Metastability for Kawasaki dynamics at low temperature with two types of particles

Metastability for Kawasaki dynamics at low temperature with

two types of particles

Citation for published version (APA):

Hollander, den, W. T. F., Nardi, F. R., & Troiani, A. (2011). Metastability for Kawasaki dynamics at low

temperature with two types of particles. (Report Eurandom; Vol. 2011007). Eurandom.

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Published: 01/01/2011

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EURANDOM PREPRINT SERIES

2011-007

Metastability for Kawasaki dynamics at

low temperature with two types of particles

F. den Hollander, F. R. Nardi and A. Troiani

ISSN 1389-2355

low temperature with two types of parti les F. den Hollander 1 2 F. R. Nardi 3 2 A. Troiani 1 January31, 2011 Abstra t

Thisistherstinaseriesofthreepapersinwhi hwestudyatwo-dimensionallatti e

gas onsistingoftwotypesofparti lessubje ttoKawasakidynami sat lowtemperature

inalargeniteboxwithanopenboundary. Ea hpairofparti leso upyingneighboring

sites hasanegativebinding energyprovidedtheirtypesaredierent,while ea hparti le

has a positive a tivation energy that depends on its type. There is no binding energy

betweenneighboringparti les ofthe sametype. Westart thedynami s from theempty

box and omputethe transition time to the full box. This transition is triggered by a

riti al droplet appearingsomewhereinthebox.

We identify the region of parameters for whi h the system is metastable. For this

region, in the limit as the temperature tends to zero, we show that the rst entran e

distributiononthesetof riti aldropletsisuniform, omputetheexpe tedtransitiontime

uptoandin ludingamultipli ativefa toroforderone,andprovethatthetransitiontime

divided byits expe tation is exponentially distributed. These results are derived under

threehypotheses,whi hareveriedinthese ondandthethirdpaperfora ertainsubregion

of the metastable region. These hypotheses involvethree model-dependent quantities 

theenergy,theshapeandthenumberofthe riti aldropletswhi hareidentiedinthe

se ondandthethirdpaperaswell.

The main motivation behind this work is to understand metastability of multi-type

parti le systems. It turnsout that fortwotypesof parti lesthe geometry of sub riti al

and riti aldropletsismore omplexthanforonetypeofparti le. Consequently,it isa

somewhatdeli atematterto apturetheproperme hanismsbehindthegrowingandthe

shrinkingofsub riti aldropletsuntila riti aldropletisformed. Ourproofsusepotential

theoryandrelyonideasdevelopedin[6℄forKawasakidynami swithonetypeofparti le.

Ourtargetistoidentifytheminimal hypothesesthatareneededformetastablebehavior.

MSC2010. 60K35,82C26.

Keywords andphrases. Multi-typeparti le systems,Kawasakidynami s,metastable

re-gion,metastabletransitiontime, riti aldroplet,potentialtheory,Diri hletform, apa ity.

A knowledgment. ATisgratefultoGabrieledallaTorreforfruitfuldis ussions.

1

Mathemati alInstitute,LeidenUniversity,P.O.Box9512,2300RALeiden,TheNetherlands 2

EURANDOM,P.O.Box513,5600MBEindhoven,TheNetherlands 3

Se tion 1.1denes the model, Se tion 1.2introdu es basi notation, Se tion 1.3identies the

metastableregion, whileSe tion1.4states themain theorems. Se tion1.5dis ussesthemain

theoremsandpla esthemintheproper ontext. Foranoverviewonmetastabilityanddroplet

growth,wereferthereadertothemonographbyOlivieriandVares[22 ℄,andthereviewpapers

byBovier[3℄,[4 ℄,den Hollander [12 ℄,and OlivieriandS oppola [21℄.

1.1 Latti e gas subje t to Kawasaki dynami s

Let

Λ ⊂ Z

2

be a largenite box. Let

∂Λ

−

= {x ∈ Λ : ∃ y /

∈ Λ : |y − x| = 1},

∂Λ

+

= {x /

∈ Λ : ∃ y ∈ Λ : |y − x| = 1},

(1.1)

be the internal boundary, respe tively, the external boundary of

Λ

, and put

Λ

−

_{= Λ\∂Λ}

−

and

Λ

+

_{= Λ ∪ ∂Λ}

+

. Withea h site

x ∈ Λ

we asso iate a variable

η(x) ∈ {0, 1, 2}

indi ating the absen e of a parti le or the presen e of a parti le of type

1

or type 2. A onguration

η = {η(x) : x ∈ Λ}

is an element of

X = {0, 1, 2}

Λ

. To ea h onguration

η

we asso iate an energy given bytheHamiltonian

H(η) = −U

X

(x,y)∈Λ

∗,−

1

{η(x)=1,η(y)=2

or

η(x)=2,η(y)=1}

+ ∆

1

X

x∈Λ

1

{η(x)=1}

+ ∆

2

X

x∈Λ

1

{η(x)=2}

,

(1.2) where

Λ

∗,−

_{= {(x, y) : x, y ∈ Λ}

−

_{, |x − y| = 1}}

is the set of non-oriented bonds inside

Λ

−

,

−U < 0

isthe binding energy between neighboring parti les ofdierent typesinside

Λ

−

,and

∆

1

> 0

and

∆

2

> 0

arethe a tivation energies of parti lesof type

1

,respe tively, 2inside

Λ

. W.l.o.g. we will assumethat

∆

1

≤ ∆

2

.

