Finite-temperature coarse-graining of one-dimensional models: mathematical analysis and computational approaches

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models: mathematical analysis and computational

approaches

Xavier Blanc, Claude Le Bris, Frédéric Legoll, Carsten Patz

To cite this version:

Xavier Blanc, Claude Le Bris, Frédéric Legoll, Carsten Patz. Finite-temperature coarse-graining of

one-dimensional models: mathematical analysis and computational approaches. Journal of Nonlinear

Science, Springer Verlag, 2010, 20 (2), pp.241-275. �10.1007/s00332-009-9057-y�. �inria-00282107v2�

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a p p o r t

d e r e c h e r c h e

Thème NUM

Finite-temperature coarse-graining of

one-dimensional models: mathematical analysis and

computational approaches

Xavier Blanc — Claude Le Bris — Frédéric Legoll — Carsten Patz

N° 6544

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(4)

approa hes XavierBlan

∗†

, Claude LeBris

‡†

, Frédéri Legoll

§†

, Carsten Patz

¶

ThèmeNUMSystèmesnumériques

ProjetMICMAC

Rapportdere her he n°6544May200840pages

Abstra t: Wepresentapossibleapproa hforthe omputationoffreeenergiesand

ensem-ble averages ofone-dimensional oarse-grainedmodels in materialss ien e. The approa h

isbaseduponathermodynami limitpro ess,andmakesuseofergodi theoremsandlarge

deviationstheory. Inadditiontoprovidingapossiblee ient omputationalstrategyfor

en-sembleaverages,theapproa hallowsforassessingthea ura yofapproximations ommonly

usedinpra ti e.

Key-words: ensemble averages, free energies, oarse-grainedmodels, materials s ien e,

thermodynami limit, omputationalstrategy

∗

Laboratoire J.-L.Lions, UniversitéPierre etMarie Curie,Boîte ourrier187,75252 ParisCedex 05,

Fran e. Conta t: blan ann.jussieu.fr

†

INRIA Ro quen ourt, MICMAC team-proje t, Domaine de Volu eau, B.P.105, 78153 LeChesnay

Cedex,Fran e.

‡

CERMICS,E oleNationaledesPontsetChaussées,6et8avenueBlaisePas al,CitéDes artes,77455 Marne-la-ValléeCedex2,Fran e. Conta t: lebris ermi s.enp .fr

§

LAMI, E oleNationale desPonts et Chaussées, 6 et 8 avenueBlaise Pas al, Cité Des artes, 77455 Marne-la-ValléeCedex2,Fran e. Conta t: legolllami.enp .fr

¶

Weierstrass-Institut fürAngewandte AnalysisundSto hastik, Mohrenstrasse 39, 10117 Berlin, Ger-many.Conta t: patzwias-berlin.de

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mathématique et appro hes numériques

Résumé: Nousprésentonsune appro hepossiblepour le al ulde moyennesd'ensemble

et d'énergies libres de modèles réduits en s ien e des matériaux, dans un adre

mono-dimensionnel. L'appro he s'appuie sur une limite thermodynamique, et utilise des

théo-rèmesergodiqueset lathéorie desgrandes déviations. Nous obtenons ainsinon seulement

unestratégienumériquee a epourle al uldemoyennesthermodynamiques,maisaussi

unmoyendevériernumériquementlavaliditédeshypothèsesquisont ourammentfaites

danslalittératurepourdévelopperdetellesappro hes.

Mots- lés: moyennesd'ensemble,énergieslibres,modèlesréduits,s ien edesmatériaux,

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1 Introdu tion

Computing anoni alaveragesis astandardtaskof omputationalmaterialss ien e.

Con-sider an atomisti , supposedly large, system onsisting of

N

parti les, at positions

u =

u

1 _{, . . . , u}

N

_{∈ R}

3N

. Providethissystemwithanenergy

E

µ

(u) = E

µ

u

1 , . . . , u

N

.

(1) Aprototypi alexampleofsu hanenergyisthepairintera tionenergy

E

µ

(u) =

1

2 X

i6=j

W u

j

− u

i

.

(2)

Thenitetemperaturethermodynami alpropertiesofthematerialareobtainedfrom

anon-i alensembleaverages,

hAi =

Z

Ω

N

A(u) exp(−βE

µ

(u)) du

Z

Ω

N

exp(−βE

µ

(u)) du

,

(3) where

Ω ⊂ R

3

is thema ros opi domainwhere thepositions

u

i

vary,

A

is theobservable of interest, and

β = 1/(k

B

T )

is the inverse temperature [15℄. The denominator of (3) is denoted by

Z

and alled thepartition fun tion. Themajor omputationaldi ultyin (3) isof oursethe

N

-foldintegrals,where

N

, thenumberofparti les, isextremelylarge. For integralsof thetype (3) to be quantitatively meaningful in pra ti e,

N

does not need to approa htheAvogadronumber,but stillneedstobeextremelylarge(

10

5

, say).

Thethreedominant omputationalapproa hesfortheevaluationof(3)areMonteCarlo

methods, Markov hains methods, and Mole ulardynami s methods respe tively(see e.g.

[13℄forareviewandamathemati aland omputational omparison). Inthepresentarti le,

weusethelattertypeofmethods, andmorepre iselytheoverdampedLangevindynami s

(also alled biasedrandowwalk). The ensemble average(3) is al ulatedasthe long-time

average

hAi = lim

T→+∞

1 T

Z

T

0 A(u(t)) dt

(4)

alongthetraje torygeneratedbythesto hasti dierentialequation

du = −∇

u

E

µ

(u) dt +

p

2/β dB

t

.

(5)

It isoften the asethat theobservable

A

a tuallydoesnot depend onthepositions

u

i

ofalltheatoms,but onlyonsomeofthem. Think forinstan eofnanoindentation: weare

espe iallyinterestedin thepositions oftheatomsbelowtheindenter, in thefor es applied

ontheseatoms,...Ourrstaim istodesignanumeri almethod thate iently omputes

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TheQuasiContinuum Method (QCM)isa ommonlyusedexample ofapproa hesthat

allow for the al ulation of ensemble averages. In its original version, the method was

fo usedon thezerotemperaturesetting. It wasoriginallyintrodu edin [44, 45℄, andthen

further developed in [27, 34, 35, 41, 42, 43℄. It has been studied mathemati ally in e.g.

[1, 2,3,7, 8,9,18,20,21,22, 30, 31,37℄. See[10℄forare entreview. Anextensionofthe

originalideahasre entlybeendevelopedin[19℄and arriesthroughtothenite-temperature

ase, onsideredin thepresentarti le. Seealso[14,29℄forpriorstudiesdevelopingideasin

thesamevein.

Let us briey detail the bottom line of oarse-grainingstrategies for the omputation

of anoni alaverages. For simpli ityof exposition,we lettheatomsvary in

Ω = R

3

. The

ideaistosubdivide theparti lesofthesystemintotwosubsets. Therstsubset onsistsof

theso- alledrepresentativeatoms (abbreviatedintheQCMterminologyasrepatoms,with

positionshen eforthdenotedby

u

r

). These ondsubsetisthatofatomsthatareeliminated in the oarse-grainedpro edure. Theirpositions aredenoted hereby

u

c

. Weassume that theobservable onsideredonlydependsonthepositions

u

r

oftherepatoms,notonthoseof theotheratoms,

u

c

. Morepre isely,onewrites

u = (u

1 _{, . . . , u}

N

_{) = (u}

r

, u

c

),

u

r

∈ R

3N

r

,

u

c

∈ R

3N

c

,

N = N

r

+ N

c

.

