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\topmatter
\title
Exact Macroscopic Description of Phase Segregation in Model Alloys with
Long Range Interactions
\endtitle
\leftheadtext\nofrills
{G.Giacomin and J.L.Lebowitz}
\rightheadtext\nofrills
{A phase segregation model}
\author
\centerline{Giambattista Giacomin}
\centerline{Institut f\"ur Angewandte Mathematik}
\centerline{Universit\"at Z\"urich--Irchel}
\centerline{Winterthurer Str. 190, CH--8057 Z\"urich Switzerland}
\bigskip
\centerline{Joel L. Lebowitz}
\centerline{Department of Mathematics}
\centerline{Hill Center, Rutgers University}
\centerline{New Brunswick, N.J. 08903, U.S.A.}
\endauthor
\abstract
We derive an exact nonlinear non-local macroscopic equation for the time
evolution of the conserved order parameter $\rho({\text{\bf r}}, t)$ of a
microscopic model binary
alloy undergoing phase segregation: a $d$--dimensional
lattice gas
evolving via Kawasaki exchange dynamics, satisfying
detailed balance for a Hamiltonian with a long range pair potential
$\gamma^d J(\gamma \vert
x \vert)$.
The
macroscopic evolution is on the spatial scale
$\gamma^{-1}$ and time scale $\gamma^{-2}$, in the
limit $\gamma \to 0$. The domain coarsening, described by
interface motion, is
similar to that obtained from the Cahn-Hilliard equation.
\hfill\break
\phantom{a}\hfill\break
\phantom{a}\hfill\break
Pacs numbers: 05.20.-y, 02.30Jr, 02.50Cw, 64.75+g, 64.70.Kb
\endabstract
\endtopmatter
\document
\baselineskip=19pt
The process of phase segregation in alloys, following a quench (sudden
cooling) from a high temperature into the miscibility gap,
is a
problem of great practical importance and theoretical interest.
The time evolution of
the local macroscopic order parameter, e.g.\ the concentration of A-atoms
in a binary A-B alloy, is commonly described by the nonlinear Cahn-Hilliard
equation (CHE)[1]. While the derivation of this equation is based on
arguments embodying deep physical
insight, there does not exist (to our knowledge) any microscopic
model system, for which the CHE, or any modifications of it
proposed so far [1], gives the exact macroscopic
description.
In this note we rigorously derive a macroscopic equation, eq.(7),
describing phase
segregation in systems with long range interactions, the same models whose
equilibrium properties are described by the van der Waals equation of state
(with the Maxwell construction) [2]. This equation, while structurally
different from the CHE,
yields similar behavior for the late stages
of the coarsening process; this regime appears to be quite universal,
see [1].
We begin by giving a simple heuristic derivation of a general CHE as an
adaptation of the nonlinear
diffusion equation
$$
{\partial \rho({\bold r}, t) \over \partial t} =
\nabla \cdot [D(\rho) \nabla
\rho] = \nabla\cdot[\sigma(\rho) \nabla \mu_{eq}(\rho)]
= \nabla \cdot\left[\sigma(\rho) \nabla \left(
{\delta F_{eq} \over \delta \rho}\right)
\right]
\tag1
$$
describing the evolution of a nonuniform macroscopic concentration
profile $\rho (\bold{r} ,t)$ toward the uniform equilibrium state.
In (1)
$F_{eq} = \int f_{eq}(\rho) d{\bold r}$,
$D$ is the diffusion coefficient,
$\mu_{eq}(\rho) = { f_{eq}^\prime (\rho)}$ is the
chemical potential, given by the derivative of the equilibrium Helmholtz
free energy per unit volume $f_{eq}(\rho)$ in a system with uniform density
$\rho$, and $\sigma$ is the Onsager mobility,
$\sigma = \chi D$, where $\chi = ({\partial
\mu_{eq} /\partial \rho})^{-1}$ is the compressibility (or
susceptibility).
Eq. (1)
can be derived
rigorously for a variety of microscopic model systems when ${\bold r}$ and
$t$ are in units scaled
by $\epsilon^{-1}$ and $\epsilon^{-2}$ respectively, compared to the
microscopic units,
in the limit $\epsilon \to 0$~[3].
The dynamics in these models are generally
stochastic (although some deterministic examples are also available),
conserve (only) particle number and have
the equilibrium Gibbs distribution
as the only stationary state.