(1.3)

The Gibbsmeasureasso iated with

H

is

µ

β

(η) =

1

Z

β

e

−βH(η)

,

η ∈ X ,

(1.4)

where

β ∈ (0, ∞)

is theinverse temperature, and

Z

β

is thenormalizingpartition sum. Kawasaki dynami s is the ontinuous-time Markov pro ess

(η

t

)

t≥0

with state spa e

X

whose transitionrates are

c

β

(η, η

′

) = e

−β[H(η

′

_)−H(η)]

+

_,

_{η, η}

′

_{∈ X , η 6= η}

′

_{, η ↔ η}

′

_,

(1.5) where

η ↔ η

′

meansthat

η

′

an be obtained from

η

byone of thefollowing moves:

•

inter hanging thestates

0 ↔ 1

or

0 ↔ 2

at neighboring sitesin

Λ

(hopping of parti lesinside

Λ

),

•

hangingthe state

0 → 1

,

0 → 2

,

1 → 0

or

2 → 0

at singlesites in

∂

−

_Λ

( reation and annihilationof parti lesat

∂

−

_Λ

and

c

β

(η, η

′

_{) = 0}

otherwise. Thisdynami s isergodi andreversiblewithrespe ttotheGibbs

measure

µ

β

. Note that the dynami s preserves parti lesinside

Λ

, but allows parti les to be reated and annihilated at

∂

−

_Λ

. Think of the parti les entering and exiting

Λ

along non-orientededgesbetween

∂

−

_Λ

and

∂

+

_Λ

. Thepairs

(η, η

′

₎

with

η ↔ η

′

are alled ommuni ating

ongurations, the transitions between them are alledallowed moves. Note thatparti les at

∂

−

_Λ

do not intera t withparti lesinside

Λ

−

.

The dynami s dened by (1.2) and (1.5) models thebehavior inside

Λ

of a latti e gas in

Z

2

, onsisting of two types of parti les subje t to random hopping with hard ore repulsion and withbinding between dierent neighboring types. We may think of

Z

2

_\Λ

as an innite

reservoir thatkeeps the parti le densitiesxedat

ρ

1

= e

−β∆

1

and

ρ

2

= e

−β∆

2

. Inour model

thisreservoirisrepla edbyanopenboundary

∂

−

_Λ

,whereparti lesare reatedandannihilated

at a rate thatmat hesthese densities. Consequently, our Kawasaki dynami s is anite-state

Markovpro ess.

Note that there is no binding energy between neighboring parti les of the same type.

Consequently,the modeldoesnot redu etoKawasakidynami sfor onetype ofparti lewhen

∆

1

= ∆

2

.

1.2 Notation

To identify themetastableregion inSe tion1.3and stateour keytheoremsinSe tion1.4, we

need some notation.

Denition 1.2.1 [Parti le numbers, a tive bonds, empty and full ongurations,

paths℄

(a)

n

i

(η)

isthe number of parti les of type

i = 1, 2

in

η

.

(b)

B(η)

isthe number ofpairsof neighboringparti les ofdierent type in

η

, i.e.,thenumber of a tive bonds in

η

.

( )



is the onguration where

Λ

isempty.

(d)

⊞

is the onguration where

Λ

islled as a he kerboard (see Remark 1.4.6 below). (e)

ω : η → η

′

isany path of allowed moves from

η

to

η

′

.

(f)

τ

A

=

inf{t ≥ 0 : η

t

∈ A}

,

A ⊂ X

, isthe rst entran e time of

A

. (g)

P

η

isthe law of

(η

t

)

t≥0

given

η

0

= η

.

Denition 1.2.2 [Communi ation heights, ommuni ation level sets, stability

lev-els, sets of stable and metastable ongurations℄

(a)

Φ(η, η

′

₎

isthe ommuni ation height between

η, η

′

_{∈ X}

dened by

Φ(η, η

′

) = min

ω : η→η

′

max

σ∈γ

H(σ),

(1.6)

and

Φ(A, B)

isits extension tonon-empty sets

A, B ⊂ X

dened by

Φ(A, B) =

min

η∈A,η

′

_∈B

Φ(η, η

′

_).

(1.7) (b)

S(η, η

′

₎

is the ommuni ationlevel set between

η

and

η

′

dened by

S(η, η

′

) = {ζ ∈ X : ∃ ω : η → η

′

, ω ∋ ζ : max

ξ∈ω

H(ξ) = H(ζ) = Φ(η, η

′

_)}.

(1.8)

( )

V

η

isthe stability level of

η ∈ X

dened by

V

η

= Φ(η, I

η

) − H(η),

(1.9)

where

I

η

= {ξ ∈ X : H(ξ) < H(η)}

isthe set of ongurationswith energy lower than

η

. (d)

X

stab

= {η ∈ X : H(η) = min

ξ∈X

H(ξ)}

is the set of stable ongurations, i.e.,the set of ongurations withminimal energy.