Ouraimisto ompute(3)forsu hobservable,thatis,

hAi = Z

−1

Z

R

3N

A(u

r

) exp(−βE

µ

(u)) du.

(6)

Weobservethat,owingtoourassumptionon

A

,

Z

R

3N

A(u

r

) exp(−βE

µ

(u)) du =

Z

R

3Nr

du

r

A(u

r

)

Z

R

3Nc

exp(−βE

µ

(u

r

, u

c

)) du

c

,

andlikewise

Z =

Z

R

3N

exp(−βE

µ

(u)) du =

Z

R

3Nr

du

r

Z

R

3Nc

exp(−βE

µ

(u

r

, u

c

)) du

c

.

Introdu ingthe oarse-grainedpotential(also alledfreeenergy)

E

CG

(u

r

) := −

1 β

ln

Z

R

3Nc

exp(−βE

µ

(u

r

, u

c

)) du

c

,

(7)

theexpression(6)rewrites

hAi = Z

−1

r

Z

R

3 Nr

A(u

r

) exp(−βE

CG

(u

r

)) du

r

,

(8) whereof ourse

Z

r

=

Z

R

3 Nr

exp(−βE

CG

(u

r

)) du

r

.

(9)

(8)

Under appropriate onditionsensuringergodi ityof thesystem,the integral(8)is in turn omputedfrom

hAi = lim

T

→+∞

1 T

Z

T

0 A(u

r

(t)) dt

with

du

r

= −∇

u

r

E

CG

(u

r

) dt +

p

2/β dB

t

.

(10)

Simulatingthedynami sin (10)isaless omputationallydemanding taskthansimulating

(5), owingto the redu eddimension

N

r

. Thissimpli ation omes at apri e: al ulating the oarse-grainedfreeenergy(7).

Remark1 The present work on entrates on the omputation of ensemble averages

us-ing oarse-grainedmodels, and free energies. Pra ti e shows that the same oarse-graining

paradigmisusedtosimulate a tual oarse-graineddynami s atnite temperature. Wewill

notgo inthis dire tion asthe physi al relevan eof thelatterapproa h isun lear tous.

Inordertoapproximatethefreeenergy(7),state-of-the-artnitetemperaturemethods

perform a Taylor expansion of the position of the eliminated atoms

u

c

. In this Taylor expansion, a linear interpolation and a harmoni approximation of the positions of the

atomsaresu essivelyperformed. Morepre isely,giventhepositions

u

r

oftherepatoms,the averagepositions

u

c

(u

r

)

oftheeliminatedatomsarerstdeterminedbylinearinterpolation betweentwo(ormore)adja entrepatoms. Thenitispostulatedthat

u

c

= u

c

(u

r

) + ξ

c

wheretheperturbation

ξ

c

issmall. Theenergyisthen al ulatedfrom aTaylorexpansion trun atedatse ondorder:

E

µ

(u

r

, u

c

) =

E

µ

(u

r

, u

c

(u

r

) + ξ

c

)

≈ E

µ

(u

r

, u

c

(u

r

)) +

∂E

µ

∂u

c

(u

r

, u

c

(u

r

)) · ξ

c

+

1

2 ξ

c

· ∂

2 _E

µ

∂u

2 c

· ξ

c

=: e

E(u

r

, u

c

).

It follows (we skip the details of the argument and refer to the bibliography pointed out

aboveforfurtherdetails) that

E

CG

(u

r

)

isapproximatedby

E

QCM

(u

r

) = −

1 β

ln

Z

R

3 Nc

exp(−β e

E(u

r

, u

c

)) du

c

,

(11)

whi hisanalyti ally omputable. Withoutsu hsimplifyingassumptions,thea tual

ompu-tationof

E

CG

forpra ti alvaluesof

N

r

and

N

c

seemsundoable. Theapproa hhasproven e ient. Reportedly,itsatisfa torilytreatsthree-dimensionalproblems oflargesize.

How-ever, from the mathemati al standpoint, it is an open question to evaluatethe impa t of

theabove oupleofapproximations(linearinterpolationofthe average positions, followed

byharmoni expansion). Thepurposeofthepresentarti leistopresentanapproa hthat,

(9)

Our approa h is based ona thermodynami limit. It wasrst outlined in [38℄ for the

spe ial aseofharmoni intera tions. Theapproa hisexa tinthelimitofaninnite

num-berof eliminatedatoms, andtherefore validwhenthis number

N

c

is largeas ompared to thenumber

N

r

of representativeatoms thatare kept expli itin the oarse-grainedmodel. Thisregime, afterall, is theregime that allee tive oarse-grainingstrategiesshould

tar-get, although,in pra ti e,

N

c

isseeminglymu h smallerthanideally, andeven sometimes ofthe sameorderof magnitudeas

N

r

. Inshort,the onsiderationof theasymptoti limit

N

c

→ +∞

makestra tablea omputationwhi his nottra tablefornite

N

c

(unless sim-pli ations, asthosementioned above,are performed). Wedo not laim for originality in

ourtheoreti al onsiderations on thethermodynami limit of the freeenergy of atomisti

systems. We providethem herefor onsisten y. However,ourspe i useofsu h

theoret-i al onsiderationsas a omputationalstrategyfor approximating oarse-grainedensemble

averagesin omputationalmaterials s ien eseems,to thebest ofourknowledge,new. We

werenotabletondany omparableendeavourintheexistingliteraturewehavea essto.

Let us on ludethis introdu tionby brieydes ribing ourapproa h. Assume for

sim-pli itythatthereisonlyonerepatom:

N

r

= 1

. Ourideato ompute

hAi

in(6)isto hange variables,thatisintrodu e

y = (y

1 , . . . , y

N

) = Φ(u)

,andre ast(6) as

hAi =

Z

R

3N

A

1 N

N

X

i=1

y

i

!

ν(y) dy

for some probability density

ν(y)

(see equation (17) below for an expli it example). We next re ognize

hAi

as the expe tation value

E

"

A

1 N

N

X

i=1

Y

i

!#

,

where

Y = (Y

1 , . . . , Y

N

)

arerandomvariablesdistributed a ordingtotheprobability

ν

. A LawofLargeNumbers providesthelimitof

hAi

when

N → +∞

(whi h orrespondsto

N

c

→ +∞

, sin e

N

r

= 1

). Therateof onvergen emayalsobeevaluatedusingtheCentralLimitTheorem.

The above approa h bypasses the al ulation of the free energy

E

CG

to ompute the ensembleaverage(8). Butitisalsointerestingtotryandevaluate

E

CG

inthesameregime, in order to, again, both provide ane ientnumeri al approa h and assessthe validity of

ommonlyused simplifyingapproa hes. First,itisto beremarkedthat

E

CG

s aleslinearly withthenumber

N

c

ofeliminatedatoms. Therelevantquantityishen ethefreeenergyper parti le

F

∞

(u

r

) :=

lim

N

c

→+∞

1 N

c

E

CG

(u

r

).

