Strict convexity of the free energy for non-convex gradient models at moderate $β$

0.22 MB·English

by  Codina Cotar

Strict convexity of the free energy for non-convex
gradient models at moderate β
Codina Cotar , Jean-Dominique Deuschel , and Stefan Mu¨ller
∗†‡ ∗§ ∗¶
8
0 February 2, 2008
0
2

n
a
J Abstract

8
We consider a gradient interface model on the lattice with interaction

] potentialwhichisanon-convexperturbationofaconvexpotential. We show
h
using a one-step multiple scale analysis the strict convexity of the surface
p
tension at high temperature. This is an extension of Funaki and Spohn’s
-
h result [10], where the strict convexity of potential was crucial in their proof.
t
a
m AMS 2000 Subject Classification. 60K35, 82B24, 35J15
[ Key words and phrases. effective non-convex gradientinterfacemodels,surface

tension, strict convexity, Helffer-Sjo¨strand representation
1
v
8
1 Introduction
0
3
1
We consider an an effective model with gradient interaction. The model describes
.
1 aphaseseparationinRd+1,eg. betweentheliquidandvaporphase. Forsimplicity
0 we consider a discrete basis Λ Zd, and continuous height variables
8 M ⊂
0
: x Λ φ(x) R.
v ∈ M −→ ∈
i
X
Thismodelignores overhangs like inIsingmodels,butgivesagoodapproximation
r in the vicinity of the phase separation. The distribution of the interface is given
a
in terms of its Gibbs distribution with nearest neighbor interactions of gradient
type, that is, the interaction between two neighboring sites x,y depends only on
the discrete gradient, φ(x,y) = φ(y) φ(x). More precisely, the Hamiltonian is
∇ −
of the form
H (φ) = V(φ(y) φ(x)) (1.1)
M
−
x,y∈ΛMX+1,|x−y|=1
∗Corresponding Author
†Supported by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physical
systems’
‡TU Berlin - Fakult¨at II, Institut fu¨r Mathematik, Strasse des 17. Juni 136, D-10623 Berlin,
Germany. E-mail: [email protected]
§TU Berlin - Fakult¨at II Institut fu¨r Mathematik Strasse des 17. Juni 136 D-10623 Berlin,
Germany. E-mail: [email protected]
¶Max Planck Institute for Mathematics in the Sciences Inselstrasse 22-26 D-04103 Leipzig,
Germany. E-mail: [email protected]
where V C2(R) is a function with quadratic growth at infinity:
∈
V(η) Aη 2 B, η R (1.2)
≥ | | − ∈
for some A > 0,B R.
∈
For a given boundary condition ψ R∂ΛM, where ∂Λ = Λ Λ , the
M M+1 M
∈ \
(finite) Gibbs distribution on RΛM+1 at inverse temperature β > 0 is given by
1
β
µ (dφ) exp( βH (φ)) dφ(x) δ (dφ(x)).
VM,ψ ≡ Zβ − M ψ(x)
M,ψ x∈YΛM x∈Y∂ΛM
β
Here Z is a normalizing constant given by
M,ψ
β
Z = exp( βH (φ)) dφ(x) δ (dφ(x)).
M,ψ RΛM+1 − M ψ(x)
Z x∈YΛM x∈Y∂ΛM
One is particularly interested in tilted boundary conditions
d
ψ (x) =< x,u>= x u
u i i
i=1
X
for some given ’tilt’ u Rd. This corresponds to an interface in Rd+1 which stays
∈
normal to the vector n = (u, 1) Rd+1.
u
− ∈
Anobject ofbasicrelevance inthiscontext is thesurfacetension orfreeenergy
defined by the limit
1
β
σ(u) = lim logZ . (1.3)
M −β M,ψu
→∞
The existence of the above limit follows from a standard sub-additivity argu-
ment. In fact the surface tension can also be defined in terms of the partition
function on the torus, see below and [10]. In case of strictly convex potential V
with
′′
c V c (1.4)
1 2
≤ ≤
where0 < c c < ,FunakiandSpohnshowedin[10]thatσ isstrictlyconvex.