(e)

X

meta

= {η ∈ X : V

η

= max

ξ∈X \X

stab

V

ξ

}

is the set of metastable ongurations, i.e., the set of non-stable ongurations withmaximal stability level.

(f)

Γ = Φ(X

meta

, X

stab

)

.

Denition 1.2.3 [Optimal paths, gates, dead-ends℄

(a)

(η → η

′

₎

opt

is the set of paths realizing the minimaxin

Φ(η, η

′

₎

.

(b) A set

W ⊆ X

is alled a gate for

η → η

′

if

W ⊆ S(η, η

′

₎

and

ω ∩ W 6= ∅

for all

ω ∈ (η → η

′

)

opt

.

( ) A set

W ⊆ X

is alled a minimal gate for

η → η

′

if it is a gate for

η → η

′

and for any

W

′

₍

_W

there exists an

ω

′

_{∈ (η → η}

′

₎

opt

su h that

ω

′

_{∩ W}

′

_{= ∅}

.

(d) A priori there may be several (not ne essarily disjoint) minimal gates. Their union is

denoted by

G(η, η

′

₎

andis alled the essentialgate for

(η → η

′

₎

opt

. (e) The ongurations

S(η, η

′

_{)\G(η, η}

′

₎

are alled dead-ends.

1.3 Metastable region

We want to understand how the system tunnels from



to

⊞

when the former is a lo al minimum and the latter is a global minimum of

H

. We begin by identifying the metastable region, i.e.,the region inparameter spa efor whi h this isthe ase.

Lemma 1.3.1 The ondition

∆

1

+ ∆

2

< 4U

is ne essary and su ient for



to be a lo al minimum but not a global minimum of

H

.

Proof. Note that

H() = 0

. We know that



is a lo al minimum of

H

, sin e assoon asa parti le enters

Λ

we obtain a onguration with energy either

∆

1

> 0

or

∆

2

> 0

. To show thatthere isa onguration

η

ˆ

with

H(ˆ

η) < 0

,we write

H(η) = n

1

(η)∆

1

+ n

2

(η)∆

2

− B(η)U.

(1.10)

Sin e

∆

1

≤ ∆

2

,we mayassumew.l.o.g. that

n

1

(η) ≥ n

2

(η)

. Indeed,if

n

1

(η) < n

2

(η)

,thenwe simplytake the onguration

η

1↔2

obtained from

η

byinter hanging thetypes1and 2,i.e.,

η

1↔2

(x) =











1

if

η(x) = 2,

2

if

η(x) = 1,

0

otherwise

,

(1.11) whi h satises

H(η

1↔2

_{) ≤ H(η)}

. Sin e

B(η) ≤ 4n

2

(η)

,we have

H(η) ≥ n

1

(η)∆

1

+ n

2

(η)∆

2

− 4n

2

(η)U ≥ n

2

(η)(∆

1

+ ∆

2

− 4U ).

(1.12)

Hen e, if

∆

1

+ ∆

2

≥ 4U

, then

H(η) ≥ 0

for all

η

and

H() = 0

is a global minimum. On the other hand, onsider a onguration

η

ˆ

su h that

n

1

(ˆ

η) = n

2

(ˆ

η)

and

n

1

(ˆ

η) + n

2

(ˆ

η) = ℓ

for some

ℓ ∈ 2N

. Arrange theparti lesof

η

ˆ

ina he kerboard squareof side length

ℓ

. Thena straightforward omputation gives

H(ˆ

η) =

1

₂

ℓ

2

∆

1

+

1

₂

ℓ

2

∆

2

− 2ℓ(ℓ − 1)U,

(1.13)

and so

H(ˆ

η) < 0 ⇐⇒ ℓ

2

(∆

1

+ ∆

2

) < 4ℓ(ℓ − 1)U ⇐⇒ ∆

1

+ ∆

2

< (4 − 4ℓ

−1

)U.

(1.14)

Hen e, if

∆

1

+ ∆

2

< 4U

, thenthere existsan

¯

ℓ ∈ 2N

su h that

H(ˆ

η) < 0

for all

ℓ ∈ 2N

with

ℓ ≥ ¯

ℓ

. Here,

Λ

mustbe taken large enough,sothat adroplet of size

¯

ℓ

tsinside

Λ

−

.



Within the metastable region

∆

1

+ ∆

2

< 4U

, we will ex lude the subregion

∆

1

, ∆

2

< U

(see Fig.1). In thissubregion, ea h timeaparti le of type

1

enters

Λ

and atta hesitselfto a parti leoftype2inthedroplet,orvi eversa,theenergygoesdown. Consequently,the riti al

droplet onsistsoftwofreeparti les, oneoftype

1

andoneof2. Thereforethis subregiondoes not exhibit propermetastablebehavior.

Figure1: Propermetastableregion.

1.4 Key theorems

The threetheorems belowwill be proved subje tto thefollowing hypotheses:

(H1)

X

stab

= ⊞

. (H2) Thereexistsa

V

⋆

_{< Γ = Φ(, ⊞)}

su hthat

V

η

≤ V

⋆

forall

η ∈ X \{, ⊞}

. Inparti ular,

X

meta

= 

.