(12) Thisenergyisrelatedtothe oarse-grained onstitutivelawofthematerialatnite

temper-ature(see[17,36℄forrelatedapproa hes). Evenif

F

∞

isagoodapproximationof

E

CG

/N

c

forlarge

N

c

,it anbeseenthat

N

c

F

∞

isnotne essarilyagoodapproximationof

E

CG

. It isnot leartoushowtousetheprobabilitymeasure

Z

−1

N

c

exp(−βN

c

F

∞

)

to omputeinan

e ientmanneranapproximationoftheaverage

hAi

(seeRemark 6below).

Wedevelopourapproa h in theone-dimensionalsetting, forsimple asesof pair

(10)

omputational strategy to approximate ensemble averages (see Se tion 2.1), and wenext

addressthe omputations of free energies (see Se tion 2.2). Numeri al onsiderationsare

olle tedin Se tion2.3.

We next turn to next-to-nearest neighbour intera tions, traditionally abbreviated as

NNN.For thismodel, wefo uson the omputation ofensembleaverages(see Se tion3.1).

AsexplainedinSe tion3.2,more ompli atedtypesofintera tionpotentialsand"essentially

one-dimensionalsystems"(in ludingpolymer hains)maybetreatedlikewise,althoughwe

donotpursuein thisdire tion.

Asimilarinterpretationofensembleaveragesastheonepresentedhere,usingaMarkov

hain formalism, shouldleadto an analogous strategyfor two-dimensional systems. Some

preliminarydevelopments,notin ludedinthepresentarti le,already onrmthis. However,

denite on lusionsareyettobeobtained,bothontheformalvalidityoftheapproa hand

onthebest possiblenumeri ale ien y a omplished. The fa tthat the two-dimensional

aseismu hmoredi ultthantheone-dimensional aseis orroboratedbytheliterature

onthissubje t: onlyverysimple ases,su hasspinsystems,orharmoni intera tions(with

zeroequilibriumlength)areknowntohaveexpli itsolutionsinthis ontext(seethereviews

[5, 39℄). Wetherefore prefer to postpone onsiderations onthe two-dimensional situation

untilafuturepubli ation[11℄.

2 The nearest neighbour (NN) ase

Asmentionedintheintrodu tion,ourapproa hisbasedontheasymptoti limit

N −→ +∞

. Wethereforerstres aletheproblemwiththeinteratomi distan e

h

,su hthat

N h = L = 1

. Theatomisti energy(2)in theres aled NN asewrites

E

µ

u

1 , . . . , u

N

=

N

X

i=1

W

u

i

_{− u}

i−1

h

.

(13) Wenowimpose

u

0 _{= 0}

to avoidtranslationinvarian e,and onsiderthat onlyatoms0and

N

are repatoms, while allthe other atoms

i = 1, . . . , N − 1

are eliminated in the oarse-grainingpro edure(seeFigure1). Ourargument anbestraightforwardlyadaptedtotreat

the aseof

N

r

≫ 2

repatoms(seeFigure2). SeetheendofSe tion2.1forthis.

L

0 N

Figure 1: We isolate a segment between two onse utive repatoms (in red). All atoms

(11)

0 _N

Figure 2: The repatoms (in red) are expli itly treated, all other atoms (in blue) being

eliminatedintheCGpro edure.

Inthissimplesituation,theaverage(6)reads

hAi

N

= Z

−1

Z

R

N

A u

N

_exp

_−β

N

X

i=1

W

u

i

_{− u}

i−1

h

!

du

1 . . . du

N

_,

(14)

where wehaveexpli itly mentionedthedependen e of

hAi

upon

N

using asubs ript. We introdu e

y

i

:=

u

i

_{− u}

i−1

h

, i = 1, . . . , N,

(15) andnextremark that

u

N

= h

N

X

i=1

y

i

=

1 N

N

X

i=1

y

i

.

(16)

Theaveragerewritesas

hAi

N

= Z

−1

Z

R

N

A

1 N

N

X

i=1

y

i

!

exp

−β

N

X

i=1

W (y

i

)

!

dy

1 . . . dy

N

_,

(17) wherenow

Z =

Z

R

N

exp

−β

N

X

i=1

W (y

i

)

!

dy

1 . . . dy

N

.

Remark2 In(14),weletthevariables

u

i

varyonthewholereal line. Wedonot onstrain

them to obey

u

i−1

_{≤ u}

i

, whi h en odes the fa t that nearest neighbours remain nearest

neighbours. The argumentprovided here and below arries through when this onstraintis

a ountedfor,basi allyrepla ing the intera tion potential

W

by

W

c

(y) =

W (y)

when

y ≥ 0

+∞

otherwise.

(18)

Likewise, we ould also imposethat all the

u

i

stayin agiven ma ros opi segment. If they

areorderedin reasingly,itsu estoimposethis onstrainton

u

0

and

u

N

. Thisisagaina

(12)

2.1 Limit of the average

Itisevidentontheexpression(17)that

hAi

N

= E

"

A

1 N

N

X

i=1

Y

i

!#

forindependentidenti allydistributed(i.i.d.) randomvariables

Y

i

,sharingthe ommonlaw

z

−1

_{exp(−βW (y))dy}

,with

z =

Z

R

exp(−βW (y))dy

. Asimple omputationthus gives Theorem1 Assumethat

A : R −→ R

is ontinuous, that for some

p ≥ 1,

there existsa onstant

C > 0

su hthat

∀y ∈ R,

|A(y)| ≤ C(1 + |y|

p

_),

(19)

andthat

Z

R

(1 + |y|

p

_{) exp (−βW (y)) dy < +∞.}

(20) Then,

lim

N

→+∞

hAi

N

= A(y

∗

_),

where

y

∗

_{:= z}

−1

Z

R

y exp(−βW (y)) dy,

(21) with

z =

Z

R

exp(−βW (y)) dy

. Inaddition, if

A

is

C

2

andif (19)-(20)hold with

p = 2

,then

hAi

N

= hAi

approx,1

N

+ o

₁

N

,

(22)

with

hAi

approx,1

N

:= A(y

∗

_{) +}

σ

2 2N

A

′′

_(y

∗

_),

where

σ

2 _{= z}

−1

Z

R

(y − y

∗

)

2 exp(−βW (y)) dy.

(23) Theproofof(21)isadire tappli ationoftheLawofLargeNumbers,andthatof(22)is

anappli ationoftheCentralLimitTheorem. Weskipthem. Thefollowing onsiderations,

formoreregularobservables

A

,indeed ontaintheingredientsforproving(21)-(22),simply bytrun atingtheexpansionatrstorder.

Note that if

A

is more regularthan stated in Theorem 1,then it is of ourse possible topro eedfurther intheexpansionof

hAi

N

in powersof

1/N

. Indeed,assumeforinstan e

(13)

that

A

is

C

6

,that

A

(6)

isgloballybounded andthat(19)-(20)holdwith

p = 6

. Then

A

1 N

N

X

i=1

Y

i

!

=

A

y

∗

₊

1 N

N

X

i=1

D

i

!

,

=

A(y

∗

) + A

′

(y

∗

)

1 N

N

X

i=1

D

i

+

1

2 A

′′

_(y

∗

₎

1 N

N

X

i=1

D

i

!

2 +

1

6 A

(3)

_(y

∗

₎

1 N

N

X

i=1

D

i

!

3 +

1

24 A

(4)

_(y

∗

₎

1 N

N

X

i=1

D

i

!