1 2
≤ ∞
The simplest strictly convex potential is the quadratic one with V(η) = η 2,
| |
which correspondsto a Gaussian model, also called gradient free field or harmonic
crystal. Models with non quadraticpotentials V are sometimes called anharmonic
crystals.
The strict convexity of the surface tension σ plays a crucial role in the deriva-
tion of the hydrodynamical limit of the Landau-Ginsburg model in [10].
Under the condition (1.4), a large deviation principle for the rescaled profile
withratefunctiongivenintermsoftheintegrated surfacetensionhasbeenderived
in [7]. Here also the strict convexity of σ is very important. Both papers [10] and
[7]usevery explicitely thecondition (1.4)intheir proof. Inparticular they rely on
the Brascamp Lieb inequality and on the random walk representation of Helffer
and S¨jostrand, which requires a strictly convex potential V.
The objective of our work is to prove strict convexity of σ also for some non
convex potential. One cannot expect strict convexity for any non convex V, see
2
below. Our result is perturbative at high temperature (small β), and shows strict
convexity of σ(u) at every u R for potentials V of the form
∈
V(η) = V (η)+g (η)
0 0
where V satisfies (1.4) and g C2(R) has a negative bounded second derivative
0 0
such that √β·kg0′′kL1(R) is sma∈ll enough.
Ourproofisbasedonthescaledecompositionofthefreefieldasthesumoftwo
independentfreefieldsφ andφ ,wherewechoosethevarianceofφ smallenough
1 2 1
to match the non-convexity of g. This particular type of scale decomposition was
used earlier by Haru Pinson in [11], who also suggested to us the use of this
β
approach. The partition function Z can be then expressed in terms of a
N,ψu
double integral, with respect to both φ and φ . We fix φ and perform first the
1 2 2
integration with respect to φ . This yields a new induced Hamiltonian, which is
1
a function of the remaining variable φ . The main point is that our choice of the
2
variance of φ and smallness of β allow us to show convexity in φ of the induced
1 2
Hamiltonian. Of course this Hamiltonian is no longer of the simple form (1.1),
in particular we lose the locality of the interaction. However an extension of the
technique introduced in [7] shows strict convexity of σ. The idea behind the proof
is that one can gain convexity via integration. This procedure is called ”one step
decomposition”, since we perform only one integration. Of course this procedure
couldbeiteratedwhichwouldallowtolowerthetemperature. However forgeneral
non convex g we do not expect that this procedure works at low temperature for
every tilt u.
At low temperature an approach in the spirit of [5], [4] looks more promising
[1].
Finally note that, due to the gradient interaction, the Hamiltonian has a con-
tinuous symmetry. In particular this implies that no infinite Gibbs state exists
for the lower lattice dimensions, d = 1,2 where the field ”delocalizes” as M ,
→ ∞
c.f. [8]. On the other hand, it is very natural in this setting to consider the
gradient Gibbs distributions, that is the image of µ under the gradient op-
VM,ψ
eration φ RZd φ. It is easy to verify that this distribution depends only
∈ −→ ∇
on ψ, the gradient of the boundary condition, in fact one can also introduce
∇
gradient Gibbs distributions in terms of conditional distributions satisfying DLR
conditions, c.f. [10]. Using the quadratic bound (1.1), one can easily see that
the corresponding measures are tight. In particular for each tilt u Rd one can
∈
construct a translation invariant gradient Gibbs state µ˜ on Zd with mean u:
u
E [φ(y) φ(x)] =<y x,u > .
µ˜u − −
Under (1.4), Funaki and Spohn proved the existence and unicity of extremal,
ie. ergodic, gradient Gibbs state, for each tilt u R. In the case of non convex V,
∈
unicity of the ergodic states can be violated, even at u = 0 tilt, c.f. [2]. However
in this situation, the surface tension is not strictly convex at u= 0.
