Article
Reference
Rigorous probabilistic analysis of equilibrium crystal shapes
BODINEAU, T., IOFFE, D., VELENIK, Yvan
Abstract
The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress
during the last decade. We review here the main results that have been obtained, both in two
and higher dimensions. In particular, we describe how the phenomenological Wulff and
Winterbottom constructions can be derived from the microscopic description provided by the
equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and
describe the central ideas of the existing approaches.
BODINEAU, T., IOFFE, D., VELENIK, Yvan. Rigorous probabilistic analysis of equilibrium crystal
shapes. Journal of mathematical physics, 2000, vol. 41, no. 3
DOI : 10.1063/1.533180
Available at:
http://archive-ouverte.unige.ch/unige:6376
Disclaimer: layout of this document may differ from the published version.
1 / 1
RIGOROUS PROBABILISTIC ANALYSIS OF EQUILIBRIUM CRYSTAL
SHAPES
T. BODINEAU, D. IOFFE, AND Y. VELENIK
Abstract.
The rigorous microscopic theory of equilibrium crystal shapes has made
enormous progress during the last decade. We review here the main results which have
been obtained, both in two and higher dimensions. In particular, we describ e how the
phenomenological Wul and Winterbottom constructions can b e derived from the mi-
croscopic description provided by the equilibriu m statistical mechanics of lattice gases.
We fo cus on the main conceptual issues and describe the central ideas of the existing
approaches.
Contents
Part 1. Introduction
4
1.1. Phenomenological Wul construction 4
1.1.1. Equilibrium crystal shap es 4
1.1.2. Variational methods 5
1.1.3. Stability prop erties 7
1.1.4. Winterb ottom problem 8
1.1.5. Microscopic justication 10
1.2. Microscopic Mo dels 11
1.2.1. Models with nite-range ferromagnetic 2-b ody interactions 11
1.2.2. 2D nearest-neighbors ferromagnetic Ising mo del 12
1.2.3. Kac models 14
1.2.4. Surface tension 15
1.3. Scope of the theory 16
1.3.1. Dobrushin-Koteck y-Shlosman Theory 16
1.3.2.
L
1
-Theory 16
1.3.3. Boundary Phenomena 17
1.3.4. Bibliographical review 17
Part 2.
L
1
-Theory
21
2.1. Results and the strategy of the pro of 22
2.1.1. Main results 22
2.1.2. Exponential tightness 23
2.1.3. Precise logarithmic asymptotics 23
2.1.4. Scheme of the pro of 24
2.2. Coarse graining and mesoscopic phase lab els 24
2.2.1. Tightness theorem for mesoscopic phase labels 25
2.2.2. Relation to magnetization proles 26
2.3. Examples of mesoscopic phase lab els 26
Date
: March 5, 2000.
1
2 T. BODINEAU, D. IOFFE, AND Y. VELENIK
2.3.1. Kac potentials 27
2.3.2. Bernoulli b ond percolation 28
2.3.3. Ising nearest neighbor. 30
2.4. Surface tension 31
2.4.1. FK representation 31
2.4.2. Extended representation 32
2.5. Lower b ound : Prop osition 2.1.2 32
2.5.1. Step 1 : Approximation procedure. 33
2.5.2. Step 2 : Lo calization of the interface. 33
2.5.3. Step 3 : Surface tension. 34
2.6. Upper b ound : Prop osition 2.1.3 34
2.6.1. Step 1 : Approximation procedure. 34
2.6.2. Step 2 : Minimal section argument. 35
2.6.3. Step 3 : Surface tension estimates. 36
2.7. Open problems 37
Part 3. Dobrushin-Kotecky-Shlosman (DKS) theory in 2D
38
3.1. Main Result 38
3.1.1. Heuristics 38
3.1.2. DKS theorem 39
3.1.3. DKS theory 40
3.2. Estimates in the phases of small contours 42
3.2.1. Structure of lo cal limit estimates 42
3.2.2. Basic local estimate on the
K
log
N
