**HAL Id: jpa-00232188**

**https://hal.archives-ouvertes.fr/jpa-00232188**

### Submitted on 1 Jan 1983

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**Complete devil’s staircase in the one-dimensional lattice**

**gas**

### S. Aubry

**To cite this version:**

### S. Aubry. Complete devil’s staircase in the one-dimensional lattice gas. Journal de Physique Lettres,

### Edp sciences, 1983, 44 (7), pp.247-250. �10.1051/jphyslet:01983004407024700�. �jpa-00232188�

### Complete

### devil’s staircase

### in the

### one-dimensional lattice

_{gas}

### Comment

on «### one-dimensional

### lattice

_{gas }

### and

### the

### universality

### of

### the devil’s staircase

»### bv S. E.

### Burkov

S.

_{Aubry}

Laboratoire Léon Brillouin, CEN

_{Saclay, }

91191 Gif sur Yvette Cedex, France
(Re~u le

_{2 fevrier }

1983, ### accepte

le_{15 fevrier }

1983)
Résumé. 2014 Nous calculons

### explicitement

la forme exacte de l’escalier du diable_{complet }

du modèle
de gaz sur réseau. Nous montrons _{ainsi que le }

### comportement

de la transition de_{phase }

commensurable-incommensurable

_{dépend }

essentiellement des _{interactions }

à _{longue }

distance. Nous mettons en
### question

les### propriétés

d’universalité_{suggérées }

_{par Burkov.}

Abstract. 2014 We calculate

_{explicitly }

the exact form of the _{complete }

devil’s staircase for _{the lattice gas}

model. Thus, we show that the behaviour at the commensurate-incommensurate

_{phase }

transition
essentially

### depends

on the_{long }distance interactions. We

_{question }

the assertion of Burkov about the
### universality properties

of the devil’s staircase. ClassificationPhysics Abstracts

05.20 - 68.20 - 64.90

The devil’s

_{staircase, }

calculated _{by }

Burkov in his _{interesting }

_{paper }

_{[1], }

can also be _{explicitly}

calculated

_{by }

the same method that we used in the first exact calculation of a _{complete }

devil’s
staircase in 1978

### [2].

In### addition,

our method has the_{advantage }

of _{having }

a _{rigorous }

mathe-matical foundation

_{[3]. }

Our initial model was a variation on the discrete Frenkel-Kontorova
model where the sine

_{periodic }

_{potential }

was _{replaced by }

a ### piece-wise

### parabolic periodic

### potential.

It was transformed into an

### integer

model which had a similar form to_{the lattice gas model}considered

_{by }

Burkov. The details of these calculations in reference _{[2] }

were _{given }

in ### appendix

A3of reference

_{[4b]. }

We also noted that the model studied _{by }

Bak and Bruinsma _{[5] }

could also be
transformed into such a lattice gas model _{[6]. }

In addition _{by usihg }

a result ### given

in reference_{[3c],}

which was

_{essentially }

devoted to _{obtaining rigorous }

results _{applicable }

to _{quite }

a more _{general}

class of

### models,

it was_{pointed }

out that our _{proofs }

and exact ### results

could be extended_{unchanged}

to models with _{any }kind of

_{long }

_{range }interactions

_{given }

some _{necessary }

### convexity

conditions.The purpose of this comment is not to

_{reproduce }

_{the substance of papers }

_{already published }

or
about to _{appear }

_{but,}

### 1)

to show how our### method,

which is_{simpler, }

can be _{applied }

to the model studied _{by }

Burkov.
### Thus,

we find### again

the### equation

of the devil’s staircase### given by

### Burkov,

### 2)

to_{question }

the _{universality properties }

of the devil’s staircase _{suggested by }

Burkov.
L-248 JOURNAL DE PHYSIQUE - LETTRES

The one-dimensional lattice _{gas }model

_{[1] }

with Hamiltonian
is included in the class of models

### (34)

of reference_{[6] }

where for each _{Um(x) }

there exists a _{positive}

constant

_{Cm }

such that for all x, ### Um(x)