2 Main result and outline of the proof
We study the convexity properties of the free energy (as a function of the tilt
u) for non-convex gradient models on a lattice. Using the results of [10], we
3
work on the torus, instead of the box Λ , see Remark 2.4 below. Thus, let
M
Td = (Z/MZ)d = Zd mod (M) be the lattice torus in Zd, let u Rd and let
M ∈
β > 0. For a function φ: Td R, we consider the discrete derivative
M →
φ(x) =φ(x+e ) φ(x) (2.5)
i i
∇ −
and the Hamiltonian
d
H(u,φ) = Vi( φ(x)+u )+gi( φ(x)+u ) , (2.6)
0 ∇i i 0 ∇i i
xX∈TdMXi=1(cid:2) (cid:3)
where Vi is convex and gi is non-convex (see (2.11) below). We consider the
0 0
partition function
Zβ (u) = e βH(u,φ)m (dφ), (2.7)
M − M
ZX
where
X = φ : Td R :φ(0) = 0 (2.8)
{ M → }
and
m (dφ) = dφ(x)δ (dφ(0)), (2.9)
M 0
x∈TYdM\{0}
and the free energy
1
β β
f (u) = logZ (u). (2.10)
M −β M
We will prove
Theorem 2.1 Suppose that Vi and gi are C2 functions on R and that there exist
0 0
constants C ,C ,C and
0 1 2
0 < C (Vi) C , C (gi) 0. (2.11)
1 ≤ 0 ′′ ≤ 2 − 0 ≤ 0 ′′ ≤
Set
C C
C¯ = max 0, 2 1,1 . (2.12)
C C −
(cid:18) 1 1 (cid:19)
If (gi) L1(R) and for i 1,2,...,d
0 ′′ ∈ ∈ { }
4 1 1
π(12dC¯)1/2 βC1C ||(g0i)′′||L1(R) ≤ 2, (2.13)
1
p
then
C
(D2fβ )(u) 1 Td Id, u Rd, (2.14)
M ≥ 2 | M| ∀ ∈
where Td = Md denotes the number of points in Td . In other words, the free
| M| M
energy per particle is uniformly convex, uniformly in M.
4
Remark 2.2 Themainpointisthattheconvexityestimate(2.14)holdsuniformly
in the size M of the torus. Indeed a direct calculation of D2f1 yields at u
M
D2f1 (u) = D2H(u, ) var D H(u, ), (2.15)
M u · H − H u ·
(cid:10) (cid:11)
where
f(φ)e H(u,φ)m (dφ)
f = X − M (2.16)
h iH e H(u,φ)m (dφ)
R X − M
and R
var f = (f f )2 . (2.17)
H −h iH H
D E
Now one might expect that a condition like (2.13) implies that (D2H(u, )
u · H ≥
cC Td Id(seeLemma4.1below). Theproblemisthatnaively thevarianceterm
1| M| (cid:10) (cid:11)
scales like Td 2 since D H is a sum of dTd terms. To get a better estimate,
| M| u | M|
one has to show that in a suitable sense, the terms
cov (D (V +g )(u+ φ(x)),D (V +g )(u+ φ(y))) (2.18)
H u 0 0 i u 0 0 j
∇ ∇
decay if x y islarge. IfH isnotconvex suchadecay ofcorrelations is,presently,
| − |
only proved for the class of potentials studied in [6]. As discussed above, the
Helffer-Sjo¨strand estimates do not apply directly. The main idea is to rewrite
β
Z (u) as an iterated integral in such a way that each integration involves a con-
M
vex hamiltonian to which the Helffer-Sjo¨strand theory can be applied (see (2.42)
below).
Remark 2.3 Instead of ||g0′′||L1(R) one can also use boundson lower order deriva-
tives. More precisely, condition (2.13) can, for example be replaced by
50 1 1
√2πdC¯(βC1)3/4C1||g0′||L2(R) ≤ 2 (2.19)
(see Remark 4.2 below). In view of the estimate
R(g0′)2(s)ds = Rg0(s)g0′′(s)ds ≤ C0||g0||L1(R), (2.20)
Z Z
we can see that (2.13) can be replaced by
1 1
cd2C¯3(βC1)3/2C ||g0||L1(R) ≤ 4 (2.21)
1
with c = 2500.