4 +

1 5!

A

(5)

_(y

∗

₎

1 N

N

X

i=1

D

i

!

5 +

1 6!

A

(6)

_(ξ)

1 N

N

X

i=1

D

i

!

6 ,

where

D

i

= Y

i

− y

∗

and

ξ

liesbetween

y

∗

and

(1/N )

P

Y

i

. Wenowtakethe expe tation valueofthis equality. Letusintrodu e

hAi

approx,2

N

:=

A(y

∗

_{) +}

1

2 A

′′

_(y

∗

₎

1 N

E

D

2

1 +

1

6 A

(3)

_(y

∗

₎

1 N

2 E

D

3

1 +

1

24 A

(4)

_(y

∗

₎

1 N

3 E

D

4

1 +

N − 1

N

3 E

(D

2

1 )

2 +

1 5!

A

(5)

_(y

∗

₎

1 N

4 E

D

5

1 +

N − 1

N

4 E

(D

2

1 )E(D

3

1 )

.

(24) Then

hAi

N

− hAi

approx,2

N

≤

_6!

1 kA

(6)

k

L

∞

E





1 N

N

X

i=1

D

i

!

6 

 .

(25) We now use the fa t that any i.i.d. variables

D

i

with mean value

0

satisfy the following bounds:

∀p ∈ N,

∃C

p

> 0,

E

"

1 N

N

X

i=1

D

i

!

p

#

≤











C

p

N

p

2

if

p

iseven;

C

p

N

p+1

2

if

p

isodd. (26)

This is proved by developping the power

p

of the sum, and then using the fa t that the variablesarei.i.dandhavemeanvaluezero. Wehen einferfrom(24),(25)and(26)that

hAi

N

= A(y

∗

) +

σ

2 2N

A

′′

_(y

∗

_{) +}

1 N

2 _a

3

6 A

(3)

_(y

∗

_{) +}

σ

4

24 A

(4)

_(y

∗

₎

+ O

₁

N

3 ,

(27) where

σ

isdened by(23)and

a

3 = z

−1

Z

R

(14)

More generally, it is possible to expand

hAi

N

at any order in

1/N

, provided that

A

is su ientlysmoothand

exp(−βW )

su ientlysmallatinnity. Inviewofthebounds(26), we anseethatusingaTaylorexpansionoforder

2p

around

y

∗

for

A

givesanexpansionof

hAi

N

oforder

p

.

The pra ti al onsequen eof Theorem 1is that, for omputational purposes, we may

taketheapproximation

hAi

N

≈ A

z

−1

Z

R

u

N

_{exp(−βW (u}

N

_{)) du}

N

.

(28)

Aspointedoutabove,itispossibletoimprovethisapproximationifne essarybyexpanding

furtherin powersof

1/N

.

We on ludethisse tionbyshowingthatour onsiderationofasingle"segment" arries

throughto the ase when there are 3repatoms, of respe tiveindex

0, M

1

and

M

1 + M

2

, with

M

1 h = L

1

,

M

2 h = L

2

,

N h = L = 1

(seeFigure3). Theaverageto omputewrites

hAi

N

= Z

−1

Z

R

N

A u

M

1 _{, u}

M

1 +M

2 _exp

−β

N

X

i=1

W

u

i

_{− u}

i−1

h

!

du

1 . . . du

N

.

Intheregime

h → 0

,

N, M

1 , M

2 → +∞

with

M

1 /N

and

M

2 /N

xed,wehave,usingsimilar arguments,

lim

N

→+∞

hAi

N

= A(L

1 y

∗

_{, L}

2 y

∗

).

Thegeneralizationto

N

r

> 3

repatoms,intheappropriateasymptoti regime,easilyfollows.

L1

L2

0 M1

M1+M2

N

Figure 3: When onsidering two onse utive segments -or more-, the argument may be

readilyadapted. Seethetext.

Remark3(The small temperature limit) It is interesting here to onsider the small

temperaturelimit ofthe above expansion, thatis, the limit

β → +∞.

In su ha ase, using the Lapla e method (see [6 ℄), it is possible to ompute the limit of the various terms that

appearinthe expansionof

hAi

N

(see(27)). Wegive asanexamplethe rstandthese ond terms:

A(y

∗

) = A(a) + O

₁

β

,

σ

2

2 A

′′

_(y

∗

_{) =}

1 β

A

′′

_(a)

W

′′

_(a)

+ O

₁

β

2 ,

(15)

where

a

isthepointwhere

W

attainsitsminimum(inthisremark,weassumeforsimpli ity that

W

attains itsminimum atauniquepoint). Itispossibleto re overthese termsby ex-pandingtheenergy

E

µ

aroundtheequilibrium onguration orresponding to

y

i

= a.

Indeed, if we assume that

W (y) = W

′′

_{(a) (y − a)}

2 _/2

in (14), then a simple expli it omputation

gives

hAi

N

= A(a) +

1 2N βW

′′

_(a)

A

′′

_{(a) + O}

₁

β

2 _N

2 .

Hen e, expanding the rst termsof (23) in powers of

1/β

for large

β

gives an expansion thatagrees withthatobtainedusingaharmoni approximation ofthe energy. Thisprovides

aquantitative evaluationof the latterapproa hin thisasymptoti regime.

2.2 Limit of the free energy

Wenowlookforamoredemanding result. For larity,letus returntothegeneral

oarse-grainedaverage(8),whi hof ourseequals(14)and(17)inoursimpleNN ase. Insteadof

sear hingforthelimitoftheaverage

hAi

forlarge

N

c

,wenowlook forthelimitofthefree energyperparti le(see(7)and(12)).

In the present se tion,

u

r

is in fa t equal to

u

N

(the right end atom) sin e

u

0 _{= 0}

,

although arepatom, isxed to avoid translationinvarian e. Thuswewish toidentify the

behaviourfor

N

largeof

E

CG

u

N

= −

1 β

ln

Z

R

N −1

exp −βE

µ

u

1 , . . . , u

N

du

1 . . . u

N−1

.

(29)

Note that

E

CG

is thefree energy orresponding to integratingout

N − 1

variables. From Thermodynami s,itisexpe tedthat

E

CG

s aleslinearlywith

N

. Thisis onrmedbythe onsiderationofanharmoni potential

W (x) =

k

2 (x − a)

2

,forwhi h

E

CG

u

N

=

kN

2 u

N

_{− a}

2 _{+ C(N, β, k),}

where

C(N, β, k) =

1 β

N −

1

2 ln N −

N − 1

2β

ln

_2π

βk

does not depend on

u

N

(see the detailsin [38℄). Therefore,weintrodu ethefreeenergyperparti le

F

N

(x) :=

1 N

E

CG

(x),

sothat

hAi

N

= Z

r

−1

Z

R

A u

N

_{exp −βNF}

N

u

N

du

N

_.

(30)

Thelimitbehaviourof

F

N

isprovidedbytheLargeDeviationsPrin iple. This laimis madepre iseinthefollowingtheorem.

(16)

Theorem2 Assumethepotential

W

satises

∀ξ ∈ R,

Z

R

exp (ξy − βW (y)) dy < +∞,

(31) and

exp(−βW ) ∈ H

1 _{(R \ {0}).}

Then the limit behaviour of

F

N

is given by the following Legendretransform:

lim

N→+∞

F

N

(x) +

1 β

ln

z

N

= F

∞

(x)

(32) with

F

∞

(x) :=

1 β

sup

ξ

ξx − ln

z

−1

Z

R

exp(ξy − βW (y)) dy

(33) and

z =

Z

R

exp(−βW (y)) dy

.