(H3) Theset

C

⋆

∂Λ

−

⊆ G(, ⊞)

ofall ongurationswheresomepath in

( → ⊞)

opt

rstenters

G(, ⊞)

onsists of a single droplet somewhere inside

Λ

−

and a free parti le of type

2 somewhere at

∂

−

_Λ

. The set

C

⋆

att

obtained from

C

⋆

∂Λ

−

by atta hing the free parti le somewhere to the droplet is su h that for every

η ∈ C

⋆

att

all paths in

(η → )

opt

pass through

C

⋆

∂Λ

−

.

Remark 1.4.1 Thefree parti le is oftype 2 only when

∆

1

< ∆

2

. If

∆

1

= ∆

2

(re all (1.3) ), thenthefreeparti le anbeoftype

1

or2. Inthelatter asethereisfullsymmetryof

S(, ⊞)

underthemap

1 ↔ 2

dened in(1.11) .

Denition 1.4.2 (a)The set of ongurations obtained from

C

⋆

∂Λ

−

by removing the free par-ti le is denoted by

P

andis alled the set of proto riti al droplets.

(b) The set of ongurations obtained from

P

by adding a free parti le of the appropriatetype anywhere in

Λ

is denoted by

C

⋆

_{(= P}

fp

₎

and is alled the set of riti al droplets.

( ) The ardinality of

P

modulo shifts of the proto riti al droplet isdenoted by

N

.

Note that

Γ = H(C

⋆

_{) = H(P) + ∆}

2

, and that

C

⋆

onsists of pre isely those ongurations

interpolating between

P

and

C

⋆

att

where a freeparti le of type 2 enters

Λ

and moves to the proto riti aldroplet.

Theorem 1.4.3

lim

β→∞

P



(τ

C

⋆

< τ

_⊞

| τ

_⊞

< τ

_

) = 1

and

lim

β→∞

P



(η

τ

C⋆

−

= ζ) = 1/|P|

for all

ζ ∈ P

, where

τ

C

⋆

−

is the time justprior to

τ

C

⋆

.

Theorem 1.4.4 There existsa onstant

K = K(Λ; U, ∆

1

, ∆

2

) ∈ (0, ∞)

su h that

lim

β→∞

e

−βΓ

_E



(τ

⊞

) = K.

(1.15) Moreover,

K ∼

1

N

log |Λ|

4π|Λ|

as

Λ → Z

2

_.

(1.16) Theorem 1.4.5

lim

β→∞

P



(τ

⊞

/E



(τ

⊞

) > t) = e

−t

for all

t ≥ 0

.

Remark 1.4.6 We will see in [15 ℄ that

X

stab

may a tually onsist of more than the single onguration

⊞

, namely, it may ontain ongurations that dier from

⊞

at

∂

−

_Λ

. In fa t,

dependingonthe hoi eof

U, ∆

1

, ∆

2

,largedropletswithminimalenergytendto haveashape thatiseithersquare-shapedorrhombus-shaped. Therefore,itisexpedientto hoose

Λ

tohave a similarform. However,sin e this aboundaryee t thatdoesnot ae tour main theorems,

we will ignore ithere. Amore pre isedes ription willbe given in[15 ℄.

Remark 1.4.7 Hypothesis (H3)is neededonly for the asymptoti sin(1.16). Aswe will see

inSe tion 2.3,the valueof

K

isgiven bya non-trivial variational formulainvolvingthesetof all ongurations where the dynami s an enter and exit

C

⋆

. This set in ludes not only the



Γ

-valleys around



and

⊞

, but also wells inside

C

⋆

with energy

< Γ

and ommuni ation height

Γ

towardsboth



and

⊞

. It ontains

P

and

C

⋆

att

,andpossiblymore(aswasshownin[6 ℄, Se tion 2.3.2,for Kawasaki dynami s withone typeof parti le). Asaresult ofthis geometri

omplexity, for nite

Λ

only upperand lower bounds are known for

K

. What (1.16) says is that these bounds merge and simplify inthe limit as

Λ → Z

2

, and that for the asymptoti s

onlythesimpler quantity

N

mattersrather thanthefullgeometryof riti aland near riti al droplets. We will see in Se tion 2.3 that, apart from the uniformity property expressed in

Theorem 1.4.3, the reason behind this simpli ation is the fa t that simple random walk on

Z

2

isre urrent.

1.5 Dis ussion

1. The rst part of Theorem 1.4.3 says that

C

⋆

is a gate for thenu leation, i.e., on its way

from



to

⊞

the path passesthrough

C

⋆

. These ond partsays thatall proto riti aldroplets

are equally likely to be seen upon rst entran e in

C

⋆

nu leation time is asymptoti to

Ke

Γβ

, whi h is the lassi al Arrhenius law. Theorem 1.4.5,

nally,saysthatthe nu leation timeisexponentially distributedonthes ale of its average.

2. Theorems 1.4.31.4.5 are model-independent, i.e., they are expe ted to hold in the same

form for a large lassof metastable sto hasti dynami s ina nite box. So far this hasbeen

veried foranumberofexamples,in ludingKawasakidynami s withone typeofparti le(see

Se tion 1.5for a dis ussion). The model-dependent ingredient of Theorems 1.4.31.4.5 is the

triple

(Γ, C

⋆

, N ).