2π
Remark 2.4 Note that the surface tensions defined in (1.3) and (2.10) coincide
(see, for example, [10]). Because of this, we will work from now on with the
definition of the surface tension on a torus, as it is easier to use.
5
Outline of the proof for Theorem 2.1
Step 1: Scaling argument
A simple scaling argument shows that it suffices to prove the result for
β = 1,C = 1. (2.22)
1
Indeed, suppose that the result is true for β = 1 and C = 1. Given β, Vi and gi
1 0 0
which satisfy (2.11) and (2.13), we define
V˜i(s)= βVi s , g˜i(s)= βgi s . (2.23)
0 0 √βC 0 0 √βC
(cid:18) 1(cid:19) (cid:18) 1(cid:19)
Then
1 (V˜i) C2, C0 (g˜i) 0,
≤ 0 ′′ ≤ C − C ≤ 0 ′′ ≤
1 1
||(g˜0i)′′||L1(R) = βC1C1 ||(g0i)′′||L1(R). (2.24)
1
p
Hence V˜i, g˜i satisfy the assumptions of Theorem 2.1 with β = 1 and C = 1.
0 0 1
Thus
D2f1 (,V˜i,g˜1) 1 Td Id. (2.25)
M · 0 0 ≥ 2| M|
On the other hand, the change of variables
φ˜(x) = βC φ(x),u˜ = βC u (2.26)
1 1
p p
yields
V˜i u˜ + φ˜(x) = Vi(u+ φ(x)) (2.27)
0 i ∇i 0 ∇i
(cid:16) (cid:17)
and thus
ZMβ (u,V0i,g0i) = (βC1)−(|TdM|−1)/2ZM1 (u˜,V˜0,g˜0). (2.28)
Hence
1
fβ (u,V ,g ) = const(β,C )+ f1 βC u,V˜,g˜ . (2.29)
M 0 0 1 β M 1 0 0
(cid:16)p (cid:17)
Thus (2.25) implies (2.13), as claimed.
Step 2: Separation of the Gaussian part
Next we separate the Gaussian part in the Hamiltonian. From now on, we will
always assume that β = 1 and C = 1. Set
1
1
V (s) = V (s) s2, g = V +g . (2.30)
1 0 1 0
− 2
Then
0 V C 1, C g C 1 (2.31)
≤ 1′′ ≤ 2− − 0 ≤ ′′ ≤ 2−
6
and the Hamiltonian can be rewritten as
d
1
H(u,φ) = (u + φ(x))2+G(u,φ), (2.32)
i i
2 ∇
xX∈TdM Xi=1
where
d
G(u,φ) = g(u + φ(x)). (2.33)
i i
∇
xX∈TdM Xi=1
Since for all functions φ on the torus and for all i 1,2,...d
∈ { }
φ(x) = 0, (2.34)
i
∇
xX∈TdM
we get
1 1
H(u,φ) = Td u2+ φ 2+G(u,φ), (2.35)
2| M|| | 2||∇ ||
where φ 2 = d φ(x)2. Let
||∇ || x TM i=1|∇i |
∈
P P
Z0 = e−12||∇φ||2mM(dφ). (2.36)
ZX
Then the measure
µ = 1 e−21||∇φ||2mM(dφ) (2.37)
Z
0
is a Gaussian measure. Its covariance C is a positive definite symmetric operator
on X (equipped with a standard scalar product (φ,ψ) = φ(x)ψ(x)) such
x Td
∈ M
that
P
(C 1φ,φ) = φ 2, φ X. (2.38)
−
||∇ || ∀ ∈
The partition function thus becomes (recall that we take β = 1)
ZM(u) = Z0e−12|TdM||u|2 e−G(u,φ)µ(dφ). (2.39)
ZX
Step 3: Decomposition of µ and Helffer-Sjo¨strand calculus
By standard Gaussian calculus, µ = µ µ , where µ and µ are Gaussian with
1 2 1 2
∗
covariances
C = λC, C = (1 λ)C, where λ (0,1). (2.40)
1 2
− ∈
More explicitly, for i 1,2
∈{ }
1 1 φ 2
µi(dφ) = e−2λi||∇ || mM(dφ), where λ1 = λ,λ2 = 1 λ. (2.41)
Z −
i
7
Thus
ZM(u) = Z0e−21|TdM||u|2 e−G(u,ψ+θ)µ1(dθ)µ2(dψ). (2.42)
ZXZX
Towritethefreeenergyinamorecompactform,weintroducetherenormalization
maps R . For f C(Rd X) we define R f by
i i
∈ ×
