Commun.
Math.
Phys.
172,
551-569
(1995)
Communications
ΪΠ
Mathematical
Physics
©
Springer-Verlag 1995
The
Analysis
of the
Widom-Rowlinson
Model
by
Stochastic
Geometric
Methods
J.T.
Chayes
1
,
L.
Chayes
1
,
R.
Kotecky
2
1
Department
of
Mathematics,
UCLA,
Los
Angeles,
CA
90024,
USA.
Email:
jchayes@math.ucla.
edu;
lchayes@math.ucla.edu
2
Center
for
Theoretical
Study,
Charles
University, Ovocny
trh 3, 116 36
Praha
1,
Czech
Republic
E-mail:
kotecky@cspunil2.bitnet
Received:
7
July
1994
Abstract:
We study the continuum Widom-Rowlinson model of interpenetrating
spheres. Using a new geometric representation for this system, we provide a simple
percolation-based proof of the phase transition. We also use this representation to
formulate the problem, and prove the existence of an interfacial tension between co-
existing
phases. Finally, we ascribe geometric (i.e. probabilistic) significance to the
correlation
functions which
allows
us to prove the existence of a sharp correlation
length in the single-phase regime.
1.
Introduction
1A.
Background
and
statement
of
results.
The Widom-Rowlinson model [WR] is
a
simple and beautiful model of continuum particles. It is of interest both because
of it applicability in the description of continuum systems, and because it is the
only continuum system for which a phase transition has been rigorously established
[R].
The Widom-Rowlinson (WR) model has two equivalent standard formulations
-
one as a binary gas and the other as a single-species model of a dense (liquid)
phase in contact with a rarefied (gas) phase. In the binary gas formulation, the
only interaction is a hard-core exclusion between the two species of particles -
call them A and B. There is no intraspecies interaction: two particles of the same
type can interpenetrate
freely.
The phase diagram of the model is a function of the
fugacities, ZA and ZB, of the two species. Clearly, there is a symmetry between A
and
B particles; hence z
A
= z
B
— z is a line of symmetry of the phase diagram. For
both
the continuum and lattice versions of the model, it has been shown via Peierls'
arguments that for z
large
enough, the symmetry is spontaneously broken, yielding
two phases: one is ^4-rich and the other is
B-ήch
[LG, R]. The transition between
these phases is first-order. It is expected, but not proved that the line ZA — z
B
= z
of first-order transitions ends in a critical point at some positive value z — z
c
of
the
common fugacity. The single-species formulation of the model is obtained by
Partly supported by the grants GACR 202/93/0499, GAUK 376, NSF-DMS 91-04487, and NSF-
DMS
93-02023
552 J.T. Chayes,
L.
Chayes,
R.
Kotecky
integrating out the coordinates
of
one
(say
the
B)
species (see below). The
effective
diameter
of
the particles
of
the remaining species
is
then twice the original diameter.
The
phase transition
in
the binary formulation maps into
a
liquid-vapor transition
in
the
single-species version.
The
purpose
of our
work
is to
show
how the
geometric ideas
can be
used
to
study
the
phase transition
and the
interfaces between
the
coexisting
A and B
or
liquid
and gas
phases
of the
Widom-Rowlinson model.
The key
ingredient
in
our
study
is the
introduction
of a new
stochastic representation
for the
Widom-
Rowlinson measure.
We
obtain our new representation
by
first
generating percolation
configurations
of
spherical particles, identifying groups
of
particles with overlapping
cores
as
being
in
the same cluster.
We
then "color" each particle either
A or
B. Given
the
hard-core constraint
of the
Widom-Rowlinson model,
the
only configurations
which
are
allowed (i.e.
the
only ones which receive non-zero
weight
in
the Widom-
Rowlinson measure)
are
those
for
which particles
in the
same cluster
are
either
all
A
or all B.
Otherwise,
the
problem
is
unconstrained.