(1.17)

This triple depends on the parameters

U, ∆

1

, ∆

2

in a manner that remains to be identied. The set

C

⋆

alsodepends on

Λ

,but insu h away that

|C

⋆

_{| ∼ N |Λ|}

as

Λ → Z

2

, withtheerror

oming fromboundaryee ts. Clearly,

Λ

mustbetaken largeenough sothat riti aldroplets t inside.

3. In[15 ℄and[16℄,wewillprove (H1)(H3),identify

(Γ, C

⋆

_{, N )}

andderive anupper boundon

V

⋆

subje tto a further restri tionon theparameters, namely(see Fig.2),

∆

1

< U,

∆

2

− ∆

2

> 2U.

(1.18)

More pre isely, in [15℄ we will prove (H1), identify

Γ

, show that

V

⋆

_{≤ 10U}

, and on lude

that (H2) holds as soon as

Γ > 10U

, whi h will orrespond to a ertain subregion. We will also see that it would be possible to show that

V

⋆

_{≤ 4U}

, provided ertain boundary ee ts

(arising when adroplet sits lose to

∂

−

_Λ

) ould be ontrolled. Sin e we will seethat

Γ > 4U

throughout theregion (1.18) ,thisupperboundwouldsettle(H2)withoutfurtherrestri tions.

In [16℄we will prove (H3)and identify

C

⋆

_{, N}

.

The simplifying features of (1.18) are the following:

∆

1

< U

implies that ea h time a parti le oftype

1

enters

Λ

andatta hesitselfto a parti le oftype2 inthedroplet theenergy goesdown, while

∆

2

− ∆

1

> 2U

implies that no parti le of type 2 sitson the boundary of a droplet that has minimal energy given the number of parti les of type 2 in the droplet. We

onje ture that (H1)(H3) holdwithout therestri tions in(1.18) . However, as we will see in

[15 ℄ and [16℄,

(Γ, C

⋆

_{, N )}

is dierent when

∆

1

> U

.

Figure 2: Subregionof the propermetastableregion onsidered in[15℄ and[16 ℄.

4. Theorems1.4.31.4.5generalizewhatwasobtainedforKawasakidynami swithonetypeof

Inthesepapers,theanaloguesof(H1)(H3)wereproved,(

Γ, C

⋆

_{, N )}

wasidentied,andbounds

on

K

werederivedthatbe omesharpinthelimitas

Λ → Z

2

. Whatmakesthemodelwithone

typeofparti lemoretra table isthatthesto hasti dynami s follows askeleton ofsub riti al

droplets that are squares or quasi-squares, as a result of a standard isoperimetri inequality

for two-dimensional droplets. For the modelwithtwo types ofparti les thistool isno longer

appli able and the geometry be omesharder, aswillbe ome lear inthetwo sequelpapers.

Similar results hold for Ising spins subje t to Glauber dynami s, as shown in Neves and

S honmann [20 ℄, and Bovier and Manzo [8℄. For this system,

K

has a simple expli it form. Theorems 1.4.31.4.5 are loseinspirit totheextension for Glauberdynami s whenan

alter-natingexternaleldisin luded,as arriedoutinNardiandOlivieri[18 ℄,andtotheextension

for Kawakasidynami s withone typeofparti lewhentheintera tionbetween parti lesis

dif-ferent inthe horizontal andtheverti aldire tion, as arriedout arriedout inNardi,Olivieri

and S oppola[19℄.

Our results an in prin iple be extended from

Z

2

to

Z

3

. For one type of parti le this

extension was a hieved in den Hollander, Nardi, Olivieri and S oppola [13℄, and Bovier, den

Hollander and Nardi [6℄. The geometry of the riti al droplet is more omplex in

Z

3

than

in

Z

2

, and this is also the ase for two types of parti les. All results arry over, ex ept that

C

⋆

and

N

have so far not been identied infull detail. Consequently, only upper and lower boundsare known for

K

and, sin e simple random walk on

Z

3

is transient, these boundsdo

not merge in the limit as

Λ → Z

3

. For Glauber dynami s the extension from

Z

2

to

Z

3

was

a hieved in Ben Arous and Cerf [2℄, and Bovier and Manzo [8℄, and

K

again has a simple expli it form.

5. InGaudillière, den Hollander, Nardi, Olivieriand S oppola[9℄, [10℄,[11 ℄, and Bovier, den

Hollander and Spitoni [7℄, the result for Kawasaki dynami s (with one type of parti le) on a

nite box with an open boundary obtained in[14 ℄ and [6℄ have been extended to Kawasaki

dynami s (with one type of parti le) on a large box

Λ = Λ

β

with a losed boundary. The volume of

Λ

β

grows exponentiallyfast with

β

,sothat

Λ

β

itselfa ts asa gasreservoir for the growing andshrinkingofsub riti aldroplets. Thefo usisonthetimeof therstappearan e

ofa riti aldropletanywherein

Λ

β

. Itturnsoutthatthenu leationtimein

Λ

β

roughlyequals the nu leation timeina nite box

Λ

divided bythe volume of

Λ

β

, i.e., spatial entropy enters into thegame. A hallengeisto derive asimilarresultforKawasakidynami s withtwo types

of parti les.