e Rif(u,a) := e f(u,a+b)dµ (b). (2.43)
− − i
ZX
Taking the logarithm of (2.42), we get
1
f (u) = const(M)+ Td u2+(R R G)(0,u). (2.44)
M 2| M|| | 2 1
The main point now is that the map
1
H (θ) = G(u,ψ+θ)+ θ 2 (2.45)
1
2λ||∇ ||
becomes uniformly convex for sufficiently small λ. This will allow us to use the
Helffer-Sjo¨strand representation to get a good lower bound for D2(R G), which
1
involves, roughly speaking, the expectation of G (θ) =g (u + ψ(x)+ θ(x))
i,x 0′′ i ∇i ∇i
with respect to e H1 (see (4.82)). This expectation can be controlled in terms of
−
||g0′′||L1(R) (see Lemma 4.1). Under the smallness condition (2.13) one then easily
obtains the lower bound for D2(R R G) (see (4.85) and (4.87)).
2 1
3 Consequence of the Helffer-Sj¨ostrand representation
LetUandX befinite-dimensionalinnerproductspaces,letC beapositivedefinite
symmetric operator on X and let µ be the Gaussian measure with covariance C
C
on X, i.e
µC(db) = 1 e−12(C−1b,b)db, (3.46)
Z
C
where db is the dim X dimensional Hausdorff measure on X (i.e db = db if
i
the b are the coordinates with respect to an orthonormal basis). For a continuous
i
Q
function f C(U X) we define R f by
C
∈ ×
e RCf(u,a) = e f(u,a+b)dµ (db). (3.47)
− − C
ZX
In the situation we will consider, b f(u,a+b)+ 1(C 1b,b) will be convex and
→ 2 −
hence bounded from below so that the right hand side of the above identity is
strictly positive.
For f C2(U X) we write D2f(u,a) for the Hessian at (u,a), viewed as an
∈ ×
operator from U X to itself. The restriction of the Hessian to X is denoted by
×
D2 f := P D2fP , where P is the orthogonal projection U X X. On the
X X X X × →
level of quadratic forms we thus have
D2 f(u,a)(u˙,a˙),(u˙,a˙) = D2f(u,a)(0,a˙),(0,a˙) . (3.48)
X
(cid:0) (cid:1) (cid:0) (cid:1)
8
From the Helffer-Sjo¨strand representation of the variance (see, e.g., [9] (2.6.15))
and the duality relation
1 1
A 1a,a = sup (a,b) (Ab,b) , (3.49)
−
2 − 2
1
b D(A2)(cid:18) (cid:19)
(cid:0) (cid:1) ∈
which holds for any positive definite self-adjoint operator A on a Hilbert space Y ,
0
one immediately obtains the following estimate:
Lemma 3.1 Supppose that H C2(X), sup D2H < and there exists a
∈ X| | ∞
δ > 0 such that
D2H(a) δ Id, a X. (3.50)
≥ ∀ ∈
Set
Y = K L2 (X) : DK 2 < , (3.51)
0 { ∈ loc | | H ∞}
(cid:10) (cid:11)
Y = K Y : D2K 2 < , (3.52)
{ ∈ 0 || ||HS H ∞}
where the derivatives are understood(cid:10) in the wea(cid:11)k sense and
∂2 2
DK2 2 := K (3.53)
k kHS ∂φ(x)∂φ(y)
x,y∈XTdM\{0}(cid:18) (cid:19)
denotes the Hilbert-Schmidt norm. Then for all G Y we have
∈
var G= sup 2(DG,DK) (DK,D2HDK) D2K 2 (3.54)
H − −k kHS H
K Y
∈
(cid:10) (cid:11)
Therefore
var G sup 2(DG,DK) (DK,D2HDK) . (3.55)
H ≤ − H
K Y
∈
(cid:10) (cid:11)
We will use (3.55) from Lemma 3.1 in the proof of the lemma below.