>_{em }

> 0. _{(Many }

_{rigorous }

results were _{given }

for this
class of models in reference _{[3].) (Note }

that if this condition is not _{satisfied, }

the devil’s staircase
### of (1)

may not_{exist.) }

When the atomic mean _{distance, }

_{I, }

which is the inverse of the atomic
### concen-tration,

c, i.e.is

### fixed,

the_{ground-state }

of model _{(1) }

is _{given by }

_{(see }

ref. _{[6] }

or ### [3a,

### 3b,

### 3c])

where a is an

_{arbitrary }

_{phase. }

The energy per atom with ### condition

### (2)

isand can be written

with

The method for

_{calculating }

_{t/lm(l) }

is the same as that for ### calculating

### (A3-11)

in reference### [4b]

or_{( 13-b) }

in reference _{[6]. By }

_{inspection }

of the results in reference _{[4b] }

or _{[6], }

we ### readily

findFor the real model

_{(with 1 }

not _{fixed) }

the concentration c = ### 1/1

is determined### by

### minimizing

the«

_{grand potential » (1). }

Thus unlike our models in references ### [2, 4

or_{6], }

the _{quantity }

which has to
be minimized is not _{the energy per }atom but the

_{grand potential }

_{per unit }

_{length }

_{(since }

the total
number of

_{particles }

is not _{conserved). }

Instead of _{having }

a devil’s staircase ### ~(~)

where 1 is the atomic mean distance_{and ~ }

is a tensile force _{(that }

is the _{opposite }

of a _{pressure) }

the devil’s staircase of
model _{(1) }

describes the concentration c = ### 1/1

of### particles

_{as a }function of the chemical

### potential

~.

The minimization of the

_{grand }

_{potential (1) }

per unit ### length,

which is from### equation (4)

This is the

_{implicit equation }

of the devil’s staircase _{c(P) }

of model _{(1) }

for which we are _{looking.}

### Using (7)

this_{equation }

becomes
which is identical to

_{equation (5) }

as ### given by

Burkov in reference### [1].

It is clear that the_{behaviour}

### ofc(~)

at the commensurate-incommensurate transitions reflects the_{properties }

of the interaction
### U.(x)

for_{large }

x and m. Since for sake of _{brevity }

we cannot _{give }

the details of the calculations ### here,

we_{suggest }

### tha~

the interested reader should convince himself_{by checking }

for _{example }

the cases
in which for all m :

and

in the

_{vicinity }

of the _{registered commensurability }

c = 1. In the first _{case, }

### (11~),

the well known### logarithmic

behaviouris

_{found, }

while in the second case
### Apparently

this behaviour is not universal and_{only }

_{depends }

on the _{properties }

of _{!7~(x) }

for
### large x

and_{large }

m. The _{universality }

_{properties }

claimed _{by }

Burkov are not _{clearly }

connected to
any ### physical

### observation

and appear### only

as a_{generic }

### property

of a mathematical nature for the irrational### numlbers.

In our

_{opinion, }

the hierarchical construction of the devil’s staircase _{proposed }

_{by }

Burkov is an
artefact of his method of calculation.

### Indeed,

we have shown above that there exists a### straight

forward calculation of the devil’s staircase

_{of (1) }

which does not _{involve any }details of continued fraction

_{expansions }

as in reference _{[1]. }

_{Moreover, }

the _{generation }

of the _{steps }

of the devil’s staircase
as _{proposed by }

Burkov does not _{appear }reasonable

### since,

for_{example, }

there exist _{steps }

_{generated}

at order two which are

_{infinitely }

small while at order 3 some of them can be _{large. }

It is necessary
to consider the metric

_{(that }

is the width of the _{steps) }

of the devil’s staircase for a ### good description

of the

_{generation }

of the _{steps, }

while these _{purely }

_{topological }

consideration _{although }

not wrong
### mathematically

are_{physically misleading. }

We _{proposed }

in reference _{[7] }

another _{hierarchy}

which _{corresponds }

_{to }

the _{generation }

of _{steps }

which _{approximately }

are in _{decreasing }

order of
width. This

_{corresponds }

to the well known construction of the set of rational numbers. We start
. 0 1 0 1 1

from the two

## rational y

### and

At first order we_{generate }

the ## sequence T

and## 1.