The
result
is
that
all
allowed
configurations have
weights
which depend exponentially
on the
number
of
clusters
within them. Anyone
who is
familiar with
the
Fortuin-Kasteleyn representation
of
the
Potts model [FK] should
see
immediately that
our
representation does
for the
Widom-Rowlinson model what
the
Fortuin-Kasteleyn representation does
for the
Potts
model. Indeed,
the
symmetries
of
the Widom-Rowlinson model
are
manifest
in
our new
representation.
We
use our new
representation
to
establish many properties
of the
Widom-
Rowlinson model.
In
Sect.
2, we
apply percolation methods
and use a
continu-
tum
analog
of
FKG-domination lemmas
to
present
a new
proof
of the
existence
of
a
phase transition
in two and
higher dimensions. This proof
is
completely
self-
contained
and represents
a
conceptual simplification
of
the
classic
works
on
the
sub-
ject
[R,
LL, GL].
In
Sect.
2, we
also describe
the
symmetry-broken phase
in
terms
of percolation
and
give
an
appropriate order parameter
for the
transition.
A new
proof
of
the FKG property used
for
this characterization
is
given
in the
Appendix;
the
original proof can
be
found
in
[LM,
CGLM].
In
Sect.
3, we use
monotonicity
properties
of
the representation
to
establish the existence
of
an interfacial
or
surface
tension
between coexisting phases. This
is the
first
proof
of
existence
of a
surface
tension
in a
continuum model.
In
Sect.
4, we
prove
the
existence
of a
correlation
length
for
the two-dimensional model
in
the single-phase regime.
In
order
to do
this,
we introduce
several
correlation functions
and
bound them
in
terms
of
each other.
Existence
of a
correlation length
is
then established using
one of
these functions.
IB.
The
model.
The
Widom-Rowlinson model
is a
classical statistical mechanics
system
of
interacting particles.
To
define the model
we
write the
interaction
energy,
UN(XI,
'
,XN),
of N
particles located
at the
points
X\,X2,...,XN
E
IR
J
as
follows:
For
any y e
IR^,
we
define
the
halo
of y to be the
ball
of
radius
2a
with
the
center
at y, h(y) = {x
eJR.
d
: \x
~ y\ < 2a}.
The halo
of a set F is
the union
of
the
halos
of
its points, h(F)
=
l)
ye
Fh(y). The energy
UN(XI,-.-,XN)
of
the configuration
(x\,...,
XN
)
is
just
the difference
of
the volume V{x\
,...,##)
of
the halo h(x\
,...,##)
and
the sum
Σf
=ι
|h(jc, )|,
U
N
(xι,...
9
x
N
)=V(xi
9
...,x
N
)-NVo
9
(1.1)
where
F
o
is the
volume
of a
ball
of
radius
la.
Although
the
interaction
(1.1) is
(pairwise) attractive, achieving
a
potential minimum
at
zero separation,
the
system
Widom-Rowlinson Model
553
is, overall, //-stable
0
£ U
N
(x
u
...,x
N
) £ -NVo, (1.2)
due
to
multiparticle effects that saturate
the
attraction.
The
grand-canonical partition function
at
fugacity
z and
inverse temperature
β
is
defined
in the
usual fashion:
Let A c
IR^
be the
interior
of
some (regu-
lar) finite
vessel.
We use the
notation
ω
N
=
(x\,...
9
x
N
) and dω^ = d
d
x\
d
d
xχ.
Depending
on the
particular "boundary condition"
η, we
write
V
η
(ωj\[)
(and
correspondingly
U^(ω
N
)).
Three natural boundary conditions
are:
free (F), attrac-
tive
(A), and
repulsive
(R).