6. The model in the present paper an be extended by introdu ing three binding energies

U

11

, U

22

, U

12

< 0

for the three dierent pairs of types that an o ur in a pair of neighbor-ing parti les. Clearly, this will further ompli ate the analysis, and both

(X

meta

, X

stab

)

and

(Γ, C

⋆

, N )

maybedierent. Themodel isinteresting evenwhen

∆

1

, ∆

2

< 0

and

U < 0

,sin e this orrespondsto asituationwhere theinnitegasreservoirisverydenseandtendsto push

parti lesinto thebox. When

∆

1

< ∆

2

,parti lesoftype

1

tendtoll

Λ

beforeparti lesoftype 2 appear, but this is not the onguration of lowest energy. Indeed, if

∆

2

− ∆

1

< 4U

, then the bindingenergyisstrong enoughtostill favor ongurationswitha he kerboard stru ture

(moduloboundary ee ts). Identifying

(Γ, C

⋆

_{, N )}

seemsa ompli atedtask.

2 Proof of main theorems

Inthis se tionweprove Theorems 1.4.31.4.5subje tto hypotheses(H1)(H3). Se tions2.1

Nardi[6℄for Kawasakidynami swithonetypeofparti le. Infa t,we willseethat(H1)(H3)

aretheminimal assumptions needed toprove Theorems 1.4.31.4.5.

2.1 Diri hlet form and apa ity

The keyingredient of thepotential-theoreti approa h tometastabilityistheDiri hlet form

E

β

(h) =

1

2

X

η,η

′

_∈X

µ

β

(η)c

β

(η, η

′

)[h(η) − h(η

′

)]

2

,

h : X → [0, 1],

(2.1)

where

µ

β

isthe Gibbsmeasure denedin(1.4) and

c

β

isthekernel oftransitionrates dened in (1.5). Given a pair of non-empty disjoint sets

A, B ⊆ X

, the apa ity of the pair

A, B

is dened by

CAP

β

(A, B) =

min

h: X →[0,1]

h

_{|A≡1,h|B≡0}

E

β

(h),

(2.2)

where

h|

A

≡ 1

means that

h(η) = 1

for all

η ∈ A

and

h|

B

≡ 0

means that

h(η) = 0

for all

η ∈ B

. Theunique minimizer

h

∗

A,B

of (2.2) is alledtheequilibrium potential ofthepair

A, B

and isgiven by

h

∗

_A,B

(η) = P

η

(τ

A

< τ

B

),

η ∈ X \(A ∪ B).

(2.3)

Be ause

Λ

isnite,foreverypairofnon-emptydisjointsets

A, B ⊆ X

thereexist onstants

0 < C

1

≤ C

2

< ∞

(depending on

Λ

and

A, B

) su h that

C

1

≤ e

βΦ(A,B)

Z

β

CAP

β

(A, B) ≤ C

2

∀ β ∈ (0, ∞).

(2.4)

See [6℄,Lemma 3.1.1. Theseboundsarereferredto asa priori estimates.

An important onsequen e of (H1)(H2) and (2.4) is that

{, ⊞}

is a metastable pair in the sense of Bovier,E kho,Gayrard andKlein [5℄:

lim

β→∞

max

η /

∈{,⊞}

µ

β

(η)[CAP

β

(η, {, ⊞})]

−1

min

_η∈{,⊞}

µ

β

(η)[CAP

β

(η, {, ⊞}\η)]

−1

= 0.

(2.5)

Indeed, (1.4) , (H1)(H2) and the lower bound in (2.4) give that the numerator is bounded

from above by

e

[V

⋆

_−H()]β

/C

1

= e

(Γ−δ)β

/C

1

for some

δ > 0

, while (1.4) , thedenition of

Γ

and the upper bound in (2.4) give that the denominator is bounded from below by

e

Γβ

_/C

2

, withtheminimumbeingattainedat



. See[6 ℄, Lemma 3.2.2.

Thepropertyin (2.5)hasa further important onsequen e, namely,

E

_

(τ

_⊞

) = [Z

β

CAP

β

(, ⊞)]

−1

[1 + o(1)]

as

β → ∞.

(2.6)

See [6℄, Proposition 3.2.3. Thus, the proof of Theorem 1.4.4 revolves around getting sharp

bounds on

Z

β

CAP

β

(, ⊞)

. The a priori estimates in (2.4) serve as a jump board for the derivation ofthese bounds.

2.2 Graph stru ture of the energy lands ape

View

X

as agraph whose verti es arethe ongurations and whose edges onne t ommuni- ating ongurations,i.e.,

(η, η

′

₎

is anedgeifand only if

c

β

(η, η

′

_{) > 0}



X

∗

thesubgraph of

X

obtained byremoving all verti es

η

with

H(η) > Γ

and alledges in ident to these verti es;



X

∗∗

thesubgraphof

X

∗

obtainedbyremovingallverti es

η

with

H(η) = Γ

andalledges in ident to these verti es;



X



and

X

⊞

the onne ted omponents of

X

∗∗

ontaining



and

⊞

,respe tively.