Remark4 The assumption

exp(−βW ) ∈ H

1 _{(R \ {0})}

allows for

W

to be pie ewise on-tinuous, with dis ontinuity at the origin. This in parti ular allows to deal with the type of

potentialsmentionedin Remark2.

Proof: Letusrstrewritethefreeenergy

F

N

(x)

asfollows:

F

N

(x)

= −

1 βN

ln

"Z

R

N −1

exp

−β

N

_X

−1

i=1

W

u

i

_{− u}

i−1

h

−βW

x − u

N

−1

h

!

du

1 . . . du

N

−1

#

= −

N − 1

_βN

ln h −

_βN

1 ln

"Z

R

N −1

exp

−βW

N x −

N

_X

−1

i=1

y

i

!

−β

N

_X

−1

i=1

W (y

i

)

!

dy

1 . . . dy

N

−1

#

= −

1 β

ln h −

1 β

ln z −

1 βN

ln µ

N

(x),

where

µ

N

is the law of the random variable

(1/N )

P

N

i=1

Y

i

, and

Y

i

is a sequen e of i.i.d. randomvariableswithlaw

µ = z

−1

_{exp(−βW (y))}

. A tually,wehave

µ

N

(x) = N µ

∗N

(N x) ,

where

µ

∗N

denotes the

(N − 1)

-fold onvolutionprodu tof

µ

(

µ

∗2

_{= µ ∗ µ}

).

Thesequen eof measures

µ

N

satisesalargedeviations property(seeforinstan e [23,

1 _{ln µ}

_.

(17)

lowerbound,whi hisasimple onsequen eoftheresultsof[46℄. Theupperboundismore

involved: weneedto reprodu ethe orrespondingproofof[46℄,andusearenedversionof

theCentralLimitTheorem[32℄.

Weintrodu ethefun tion

G

N

(x) = −

1 βN

ln µ

N

(x),

(34) whi h satises,inviewoftheabove omputation,

F

N

(x) = −

1 β

ln

z

N

+ G

N

(x).

(35) Firststep: lower bound. Wewrite

µ

N

+1

(x) = (N + 1)

Z

R

µ(N (x − t) + x) µ

N

(t) dt.

(36) Letus dene

J

N

(t) = −

1 N

ln µ (N (x − t) + x) .

Thisfun tion learlysatisesthefollowing onvergen e:

lim inf

u→t,N →+∞

J

N

(u) = J

∞

(t) :=

(

+∞

if

t 6= x,

0

if

t = x.

Hen e,wemayapplyTheorem2.3of[46℄,whi himpliesthat

lim inf

N

→+∞

−

1 N

ln

Z

R

exp (−NJ

N

(t)) µ

N

(t)dt

≥ inf

t∈R

(J

∞

(t) + βF

∞

(t)) = βF

∞

(x).

(37)

Sin etheleft-handsideof(37)is equalto

β(N + 1)

N

G

N+1

(x) +

ln(N + 1)

N

,

weinfer

lim inf

N

→+∞

G

N

(x) ≥ F

∞

(x).

(38)

Se ondstep: upper bound. Wenowaim at bounding

G

N

from above. Were all that the fun tionwemaximizein (33)is on ave,sothereexistsaunique

ξ

x

∈ R

su hthat

F

∞

(x) =

1 β

ξ

x

x − ln

z

−1

Z

R

exp (ξ

x

y − βW (y)) dy

.

TheEuler-Lagrangeequationofthemaximizationproblemalsoimplies

x =

Z

R

y exp (ξ

x

y − βW (y)) dy

Z

exp (ξ

x

y − βW (y)) dy

.

(39)

(18)

Weintrodu ethenotations

˜

µ(t) =

Z

exp(ξ

x

t − βW (t))

R

exp(ξ

x

t − βW (t)) dt

and

M (ξ) = z

−1

Z

R

exp(ξt − βW (t)) dt,

and ompute

µ

N

(x)

= N

Z

R

N −1

µ

N x −

N−1

_X

i=1

y

i

!

µ(y

1 ) . . . µ(y

N−1

) dy

1 . . . dy

N−1

= N M (ξ

x

)

N

−1

Z

R

N −1

µ

N x −

N

_X

−1

i=1

y

i

!

exp

−ξ

x

N

_X

−1

i=1

y

i

!

×˜

µ(y

1 ) . . . ˜

µ(y

N−1

) dy

1 . . . dy

N

−1

≥ N M(ξ

x

)

N

−1

Z

|N x−P y

i

|≤δ

µ

N x −

N

_X

−1

i=1

y

i

!

× exp

−ξ

x

N

_X

−1

i=1

y

i

!

˜

µ(y

1 ) . . . ˜

µ(y

N

−1

) dy

1 . . . dy

N

−1

≥ N M(ξ

x

)

N

−1

inf

[−δ,δ]

µ

exp(−ξ

x

N x − |ξ

x

|δ)

×

Z

|N x−P y

i

|≤δ

˜

µ(y

1 ) . . . ˜

µ(y

N

−1

) dy

1 . . . dy

N−1

.

Hen e,

G

N

(x)

≤ −

1 βN

ln N −

N − 1

βN

ln(M (ξ

x

)) +

ξ

x

β

+ |ξ

x

|

δ

βN

−

1 βN

ln

inf

[−δ,δ]

µ

−

_βN

1 ln P

1 N

N

_X

−1

i=1

Y

i

− x

≤

δ

N

!

,

(40)

where therandomvariables

Y

i

are i.i.d. of law

µ.

˜

Theequation (39)impliesthat

E

(Y

i

) =

x.

A ording to the hypotheses on

W

, we have

µ ∈ H

˜

1 _{(R \ {0}) ,}

hen e we may apply

Theorem5.1of[32℄. Itimpliesthatthelaw

θ

N

ofthevariable

N

X

i=1

Y

i

− Nx

!

/

√

N

onverges in

H

1 _(R)

tosomenormallaw. Inparti ular,wehave onvergen ein

L

∞

_,

hen e

P

1 N

N−1

_X

i=1

Y

i

− x

≤

δ

N

!

=

Z

x+δ

√

N −1

x−δ

√

N −1

θ

N

−1

(t)dt ≥

2γδ

√

N − 1

,

(19)

for

N

largeenough,where

γ > 0

doesnotdepend on

N

. Insertingthisinequalityinto(40), wend

G

N

(x)

≤ −

1 βN

ln N −

N − 1

βN

ln(M (ξ

x

)) +

ξ

x

β

+ |ξ

x

|

δ

βN

−

1 βN

ln

inf

[−δ,δ]

µ

−

1 βN

ln

_2γδ

√

N − 1

.

(41) Hen e,

lim sup

N

→+∞

G

N

(x) ≤ −

1 β

ln(M (ξ

x

)) +

ξ

x

β

,

whi h implies,a ordingtothedenitionof

M

and

ξ

x

,that

lim sup

N

→+∞

G

N

(x) ≤ F

∞

(x).