Lemma 3.2 Suppose that f C2(U X) and sup D2f < . Suppose
moreover that there exists a δ >∈ 0 such×that U×X| | ∞
D2f(u,a)+C 1 δ Id, (u,a) U X. (3.56)
−
≥ ∀ ∈ ×
Then Rf C2(U X) and for all u,u˙ U,a,a˙ X
∈ × ∈ ∈
(D2Rf)(u,a)(u˙,a˙),(u˙,a˙)
inf D2f(u,a+ )(u˙,a˙ DK(),(u˙,a˙ DK())
(cid:0) ≥ K Y · (cid:1) − · − · H,a
∈
+ (C (cid:10)1(cid:0)DK(),DK()) (cid:1)(cid:11) (3.57)
−
· · Hu,a
where (cid:10) (cid:11)
1
H (b) = f(u,a+b)+ (C 1b,b), (3.58)
u,a −
2
g(b)e Hu,a(b)db
−
g = . (3.59)
h iHu,a e Hu,a(b)db
R −
R
9
Proof
We have
e−Rf(u,a) = e−[f(u,a+b)+12(C−1b,b)]db. (3.60)
ZX
It follows from (3.56) that
1
f(u,a+b)+ C 1(a+b),(a+b) δ a+b 2 c (3.61)
−
≥ 2 | | −
(cid:0) (cid:1)
and standard estimates yield
1
f(u,a+b)+(C 1b,b) δ b 2 c 1+ a2 . (3.62)
−
≥ 4 | | − | |
(cid:0) (cid:1)
Hence, by the dominated convergence theorem, the right-hand side of (3.60) is
a C2 function in (u,a) and the same applies to Rf since the right-hand side of
(3.60) does not vanish.
Toprovetheestimate (3.57)forD2Rf,wemayassumewithoutlossofgeneral-
ity thata= 0,u = 0(otherwisewecanconsidertheshiftedfunctionf( u, a)).
·− ·−
Set
h(t) := Rf(tu˙,ta˙). (3.63)
Then
h (0) = D2(Rf)(0,0)(u˙,a˙),(u˙,a˙) . (3.64)
′′
Now (cid:0) (cid:1)
h(t) = log e f(tu˙,ta˙+b)µ (db), (3.65)
− C
−
ZX
e f(tu˙,ta˙+b)Df(tu˙,ta˙ +b)(u˙,a˙)µ (db)
h(t) = X − C (3.66)
′
e f(tu˙,ta˙+b)µ (db)
R X − C
and R
h (0) = D2f(0, )(u˙,a˙),(u˙,a˙) var Df(0, )(u˙,a˙), (3.67)
′′ · H − H ·
where (cid:10)(cid:0) (cid:1)(cid:11)
1
H(b) = f(0,b)+ (C 1b,b). (3.68)
−
2
By assumption,
D2H(b) δ Id, (3.69)
≥
i.e. H is uniformly convex. (cid:3)
Hence by (3.55) from Lemma 3.1
var g inf 2(Dg,DK)+(DK,D2HDK) . (3.70)
− H ≥ K Y − H
∈
(cid:10) (cid:11)
10