At second order0 1 1 ~ 1 . 0 l~l 2 1 3 3 1 .

we

### generate T’ ’3 ’ ’2 ’ "3 ’ T ’ at

thud order### T’4’~5’2’5’4’T

and so on. The next sequence is obtained from the_{previous }

one _{by inserting }

between two consecutive rationals of the _{sequence}

### ~~+1/~+1 ~

new_{rational r" }

+ _{r. - , Is. }

+ _{~Sn 1 ~}

Let us also note that Burkov considers it obvious that the Cantor set called

_{M1 }

in _{his paper is}

self-similar. This is a metric

_{property }

from which he concludes that _{M1 }

has measure zero which
L-250 JOURNAL DE _{PHYSIQUE - }LETTRES

could be

_{incomplete }

_{[2] }

and then this set _{M, }

has finite measure. Of course in this case the Cantor
set

_{M, }

cannot be self-similar. It is not self-similar even when _{M, }

has zero measure, at least in the
case when the

_{long }

_{range interactions decrease fast }

_{enough. }

In a _{forthcoming }

_{paper }we will

### present

the_{proof }

for the existence of an _{incomplete }

devil’s staircase in the Frenkel-Kontorova
model for a small

_{periodic potential [8] }

which confirms our _{early predictions [2]. }

Thus a _{rigorous}

### proof

for the_{completeness }

of the devil’s staircase in the case of the lattice _{gas }model is needed. This

_{proof, }

which is in fact _{very }

_{simple, }

can be found in reference _{[6].}

### Finally, although

we do not believe the_{physical interpretation }

of the hierarchical construction
in the form

_{proposed }

_{by }

Burkov _{by using }

the continued fraction _{expansion }

of irrational ### numbers,

this method of

_{expansion }

turns out to be _{very }useful for

_{understanding }

the « transition _{by breaking}

of _{analyticity }

» _{[2] }

which is _{closely }

related to the _{stochasticity }

threshold _{[9] }

but such a transition
does not exist in model ### (1).

The_{physical meaning }

of this _{expansion }

_{becomes very }

_{transparent }

_{by}

### considering

the renormalization group### approaches

used to_{investigate }

this kind of transition _{[10].}

### (I apologize

to Prof. Burkov and to_{the reader that my }

_{early }

works on the devil’s staircase are
not

_{easily }

available in the literature because ### they

are### incompletely published

in### journal

whichhave

_{only }

a limited _{readership. }

This was due _{largely }

to serious difficulties with their _{publication.)}

References

[1] BURKOV, S. E., One-dimensional lattice gas and the universality of the devil’s staircase, J. _{Physique }Lett.
44 _{(1983) }L-179.

### [2]

AUBRY, S., Soliton in Condensed Matter, Solid State Sci. 8 (1978) 264.### [3]

a) AUBRY, S., On modulated_{crystallographic }

structures. Exact results on the classical _{ground-state}

### of a one-dimensional model,

### preprint unpublished (rejected for

### publication)

### (1978).

b) AUBRY, S. and ANDRÉ, G., Ann. Israel. _{Phys. }Soc. 3 (1980) 133.

c) AUBRY, S. and LE _{DAERON, }P. _{Y., }

_{preprint }

_{(1982). }The discrete Frenkel-Kontorova model and its extension. I. Exact results for the

_{ground-state, }

submitted to _{Physica }

D.
d) AUBRY, S., The twist map, the extended Frenkel-Kontorova model and the devil’s _{staircase, }to

appear in

### Physica

D_{(1983).}

[4] a)

### AUBRY, S., Ferroelectrics 24 (1980) 53.

b) AUBRY, S., The devil’s staircase _{transformation }in commensurate lattices, in _{Proceeding }of IHES and

Columbia 1979-1980 _{(Eds. }D. and G. _{Chudnovsky), }Lectures Notes in Mathematics _{925, 221.}

### [5]

BAK, P. and BRUINSMA, R., Phys. Rev. Lett. 49_{(1982) }249.

### [6]

AUBRY, S., Exact models with a_{complete }

devil’s staircase, to _{appear in J. }Phvs. C

### (1983).

### [7]

AXEL, F. and AUBRY, S., J._{Phys. }

C 14 _{(1981) }

5433 (third ### section).

### [8]

DE SEZE, L. and AUBRY, S., in_{preparation.}

### [9] GREENE,

J., J. Math._{Phys. }

20 _{(1978) }

1183.
### [10]

PEYRARD, M. and_{AUBRY, S., }Critical behavior at the transition