(42)

Estimates(38)and(42)imply

lim

N

→+∞

G

N

(x) = F

∞

(x)

. Inviewof(35),thisimplies(32).

♦

Remark5(The small temperature limit) As in Remark 3, it is possible to ompute

the expansion of

F

∞

(x)

as

β → +∞

. Using the Lapla e method, and assuming that

W

is onvex, onendsthat

F

∞

(x) = W (x) +

1 2β

ln W

′′

_{(x) + O}

₁

β

2 .

Let us now onsider another strategy to nd an approximation of

F

N

. In the spirit of the QuasiContinuum Method, we expand

E

µ

(u

1 _{, . . . , u}

N

₎

around the equilibrium onguration

u

i

_{= iu}

N

_/N

,for agiven

u

N

. More pre isely, weset

u

i

_{= u}

i

_{+ ξ}

i

,assume that

ξ

i

issmall, and expand the energy at se ond order with respe t to

ξ

i

, as explained in the Introdu tion (see (11)). We next insert this approximated energy

E

e

in (29). Due to the harmoni approximation, the resulting oarse-grained energy, that we denote

E

QCM

, is analyti ally omputableandwrites

E

QCM

(x) = N W (x) +

N − 1

2β

ln W

′′

_{(x) +}

N − 1

2β

ln

β

2π

+

1 2β

ln N.

(43) Hen e,

F

QCM

(x) :=

lim

N→+∞

1 N

E

QCM

(x) = W (x) +

1 2β

ln W

′′

_{(x) +}

1 2β

ln

β

2π

.

(44) Thus,uptoa onstant,

F

QCM

(x)

orrespondstotherst-orderapproximation (inpowersof

1/β

)of

F

∞

(x)

.

Slightly improving theproof of Theorem 2above, it is also possibleto provethe

on-vergen eofthederivativeofthefreeenergy,aquantitywhi hisindeedpra ti allyrelevant

(20)

Corollary1 Assumethat thehypotheses ofTheorem2aresatised. Then,wehave

F

N

(x) +

1 β

ln

z

N

−→ F

∞

(x)

in

L

p

loc

,

∀p ∈ [1, +∞).

(45) Inparti ular, this impliesthat

F

′

N

onvergesto

F

′

∞

in

W

−1,p

loc

.

Proof: A ordingto Theorem 2,wealreadyknowthepointwise onvergen eof

G

N

(x) =

F

N

(x) + β

−1

ln(z/N )

. Wethereforeonlyneedtoprovethat

G

N

isboundedin

L

∞

loc

toprove our laim.

Lowerbound: Wegoba kto(36),andpointoutthat

µ ≤ 1/z

. Hen e,

µ

N+1

(x) ≤

N + 1

z

Z

R

µ

N

=

N + 1

z

,

whi h implies,using (34),that

G

N

+1

(x) ≥ −

1 β(N + 1)

ln

N + 1

z

,

whi h isbounded frombelowindependentlyof

N

.

Upper bound: We return to (41), and noti e that a ording to the denition of

ξ

x

, the fun tion

x 7→ ξ

x

is ontinuous. Inaddition,the onstant

γ

in (41)isa ontinuousfun tion of

ξ

x

. Therefore,(41)providesanupperbound on

G

N

.

Asa on lusion,

G

N

isboundedin

L

∞

loc

,whi hallowsto on lude.

♦

Remark6 Considering the above theoreti al results, it ould be temptingto approa h the

average (30),that is,

hAi

N

= Z

r

−1

Z

R

A u

N

exp −βNF

N

u

N

du

N

,

by

Z

∞

−1

Z

R

A u

N

_{exp −βNF}

∞

u

N

du

N

_.

(46)

Notethat

F

N

hasbeenrepla edby

F

∞

intheexponentialfa tor. Thisstrategyisnote ient sin e this approximation does not provide the expansion (22)-(23) of

hAi

N

in powers of

1/N

. Indeed, itispossible touse the Lapla e method to ompute the expansion of (46)as

N → +∞

. Itreads

A(y

∗

) +

1 2N

σ

2 A

′′

(y

∗

) +

d

3 σ

2 A

′

_(y

∗

₎

+ o

₁

N

,

where

σ

isdenedby (23) and

d

3 = z

−1

Z

R

(y − y

∗

₎

3 exp(−βW (y)) dy

. This expansion o-in ideswith (22)-(23) onlyfor the rst term,that is

A(y

∗

₎

. The se ondonediers, unless

(21)

Toimprove the approximation (46), onemay usethe pre isedlarge deviationsprin iple

(see[16,Th.3.7.4℄ or [4 ℄). In su ha ase,onerepla es (46)by

Z

∞

−1

Z

R

A u

N

q

_F

′′

∞

(u

N

) exp −βNF

∞

u

N

du

N

_.

(47)

Thisquantityiswell-denedsin e

F

∞

isa onvexfun tion. Thenitisseenthattheexpansion of (47) in powers of

1/N

agrees with (22)-(23) up tothe se ond term. Note however that using (47)leads toamu hmore expensive omputationthan using (23).

The above onvergen e of thefreeenergy

F

N

isuseful e.g. forthe omputation ofthe freeenergyofa hainofatomswithapres ribedlength. Indeed,insu ha ase,weimpose

u

N

_{= ℓ,}

where

ℓ

is xed, and aimat omputing thefreeenergy

F

N

asafun tion of

ℓ

, in thelimit

N → +∞.

Wehave

F

N

(ℓ) = −

1 βN

ln

"Z

R

N −1

exp

−β

N

X

i=1

W

u

i

_{− u}

i−1

h

!

du

1 . . . du

N−1

#

,

where

u

N

_{= ℓ}

. Thelimitof

F

N

isprovidedbyTheorem 2.

Another interest of the approa h is to provide an approximationof

F

′

N

(ℓ)

, aquantity relatedtothe onstitutivelawofthematerialunder onsideration,atthenitetemperature

1/β

. Indeed,notethat

F

_N

′

(ℓ) =

hA

N

i

N−1

hB

N

i

N−1

,

(48)

where

h·i

N

−1

is the average with respe t to the Gibbs measure asso iated to the energy

N−1

_X

i=1

W

_u

i

_{− u}

i−1

h

,andtheobservables

A

N

and

B

N

aredenedby

B

N

u

1 , . . . , u

N

−1

= exp −βW N ℓ − u

N−1

,

A

N

u

1 , . . . , u

N

−1

= W

′

N ℓ − u

N−1

exp −βW N ℓ − u

N

−1

_.

Hen e

F

′

N

(ℓ)

anbeinterpretedastheaveragefor ebetweenatoms

N − 1

and

N

,whenthe positon ofatom

N

is pres ribedat

u

N

_{= ℓ}

. Corollary1providesthe onvergen eof

F

′

N

(ℓ)

to

F

′

∞

(ℓ)

in aweaknorm.

Remark7 Note that, in (48), both observables

A

N

and

B

N

depend on

N

. Hen e, the resultsofSe tion2.1(obtainedusingthe Law ofLargeNumbersandnotinvolvingthe Large

(22)

2.3 Numeri al tests

Forournumeri altests,we hoosethepairintera tionpotential

W (x) =

1

2 (x − 1)

4 ₊

1

2 x

2

(49)

shownonFigure 4. Notethat

W (x)

growsfastenoughto

+∞

when

|x| → +∞

, su h that assumptions (20) and (31) are satised. Note also that wehave made no assumption on

the onvexityof

W

in Theorems1and2. We onsiderherea onvexpotential. Attheend ofthisse tion, wewill onsideranon- onvexexample(see(50)),andshowthatweobtain

similar on lusions.