By (H1)(H2)and the denitionof

Γ

,we have

X



= {η ∈ X : Φ(η, ) < Γ = Φ(η, ⊞)},

X

⊞

= {η ∈ X : Φ(η, ⊞) < Γ = Φ(η, )},

(2.7)

with

X



and

X

⊞

dis onne ted in

X

∗∗

.

Wenowhaveall thegeometri ingredients fortheproofof Theorems1.4.31.4.5 alongthe

lines of [6℄, Se tion 3. Our hypotheses (H1)(H3) repla e Propositions 2.3.72.3.8, Theorem

2.3.10 and Propositions 2.4.12.4.2in[6 ℄.

2.3 Proof of Theorem 1.4.4

Proof.Ourstarting point is(2.6) . Re alling (2.12.3),our taskisto show that

Z

β

CAP

β

(, ⊞) =

1

₂

X

η,η

′

_∈X

Z

β

µ

β

(η)c

β

(η, η

′

) [h

∗

,⊞

(η) − h

∗

,⊞

(η

′

)]

2

= [1 + o(1)] Θ e

−Γβ

as

β → ∞,

(2.8)

and to identify the onstant

Θ

,sin e (2.8) will imply (1.15) with

Θ = 1/K

. Thisis done in four steps.

1. For all

η ∈ X \X

∗

we have

H(η) > Γ

, and so there exists a

δ > 0

su h that

Z

β

µ

β

(η) ≤

e

−(Γ+δ)β

. Therefore, we an repla e

X

by

X

∗

in the sum in (2.8) at the ost of a prefa tor

1 + o(e

−δβ

)

. Moreover, asshownin[6 ℄,Lemma 3.3.1, thereexist

C < ∞

and

δ > 0

su h that

min

η∈X

_

h

∗

,⊞

(η) ≥ 1 − Ce

−δβ

,

_η∈X

max

⊞

h

∗

_,⊞

(η) ≤ Ce

−δβ

.

(2.9)

(The proof given in [6℄ uses a renewal argument in ombination with (2.3) and the a priori

estimates in(2.4) .) Therefore, on theset

X



∪ X

⊞

,

h

∗

,⊞

istrivial and its ontribution to the sumin(2.8) analso be putinto theprefa tor

1 + o(1)

. Consequently,all thatisneededisto understand what

h

∗

,⊞

lookslikeon theset

X

∗

\(X



∪ X

⊞

) = {η ∈ X

∗

: Φ(η, ) = Φ(η, ⊞) = Γ}.

(2.10)

However,

h

∗

,⊞

isalso trivialon theset

X

∗∗

\(X



∪ X

⊞

) =

I

[

i=1

X

i

,

(2.11)

whi hisaunionofwells

X

i

,

i = 1, . . . , I

,in

S(, ⊞)

forsome

I ∈ N

. (Ea h

X

i

isaminimalset of ommuni ating ongurations withenergy

< Γ

and with ommuni ation height

Γ

towards both



and

⊞

.) Namely,asshownin[6℄, Lemma 3.3.2,insideea h

X

i

we have

for some onstant

C

i

∈ [0, 1]

, and thereforethe ontribution to thesum in (2.8) of the tran-sitions inside awell an also be putinto theprefa tor

1 + o(1)

. Thus,only the transitionsin and out of wells ontribute.

2. Inview of the above observations, theestimation of

Z

β

CAP

β

(, ⊞)

redu es to the study of asimpler variationalproblem, namely,

Z

β

CAP

β

(, ⊞) = [1 + o(1)] Θ e

−Γβ

(2.13) with

Θ = min

C

1

...,C

I

min

h: X ∗→[0,1]

h

_|X



≡1, h|X

⊞

≡0, h|X

i

≡Ci ∀ i=1,...,I

1

2

X

η,η

′

_∈X

∗

1

{η↔η

′

_}

[h(η) − h(η

′

)]

2

.

(2.14)

See [6℄,Proposition 3.3.3. Thisredu tionuses that

• Z

β

µ

β

(η)c

β

(η, η

′

) = 1

{η↔η

′

_}

e

−Γβ

forall

η, η

′

_{∈ X}

∗

thatarenot eitherbothin

X



orboth in

X

⊞

or bothin

X

i

for some

i = 1, . . . , I

.

To he k the latter, note that there are no allowed moves between these sets, so that either

H(η) = Γ > H(η

′

)

or

H(η) < Γ = H(η

′

₎

for allowed moves in and out of these sets. In

(2.14) , the approximate boundary onditions given by (2.9) and (2.12) arerepla ed by sharp

boundary onditions. Combining (2.6) with (2.132.14), we see that we have ompleted the

proofof (1.15)with

K = 1/Θ

. Thevariationalformulafor

Θ

isnon-trivial be auseitdepends on thegeometry ofthe wells

X

i

,

i = 1, . . . , I

.