These
are
introduced
by
taking
V
F
(ωχ)
=
|h(ω#)
Π
Λ\,
V
R
(ω
N
)
=
\h(ω
N
)\,
and
V
A
(ω
N
)
=
\h(ω
N
)Π
A
a
\,
where
A
a
C A is
obtained
by
deleting from
A all
points within
a
distance
α of the
boundary
dΛ. It is not
hard
to
see
that attractive boundary conditions actually
favor
the
presence
of
particles
near
the
boundary while repulsive boundary conditions tend
to
repel them. Then
the
grand-canonical partition function
is
s?\*,/ϊ) = Σ**z?;(/0,
(i 3)
N
where
(1.4)
iV \
Λ
A
priori
this model does
not
seem
to
have
any
distinguishing features that would
make
the
study
of
the liquid-gas transition particularly tractable.
In
an
equivalent formulation
of
the model (also introduced
in
[WR]), one consid-
ers
two
species
of
particles,
A and B.
Here, there
is a
hard-core exclusion between
A
and B
particles
but no
interaction between pairs
of A or
pairs
of B
particles.
Formally,
if
there
are
particles located
at the
positions
x\
and X2,
we may put
and
f
0 if
\x\
—
xi\
> 2a ,
VAB{XI,XΪ)=\
' . (1-6)
^
oo
otherwise
.
The
grand canonical partition function,
at
fugacities
ZA
and
z
B
in a
volume
A is
given
by
where
4) (1.8)
with
χηicoffiωff)
defined
to be
zero
if the
above described hard-core condition
is
violated
in the
configuration (ω^,ω^) with
the
boundary condition
η, and
unity
otherwise. Relevant here
are the
",4-only" boundary conditions, where each point
of
dA
is
deemed
to be
occupied
by an A
particle,
the
"/?-only,"
and the
free boundary
conditions.
Equations
(1.3) and (1.7)
enable
one to
define
the
grand-canonical Gibbs
mea-
sures
μ®l
β
and μfj^.
554 J.T. Chayes, L. Chayes, R. Kotecky
Equivalence of the two-component system (1.5) - (1.8) to the one-component
model
(1.1)—(1.4) is readily demonstrated. Simply fix the configuration ω^ of A
particles and integrate over configurations ωjy of B particles. It is seen that each
of the B particles moves freely through the region A
\h(ω^),
yielding a factor of
|/l\h(ω^)|
to the N
th
power. Summing over N
gives,
up to irrelevant constants,
the
partition function (1.4) with interaction (1.1) at an effective temperature given
by ZB = β. Here, as it turns out, the measure μχ
Ά
z z
transforms, after integrating
out
the 2?-particles, into the measure μ^
Ά
e
_
βVo β
, where η = A boundary condi-
tions
transform into the attractive boundary conditions, ή — A, B into repulsive, and
free into free boundary conditions. In the next section we
will
define yet a third
representation
for this model.
2.
The
Phase
Transition
2A. The
Gray
Representation.
We call the proposed new description of the Widom
Rowlinson system the
gray
or
color-blind
representation. Let A c Wi
d
and, for the
moment,
let us ignore the boundary conditions on A.
Also,
in order to simplify this
section, let us consider only hypercubic
1
subsets of IRΛ Denote by ω
N
= (xi,... ,*#)
any configuration of N points in A and by s^ any of the 2^ conceivable colorings
(i.e.
assignments of the A and B labels) of the N given particles. Weighting A and
B particles equally, we may write the configurational partition function for the TV
particles as
Z
Λ,N
= 77y / dω
N
ds
N
χ
η
N
(ω
N
,s
N
) , (2.1)
^
-
s
N
,ω
N
where
Γ 1 if the configuration
{ω
N
,s
N
}
is "allowed"
XN(CO
N
,S
N
)=< „ (2.2)
t
0 if the configuration
{CON,S
N
}
IS forbidden
with given boundary condition η. For any particle k— 1,...,N, centered at any
point
jty e /t, we may define the core region
c(x
k
)
=
{xeti
d
\\x-Xk\
<a}. (2.3)
If
a>N
is a configuration, the set
=
U c(**) (2.4)
consists of distinct components or clusters. Two particles in ω# are said to be core
connected
if they belong to the same component.