W (x)

x

2.5 2 1.5 1 0.5 0 -0.5 6 5 4 3 2 1 0

Figure4: Thepotential

W

hosenforthetests.

Werst onsiderthe omputationofensembleaverages,andweagainrestri tourselves

to the aseoftworepatoms

u

0 _{= 0}

and

u

N

. This isjust for simpli ityand forthesakeof

demonstratingthefeasibilityandtheinterestofourapproa h. The aseof

N

r

repatomsmay betreatedlikewise. It isof oursemore omputationallydemanding,althoughaordable.

We hooseanobservable

A(x)

,andwe omparethefollowingfourquantities:

(i) theexa taverage

hAi

N

dened by(14). Following(4)-(5), thisquantityis omputed asthelong-timeaverageof

A(u

N

_(t))

alongthefullsystemdynami s

du = −∇

u

E

µ

(u) dt +

p

2/β dB

t

in

R

N

_.

This equationis numeri allyintegratedwith theforwardEuler s heme, withasmall

timestep. Inpra ti e,wehavesimulatedmanyindependentrealizationsofthisSDE,

in orderto omputeerrorbarsfor

hAi

N

.

(ii) a QuasiContinuumtype approximation of

hAi

N

, based on the 'interpolation + har-moni expansion'pro edure outlinedabove. That is, weintrodu e

E

QCM

dened by

(23)

(43),andweapproximate

hAi

N

by

hAi

QCM

N

:=

Z

R

A(x) exp [−βE

QCM

(x)] dx

Z

R

exp [−βE

QCM

(x)] dx

.

(iii) a Law of Large Numbers (LLN) type approximation of

hAi

N

, whi h onsists in ap-proximating

hAi

N

by

A(y

∗

₎

, followingTheorem1.

(iv) arenedapproximation,whi h onsistsinapproximating

hAi

N

by

hAi

approx,1

N

dened by(23),followingTheorem1.

Notethat onlyone-dimensional integralsare neededforapproximations(ii), (iii) and(iv).

They anbe omputedwithahigh a ura y.

We plotonFigure 5these four quantities, forin reasing valuesof

N

(the temperature isxedat

1/β = 1

),fortheobservable

A(x) = exp(x)

. OnFigure 6,we omparethesame quantities,nowasfun tionsofthetemperature,for

N = 100

andfor

N = 10

. Weherework with

A(x) = x

2

,forwhi h

hAi

N

= hAi

approx,1

N

. QCM renedLLN LLN exa t

N

100 80 60 40 20 2.3 2.2 2.1 2 1.9 renedLLN LLN exa t

N

100 80 60 40 20 1.95 1.94 1.93 1.92 1.91 1.9

Figure 5: Convergen e, as

N

in reases, of

hA(u

N

_)i

N

(exa t), of

hA(u

N

_)i

approx,1

N

(rened LLN)andof

hA(u

N

_)i

QCM

N

(QCM)and omparisonto

A(y

∗

₎

(LLN) (temperature

1/β = 1

, observable

A(x) = exp(x)

; wehaveperformed omputationsfor

N = 10

,25,50and100;on therightgraph,weshowerrorbarsfor

hA(u

N

_)i

N

).

Asexpe ted,thethermodynami limitstrategies(iii)and(iv)betteragreewiththefull

atom al ulation, whateverthe temperature, provided the number of eliminated atoms is

large (note that the strategy (iv) is very a urate even for the small value

N = 10

, at thetemperature

1/β = 1

). Approximation(ii) is learlyinee tive forhigh temperatures. Onthe otherhand,for asu ientlysmall temperatureand asu ientlysmall numberof

(24)

QCM LLN exa t

1/β

1 0.8 0.6 0.4 0.2 0.7 0.6 0.5 0.4 0.3 0.2 QCM LLN exa t

1/β

1 0.8 0.6 0.4 0.2 0.7 0.6 0.5 0.4 0.3 0.2

Figure 6: We plot

hA(u

N

_)i

N

= hA(u

N

)i

approx,1

_N

(exa t),

hA(u

N

_)i

QCM

N

(QCM) and

A(y

∗

₎

(LLN) as fun tions of the temperature

1/β

: on the left,

N = 100

; on the right,

N = 10

(observable

A(x) = x

2

).

eliminatedatoms,thisapproximationis losetothefullatomresult. However,evenforthe

small values

N = 10

and

1/β = 0.2

, our asymptoti result

hA(u

N

_)i

approx,1

N

= 1.6299

(for

A(x) = exp(x)

) is loser to the exa tresult

hA(u

N

_)i

N

= 1.6303 ± 0.0008

than the QCM result

hA(u

N

_)i

QCM

N

= 1.6469

.

Remark8 As in Remark 2, we emphasize that the omputations reported on here do not

a ountfor the onstraintsonthepositionsofatoms. Analogous omputations,thata ount

for onstraints, may be performed. They provide similar on lusions, as an be seen on

Figure7,whi h isvery similar toFigure 6.

Wenow onsider the omputation of freeenergies,morepre isely,of thederivativesof

freeenergies. Thefull atom value

F

′

N

(x)

is omputedasaratioof ensembleaverages(see (48)). We omparethisquantitywith

(i) itslarge

N

limit

F

′

∞

(x)

,where

F

∞

isdenedby(33),ontheonehand, (ii) and, on the other hand, its QuasiContinuum type approximation

F

′

QCM

(x)

, where

F

QCM

isdenedby(44). Itreads

F

QCM

′

(x) = W

′

(x) +

1 2β

W

′′′

_(x)

W

′′

_(x)

.

Webriey detailhowwe ompute

F

′

∞

(x)

. Let

ξ

x

bethe uniquerealnumberat whi hthe supremumin(33)isattained. Wehave

F

′

∞

(x) =

ξ

x

(25)

QCM LLN exa t

1/β

1 0.8 0.6 0.4 0.2 0.7 0.6 0.5 0.4 0.3 0.2

Figure 7: We plot

hA(u

N

_)i

N

= hA(u

N

)i

approx,1

N

(exa t),

hA(u

N

_)i

QCM

N

(QCM) and

A(y

∗

₎

(LLN)asfun tionsofthetemperature

1/β

(

N = 100

,observable

A(x) = x

2

). Thepotential

energyisoftype(18): itis equalto

W (x)

denedby(49)if

x > 0

,and

+∞

otherwise. TheEuler-Lagrangeequationsolvedby

ξ

x

is(39),thatwere astas

z

−1

Z

R

(x − y) exp(ξ

x

y) exp(−βW (y)) dy = 0.

Letusintrodu e

G(y, ξ) = (x−y) exp(ξy)

. Wehen elookfor

ξ

x

su hthat

E

µ

[G(y, ξ

x

)] = 0

, where the random variable

y

is distributed a ordingto the probability measure

µ(y) =

z

−1

_{exp(−βW (y))}

. TheRobbins-Monroealgorithm [28℄ anbeused to ompute

ξ

x

, hen e

F

′

∞

(x)

.