3. So far we have only used (H1)(H2). In theremainder of the proof we use (H3) to prove

(1.16). Thisgoesasfollows. Whenthefree parti leatta hesitself to theproto ritial droplet,

thedynami senters theset

C

⋆

att

. Theentran e ongurationsof

C

⋆

att

areeitherin

X

⊞

or inone of the

X

i

's. Inthe former ase the path an rea h

⊞

while staying below

Γ

inenergy, in the latter aseit annot. By(H3), ifthe path exitsan

X

i

,thenforittoreturn to

X



itmustpass through

C

⋆

,i.e.,itmustgothroughaseriesof ongurations onsistingofasingleproto riti al

droplet and a free parti le moving away from that proto riti al droplet towards

∂

−

_Λ

. Now,

this ba kward motionhasa smallprobability be ause simple randomwalk in

Z

2

isre urrent,

namely, the probability is

[1 + o(1)] 4π/ log |Λ|

as

Λ → Z

2

(see [6 ℄, Eq. (3.4.5)). Therefore,

the free parti le is likely to re-atta h itself to the proto- riti al droplet before it manages to

rea h

∂

−

_Λ

. Consequently,witha probabilitytending to 1as

Λ → Z

2

,before thefreeparti le

manages to rea h

∂

−

_Λ

it will re-atta h itself to theproto riti al droplet inall possible ways,

whi h in ludes a way su h that the dynami s enters

X

⊞

. In other words, after entering

C

⋆

att

the path is likely to rea h

X

⊞

beforeit returnsto

X



, i.e., itgoes over thehill. Note that in thelimit as

Λ → Z

2

the wells

X

i

be omeirrelevant,and onlythetransitionsinand outof

X



and

X

⊞

areimportant.

Remark 2.3.1 Theproto riti aldroplet may hange ea h timethepath enters and exitsan

X

i

. There are

X

i

's from whi h the path an rea h

⊞

without going ba k to

C

⋆

and without

ex eeding

Γ

in energy (see[6℄,proof ofTheorem 1.4.3).

4. Asaresultof the above observation,

Θ

an besandwi hed between two sumsof apa ities involvingsimplerandomwalkon

Λ

+

startingon

∂

+

_Λ

The sum runs over all possible lo ations and shapes of the proto riti al droplet. See [6 ℄,

Proposition 3.3.4, for details. In thelimit as

Λ → Z

2

, ea h of these apa ities hasthe same

asymptoti behavior, namely,

[1 + o(1)] 4π/ log |Λ|

,irrespe tiveofthe lo ationandshape ofthe proto riti al droplet (provided it is not too lose to

∂

+

_Λ

,whi h is a negligible fra tionof the

possiblelo ations). Therefore the on lusion isthat

Θ = K

−1

growslike

[1 + o(1)] 4π/ log |Λ|

timesthenumberof proto riti aldropletsin

Λ

. Thelatternumbergrows like

[1 + o(1)] N |Λ|

,

and sowe have proved (1.16) .



2.4 Proof of Theorem 1.4.5

Proof. Theproof isimmediate from Bovier,E kho, Gayrard and Klein[5℄, Theorem1.3(iv),

andthe observationsmadeinSe tions2.12.3. Themainideaisthat,ea htimethedynami s

rea hes the riti aldroplet but fails to go over thehilland falls ba kinto thevalley around



, it has a probability exponentially lose to 1 to return to



be ause, by (H2),



lies at the bottom of its valley(re all (2.3) and (2.9))and start froms rat h. Thus, thedynami s

manages to grow a riti al droplet and go over the hill only after a number of unsu essful

attempts that tendsto innityas

β → ∞

,ea h having asmall probabilitythat tendsto zero as

β → ∞

. Consequently,the timeto gooverthehillisexponentiallydistributedonthes ale

of itsaverage.



2.5 Proof of Theorem 1.4.3

Proof. See [6℄,Se tion 3.5. The proofrelies onthefa tthat the onguration spa e

X

an be partitioned into surfa eswitha xednumber ofparti lesof type 2,i.e.,

X =

|Λ|/2

[

n=0

V

_n

2

,

V

_n

2

= {η ∈ X : n

2

(η) = n}.

(2.15)

Clearly,allpathsfrom



to

⊞

mustpassthroughallthesesurfa es,and

G(, ⊞) ⊆ V

2

n

∗

with

n

∗

thenumberofparti lesoftype2in

C

⋆

. Theoptimaltransitionsfrom

V

2

n

to

V

2

n+1

o urwhento a ongurationwithminimalenergyin

V

2

n

aparti leoftype2isaddedsomewhereat

∂

−

_Λ

. For

the ase

n = n

∗

the transitionsmustall beoptimal, sin e

P

onsistsofthose ongurations in

V

_n

2

∗

₋₁

with minimalenergy

Γ − ∆

2

,andthe ongurationsin

C

⋆

∂

−

_Λ

areobtained fromthosein

P

byaddingaparti le oftype

2

at

∂

−

_Λ

. Therefore,withea h onguration

η ∈ P

ˆ

, onsisting ofaproto riti aldropletwith

n

∗

_{− 1}

parti lesoftype

2

,we anasso iateasetof ongurations in

C

⋆

∂

−

_Λ

, onsisting of the same proto riti al droplet asin

η

ˆ

and a free parti le anywhere at

∂

−

_Λ

. Sin e this freeparti le is equallylikely to appearat anylo ation in

∂

−

_Λ

,theentran e

into

C

⋆

∂

−

_Λ

from

P

o ursalong anoptimal path,and theentran e distribution isuniform. The rst laim in Theorem 1.4.3 also follows from the general analysisin Manzo, Nardi,

Olivieri andS oppola[17℄, Se tion5.



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