It
is evident that if c(x;) and c(x
y
) overlap for some / and j, then
χ
η
N
((ύN,s
N
)
will
vanish unless / and j belong to the same species. This observation obviously
generalizes: each separate cluster of ω# must be composed of a single species. The
number
of
ways
in which this can be arranged is clearly 2^^
ωN
\ where
^(CUN)
= # of components of c(ω v) with the boundary condition η . (2.5)
1
It turns out, however, that all our results hold in more generality, e.g. in the case in which we take
the
thermodynamic limit in the sense of van Hove.
Widom-Rowlinson Model
555
Of course, Eq. (2.5) must
be
carefully interpreted
in
the presence
of
specific
boundary conditions.
In
the simplest cases, the interpretation
is
straightforward
—
in
particular,
for
free
(or
periodic) boundary conditions, ^
F
(ω
N
)
(or
^
p
(ω
N
))
is
just
the number
of
clusters.
Also
of
interest
in
this section
are the
"^4-only"
boundary conditions, where each point
of dΛ is
deemed
to be
occupied
by an
A
particle. Here the number
of
clusters
^(ω^)
counts
all
clusters connected
to
dΛ
a
= {x
G
A,
dist(jc, dA)
^ a} as a
single
cluster. For this system,
we
will
denote
the
associated
gray
measure
by
μ^
z
( ).
Hence
we
can write (2.1)
as
ZΪJ
=
jji I
dω
N
2*™
. (2.6)
In
particular,
we
may express the relative probability
of
the
"gray"
configuration,
ω#,
as
Prfj(ω
N
)<x^^dω
N
.
(2.7)
We may also define
a
corresponding grand-canonical
gray
measure
at
fugacity
z: μj
5
/. From Eq. (2.7)
it is
seen that
if we
consider
μj*
conditioned
on the
jV-particle
state, then
the
Radon-Nikodym derivative
of
this conditional measure
relative
to
the Poisson point process
at
intensity
z
(also conditioned on
N
particles)
is precisely 2^
η(a)N
\
A major advantage
of
the color-blind formalism
is
that
it
allows
a
compari-
son between the
WR
system and
an
ideal gas. The comparison
we
have
in
mind
is the continuum analog
of
FKG-type dominations that
are
widely
used
in
lattice
systems. For continuum problems
we
proceed
as
follows.
Let
CON
=
(x\,...,x
N
) and
%
=
(y\,...,
JVM)
denote
N
and
M
particle configurations, respectively. We say that
(DM
>-
OJN
if
for each
k we
can find
a j
such that Xk
= yj.
Somewhat
less
precisely,
ω
M
>-
ω
N
if ω
M
D ω
N
.
An event
stf is
said to be increasing
if, for
any
ω
G «s/,
it is
the case that
η
G
si
for
all η
>-
ω. In other words, the event
si is
never destroyed
by
adding particles
to
a configuration
in
which
it
occurs.
If
μ&
and
v
Λ
are two (grand-canonical) measures,
we
say
that
μ
Λ
(-)
£ v
Λ
(-) (2.8)
FKG
if
μ^{si)
^
VA(S/)
whenever
si is an
increasing event.
The
following
is a
continuum analog
of a
result that
is
standard
in
discrete
systems. (It can also
be
obtained
as a
special case
of
Theorem 9.1
of
[P] or Lemma
2.1
of
[J].)
Proposition
2.1. Let
μ
Λ
( )
denote
a
grand-canonical
measure
for
indistinguishable
particles
on
some
A C
IR
J
with
N-particle
conditional
measures
dμΛ,N(
ω
N)
=
j7-
}
W
N
(ω
N
)dω
N
with
W
N
(ω
N
) a.e.
continuous,
positive
and
satisfying
the
usual
stability
hypothesis
(i.e. WM
^ e
bN
for
some
b <
oo).
For y
G
A,
regard
(co^,y)
as
an
N
-f-1
particle