Werststudythe onvergen eof

F

′

N

(x)

to

F

′

∞

(x)

as

N

in reases,foraxed hainlength

x = 1.4

andaxedtemperature

1/β = 1

. ResultsareshownonFigure8. Weindeedobserve that

F

′

N

(x) → F

∞

′

(x)

when

N → +∞

.

Wenow omparethetwoapproximations(i)and(ii)of

F

′

N

(x)

,for

N = 100

and

1/β = 1

. Results are shown on Figure 9. We observethat

F

′

∞

(x)

is a very good approximationof

F

′

N

(x)

. Asexpe ted,thetemperatureistoohighfortheharmoni approximationtoprovide ana urateapproximationof

F

′

N

(x)

. On Figure 10, weplot

F

′

∞

(x)

forseveral temperatures,aswellasits zerotemperature limit,whi his

W

′

_(x)

(seeRemark5).

Up to here,wehaveused the onvexpotential (49). Forthe sakeof ompleteness, we

nowbriey onsiderthe aseofanon- onvexpotential

W

. We hoosethetoy-model

W (x) = (x

2 − 1)

2 ,

(50) whi h orresponds to a double-well potential. On Figure 11, we plot

F

′

∞

(x)

for several temperatures, for this double-well potential. Although we have not yet ompared these

(26)

F

′

N

(x = 1.4)

F

′

∞

(x = 1.4)

N

50 40 30 20 10 2.15 2.14 2.13 2.12 2.11 2.1 Figure8: Convergen eof

F

′

N

(x)

(shownwitherrorbars)to

F

′

∞

(x)

as

N

in reases (temper-ature

1/β = 1

,xed hainlength

x = 1.4

).

F

′

∞

(x)

F

′

N

(x)

F

′

QCM

(x)

x

1.6 1.4 1.2 1 0.8 0.6 0.4 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 Figure9: Weplot

F

′

N

(x)

,

F

′

∞

(x)

and

F

′

(27)

F

′

∞

(x)

,

1/β = 1

F

′

∞

(x)

,

1/β = 0.5

W

′

_(x)

x

1.4 1.2 1 0.8 0.6 0.4 2.5 2 1.5 1 0.5 0 -0.5 -1 Figure10:

F

′

∞

(x)

fordierenttemperatures.

oarse-grained omputationswiththefullatom omputations,thenumeri alresultsreported

hereare onsistent withthe small temperature limit

lim

T

→0

F

′

∞

(x) = (W

∗

)

′

(x)

, where

W

∗

is

the onvexenvelopof

W

.

F

′

∞

(x)

,

1/β = 0.5

F

′

∞

(x)

,

1/β = 1

(W

W

∗

₎

′

_(x)

′

_(x)

x

1.5 1 0.5 0 -0.5 -1 -1.5 6 4 2 0 -2 -4 -6 -8 Figure11:

F

′

∞

(x)

fordierenttemperatures,inthe aseofthedouble-wellpotential(50).

3 The NNN ase and some extensions

In this Se tion, we rst onsider the ase of a NNN intera ting system. The analysis is

(28)

NNNN ase(stillforone-dimensionalsystems)andse ondthe aseoflinearpolymer hains,

whereatomssamplethephysi alspa e

R

3

.

3.1 The next-to-nearest neighbour (NNN) ase

Wenow onsiderthenext-to-nearestneighbour ase. Itturnsoutthat,forthe omputation

ofensembleaveragesaswell asfor otherquestions,this aseissigni antlymoreintri ate

thantheNN ase. Ourstrategy,basedontheLawofLargeNumbers,willbesimilartothat

usedfortheNN ase,buttheobje tmanipulatedarenotindependentrandomvariablesany

longer. Markov hainsaretherightnotionformalizingthesituationmathemati ally.

Webeginbyintrodu ingtheres aledatomisti energy,similarly to(13):

E

µ

u

1 , . . . , u

N

=

N

X

i=1

W

1 u

i

_{− u}

i−1

h

+

N

_X

−1

i=1

W

2 u

i+1

_{− u}

i−1

h

.

(51)

As above, weintrodu e the hange ofvariables (15), repla ing

(u

i

_{− u}

i−1

_)/h

bythe

inter-atomi distan es

y

i

. Re allfrom (16)that

u

N

₌

1 N

N

X

i=1

y

i

. Theensembleaverage

hAi

N

of anobservablethat dependsonlyontheright-endatomthereforewrites

hAi

N

=

Z

−1

Z

R

N

A u

N

_{exp −βE}

µ

u

1 , . . . , u

N

du

1 . . . du

N

=

Z

−1

Z

R

N

A

1 N

N

X

i=1

y

i

!

e

−β P

i

W

1 (y

i

)

_e

−β P

i

W

2 (y

i

+y

i+1

)

_dy

1 . . . dy

N

.

(52)

Thekeyingredientisnowtoseetheaboveexpression,as

N

goestoinnity,asanasymptoti s foradis rete-timeMarkov hain. Theasymptoti sofMarkov hainsbeingamathemati al

problem mu h more involved than that of i.i.d. sequen es, we restri t ourselves to the

omputationof theaverage ofanobservable. Theasymptoti behaviourofthefreeenergy

may bestudied, applying aLargeDeviations Prin iple forMarkov hains(see forinstan e

[26,Th.IV.3℄). Wewillnotpursueinthis dire tion.

Se tion 3.1.1deals withthe aseoftworepatoms (namely

u

0 _{= 0}

and

u

N

), while

Se -tion3.1.2 indi ate the hanges in order to dealwith morethan tworepatoms. Numeri al

resultswill bereportedin Se tion3.1.3.

3.1.1 Limit ofthe average,the aseof tworepatoms

Inorderto ompute

lim

N

→+∞

hAi

N

,weintrodu ethenotation

(29)

Equation(52)rewrites

hAi

N

= Z

−1

Z

R

N

A

1 N

N

X

i=1

y

i

!

e

−βW

1 (y

1 )

_{f (y}

1 , y

2 ) . . . f (y

N

−1

, y

N

) dy

1 . . . dy

N

.

(53)

Ourmethod onsistsin onsideringthesequen eofvariables

(y

1 , . . . , y

N

)

in(53)asa real-izationofaMarkov hainwithkernel

f (·, ·)

. However,theslightte hni aldi ultyatthis stageisthat thekernel

f

isnotnormalized,sin eingeneral

Z

R

f (y

1 , y

2 ) dy

2 =

Z

R

exp(−βW

2 (y

1 + y

2 )) exp(−βW

1 (y

2 )) dy

2 6= 1.

Astandardtri kofProbabilitytheoryallowsto ir umventthisdi ulty. Introdu e

¯

f (x, y) := exp

−βW

2 (x + y) −

β

2 W

1 (x) −

β

2 W

1 (y)

.

Notethat

f

¯

isasymmetri fun tion (whereas

f

isnot), hen etheoperator

P φ(y) =

Z

R

¯

f (y, z)φ(z) dz

(54) isself-adjointon

L

2 _(R)

. Considerthen

ψ

1 :=

argmax

Z

R

2 ψ(y) ψ(z) ¯

f (y, z) dy dz;

Z

R

ψ

2 _{(y) dy = 1}

,

(55) andset

λ =

Z

R

2 ψ

1 (y) ψ

1 (z) ¯

f (y, z) dy dz.

(56) Using standardtoolsof spe tral theory ofself-adjoint ompa t operators, it is possible to