _{ways of}

_partitioning

the set of the four corners into sets of diameter

d,

Fig.

2. - The two

ways in which the corners of a _square can be united to form subsets with a diameter

equal

to _{the side of the square.}

together

the left and the

_{right pairs. Clearly}

in both cases there are

_pairs

of

_{points belonging}

to

different subsets which have a distance d

_(see

_Fig.

_2).

The situation is much

_simpler

when ruz is ultrametric for the distances

dij.

In this case it is

easy to

_verify

that the

_{sequence {17n}}

is determined

_uniquely.

Furthermore

_{ultrametricity}

implies

that the distance between two elements

H

E

_En

and

Ei

E

_En

is

_always

_greater

than

dn

and does not

_{depend upon i, j.}

In this case we have

Thus

_{if q}

is ultrametric

_{each Tl n}

is also ultrametric.

Remark

_that,

if

_{En + 1 c En}

and

_{E’ , 1 r-- E’,}

we

_get

for the

_entropic

distance that we

introduced in section 3

where P is the common ancestor of

_{En +}

1 and

Es n

that is we find the silhouette of the

subtree which

_originates

from P.

Instead of

_ordering

the

_{partitions {’Tln}}

_according

to

_decreasing

diameters it is

_customary

in the

_theory

of disordered

_systems

to order them

_according

to

_increasing

maximum

_overlaps

of the subsets within their elements :

_{0 = qo -- q, « - - - ...: -- qn - - « ...}

_{qMax. The} exact relation between distances and

_overlaps

is not

_important.

It is

_enough

to assume that the

_overlap

qij

= q

(Ei , Ej)

between two elements

Ei

and

Ei

ouf 17

is a

_decreasing

function of their distance. The distance and the

_overlap

considered in this section are of a _pure

_geometrical

nature in

contrast with that discussed in section 3

_(and

_generalized

in Sect. 4 for a e

_[0,1))

which

depends

upon the

probability

measure. To establish the relation between these two distances

we shall

_introduce,

_following

_Parisi,

the

_probability

distribution of the

_overlaps

_qij

in the

partition q :

P

_{(q )}

is a measure of the

_complexity

of the

_{partition q}

which Parisi has called order

_parameter

for the

_glassy

transition

_[15].

A measure of the

_departure

from

_{ultrametricity}

of the

_partition

~ n is

given by

the

quantity

Indeed,

we can _{prove the}

_{following :}

Proposition

3. àn

is non

_negative

and it vanishes

if

~ is ultrametric.

Conversely, if

làn

vanishes,

for

all choices

_{o f qn,}

then the elements

_{o f}

~ with non

_vanishing

_{probability form}

an

ultrametric set.

Proof :

where the first term arises from the case r = s. Therefore

If q is ultrametric the sum vanishes because elements

E‘ ,

Ei

with

_overlaps

_greater

than qn are contained in the same element of _17n. This concludes the first

_part

of the

_proposition.

To prove the second statement _{suppose that the elements}

_{ouf 17}

with non

_vanishing

probability

are not an ultrametric set. Then there exists at least three elements

_Ei,

Ej,

Ek

with non

_{vanishing probability}

for which

_{qij >}

min

(qik,

_qkj)

=

qjk.

Choose qn

=

qij-Then the three elements

_{E’, Ei,}

Ek

cannot be contained in the same element of ~n, because this would have a diameter

_larger

than

_dn.

But if

Ei

is not contained in the same

element _{of 17n that contains}

E’

or

Ek

there is a non

_negative

contribution to the r.h.s. of

_(5.15).

Q.E.D.

In the

_theory

of

_{spin glasses ultrametricity}

occurs

_only

after the average over disorder.

Indeed,

disorder is

_together

with frustration a _necessary

_ingredient

of the

theory

of

spin

glasses,

which is taken into account

_by

a

_probability

distribution of

couplings

between

spins.

In our

_description

disorder is instead

_{represented by}

a

probability

distribution for the

coefficients

_ai’s.

_Usually

the average over the

_equilibrium

measures is denoted

_by

a bar

(3)

and one can consider the

_averaged

versions of the

previous

relations.

The statement of

_{ultrametricity}

in the mean field

_theory

of

_spin

_glasses

is

_{expressed by}

the

weaker relation

which means that the

_probability

of

finding

three elements

of TI n

with non zero measure that

do not

_satisfy

the

_{ultrametricity}

_inequality

is zero.

Specifically,

let

_{f n (W )}

be the

_probability

that an event of the

_partition

(n) has

_probability

W,

that is

Wfn(W)

is the

_probability

for the

_weight

W and we

_get

as a _{consequence of}

(5.16)

Similarly

the average of the von Neumann-Shannon

entropy

is

_{given by}

For

_example,

a Poisson distribution of

_{exponential weights}

_[16],

which is

_typical

of the random energy model

[17],

will

_give

rise to

with xn

a

_parameter

_{characterizing}

the

_partition.

Such a distribution also

_applies

to

_{spin glasses}

[11, 18]

and can be

_interpreted

as a distribution of random free

_energies.

It follows that

where as a _{consequence of}

_{equation (4.11)}

0 _{Xn -1 xn _ 1. The average value of the} von

Neumann-Shannon

_entropy

is instead.

where

is

_increasing

in the interval

_{(0, 1)}

and

_diverges

at minus

_infinity

when z

_approaches

zero. All the

_entropy

distances we have introduced are well defined

_{when q}

_qMaX or

_equivalently

xn «-- 1.

Equations

(5.18), (5.21)

and

_(5.22)

_provide

the connection between the order

_parameter

and the average value of the

complexity

of the

_system

at _{the scale qn.}

Acknowledgment.

One of us

_(L.

A.

_R.)

wishes to thank the

_hospitality

at the Institute des Hautes

Études

Scientifiques,

Bures-sur-Yvette,

where

_part

of this work has been done.

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