scale 43
3.2.3. Super-surface estimates in the restricted phases 43
3.3. Bulk Relaxation in Pure Phases 44
3.3.1. Non-positive magnetic elds
h
0. 44
3.3.2. Positive magnetic elds
h >
0. 45
3.3.3. Phases of small contours 45
3.4. Calculus of Skeletons 46
3.4.1. Denition 46
3.4.2. Energy estimate 46
3.4.3. Calculus of skeletons 47
3.4.4. Skeleton lower b ound 48
3.5. Structure of The Pro of 49
3.5.1. Lower b ound 49
3.5.2. Upper b ounds 50
3.6. Open Problems 50
Part 4. Boundary eects
52
4.1. Wall free energy 52
4.2. Surface phase transition 54
4.3. Derivation of the Winterbottom construction 55
4.3.1. 2D Ising model 57
4.3.2. Ising model in
D
>
3 59
4.4. The tools 59
4.4.1. 2D Ising model 59
4.4.2. Ising model in
D
>
3 67
3
4.5. Open problems 67
Part 5. Appendix
69
5.1. Appendix A : Pro of of Theorem 2.2.1 69
5.1.1. Estimate on the volume of zero
u
k
-blocks. 69
5.1.2. Peierls estimate on the size of large contours. 70
5.1.3. Estimate in the phase of small contours. 70
5.2. Appendix B : Pro of of the three-p oint lower b ound Lemma 3.4.3 71
References 73
4 T. BODINEAU, D. IOFFE, AND Y. VELENIK
W
(
V
) =
R
@ V
(
~n
x
) d
H
(
d
1)
x
x
~n
x
@ V
Vapor
Crystal
Figure 1.
The free energy of the crystal-vap or interface is given by the integral of the
anisotropic surface tension
over
@ V
.
H
(
d
1)
is the (
d
1)-dimensional Hausdor
measure.
Part
1.
Introduction
1.1.
Phenomenological Wulff construction
1.1.1.
Equilibrium crystal shapes.
The phenomenological theory of equilibrated crys-
tals dates back at least to the b eginning of the century [Wu]. Suppose that two dierent
thermodynamic phases (say crystal and its vap or) coexist at a certain temperature
T
.
Assuming that the whole system is in equilibrium, in particular that the volume
v
of the
crystalline phase is well dened, what could b e said ab out the region this phase o ccupies?
Of course, the issue cannot b e settled in the language of bulk free energies - these do not
depend neither on the shap e, nor even on the prescribed volume
v
of the crystal. Instead,
possible phase regions are quantied by the value of the free energy of the crystal-vap or
interface, or by the total surface tension b etween the crystal and the vap or
1
. Equilibrium
shapes corresp ond, in this way, to the regions of minimal interfacial energy. This is an
isoperimetric-type problem: The surface tension
(where, throughout the article,
de-
notes the inverse temp erature,
= 1
=T
) is an anisotropic function of the lo cal direction of
the interface. Thus, assuming that the crystal occupies a region
V
R
d
, the correspond-
ing contribution
W
(
V
) to the free energy is equal to the integral of
over the b oundary
@ V
of
V
(Fig. 1).
The Wul variational problem could then b e formulated as follows:
(
WP
)
v
W
(
V
)
!
min Given : vol(
V
) =
v
As in the usual isop erimetric case (WP)
v
is scale invariant,
8
s >
0
;
W
@
(
sV
)
=
s
d
1
W
@ V
:
Consequently, any dilatation of an optimal solution is itself optimal, and one really talks
here in terms of optimal shap es.
The canonical way to produce an optimal shap e is given by the following Wul con-
struction (Fig. 2): Dene
K
=
\
~n
2
S
d
1
n
x
2
R
d
:
x
~n
(
~n
)
o
=
\
~n
2
S
d
1
H
(
~n
)
:
(1.1.1)
1
In this review, our p oint of view is that of mathematical physics; for an exposition of the problem from
the viewp oint of theoretical physics, we refer to [RW] and references therein.