This includes the Ising model with Kawasaki dynamics
for $T>T_c$ in $d=2$, [4], and the continuous spin Ginzburg--Landau model
with conservative Langevin dynamics,
for all $T$ and all dimensions [5]. In the coexistence region
the mobility $\sigma(\rho)$, which is always given by a Green-Kubo formula ,
remains finite while
$\mu_{eq} (\rho)$ is constant, so $\chi^{-1} = 0$ and
$D(\rho) = \sigma(\rho)/\chi (\rho) = 0$ there [6]. Hence,
when the range of densities
in a nonuniform system lies
within the miscibility gap, the right--hand side of (1) vanishes. This may
seem at first paradoxical but one should remember that we are considering
in (1) density variations on the macroscopic
spatial scale $\epsilon^{-1}$ which is very large compared to the
interaction range. This means that regions of size $\epsilon^{-1}$
can contain
essentially equilibrated domains of both
phases. Variations of the density on the spatial scale $\epsilon^{-1}$ can
therefore be compensated for by changes in the volume fraction of the two
phases. Of course the system will still try to aggregate further to reduce
interfacial area but this evolution will occur on a time scale larger than
$\epsilon^{-2}$, so it is not seen in (1).
The CHE with some mobility $\sigma(\rho)$ can be obtained formally [1] by
replacing $F_{eq}$ (and thus $\mu_{eq}(\rho)$) in the right
side of (1) by a free energy functional
$\tilde F(\{\rho({\bold r}, t)\})$
$$
\tilde F = \int d{\bold r}\left[\tilde f(\rho) + {1 \over 2} \zeta ({\bold
\nabla} \rho)^2\right],
\ \ \ \ \ \ \mu = \tilde f^{\prime} (\rho) - \zeta \nabla^2 \rho
\tag2
$$
where $\tilde f (\rho)$ is a {\sl constrained equilibrium} free energy
density which
has a single minimum above $T_c$ and a double well shape at $T < T_c$, and
$\zeta$ is a constant related to the surface tension between the
phases. This gives the general CHE
$$
{\partial \rho \over \partial t} = \nabla \cdot
\left\{\tilde \sigma(\rho) \nabla
\left[ \tilde f^{\prime} (\rho) - \zeta
\nabla^2 \rho\right]\right\}.
\tag3
$$
$\tilde f(\rho)$ is often taken to be a quadratic polynomial in $\rho$ and
$\tilde \sigma(\rho)$ set equal to a constant: one then obtains the
{\it standard} CHE [1,7].
As noted, however, the model systems yielding (1) give ${\partial
\rho /\partial t} =
0$ in the miscibility gap, where $f_{eq}^{\prime \prime}(\rho) = 0$, so (3)
does not seem to have any microscopic derivation.
To overcome this problem and find a macroscopic equation which
describes exactly the phase segregation of some microscopic model,
we consider the dynamics of a system
in which the interaction responsible for phase
segregation has a range, $\gamma^{-1}$,
which is large compared to the interparticle
spacings. The equilibrium properties
of such a system are well known [2]: here we consider
the evolution of the order parameter on the spatial
scale $\gamma ^{-1}$ (or a hundred times as large).
The particles live in a cube
$\Lambda_{\gamma}$
of side $\ell \gamma^{-1}$, on
the $d$-dimensional lattice
${\Bbb Z}^d$.
The time evolution of the configuration $\eta_t$
at time $t$,
specified by $\eta_t ({x}) = 0$ or $1$ for all ${ x} \in
\Lambda_{\gamma}$,
proceeds via a Kawasaki type exchange dynamics,
specified by giving the rate for the exchange of the occupancy of
nearest neighbor sites ${ x}$ and ${ y}$ (using periodic boundary
conditions) in the configurations $\eta$, i.e.\ for the transition
$\eta \to \eta^{x,y}$. We take this rate to be
$$
c_{\gamma}(x,y;\eta) = \exp\left\{-{1 \over 2} \beta \left[H_{\gamma}
(\eta^{x,y}) -
H_{\gamma} (\eta)\right]\right\}
\tag4
$$
where $\beta \geq 0$ is the reciprocal temperature, and
$$
H_{\gamma} (\eta) = -{1 \over 2} \sum_{x,y \in \Lambda_{\gamma}} \gamma^d
J(\gamma(x-y)) \eta(x) \eta(y)
\tag5
$$
and $J({\bold r}) \geq 0$ is a smooth function
with $\int J({\bold r}) d{\bold r} =
\alpha$. We choose $\ell$ to be much larger than the range of
$J({\bold r})$, so we don't have to worry about
periodicity, and for simplicity let $J({\bold r})$
depend only on $\vert {\bold r}\vert$; the
anisotropic case can be treated too [8].
Assume furthermore that $J$ is decreasing in $\vert
\bold{r} \vert$. The dynamics (4) satisfies
detailed balance w.r.t.,
the Gibbs measure $Z^{-1} _N \exp[-\beta H_{\gamma}
(\eta)]$, with any fixed total number of particles $N = \sum_{x \in
\Lambda_{\gamma}} \eta({\bold x})$, which is the unique stationary measure
(for $N$ fixed).
We are interested in initial conditions
described by a probability distribution
($\text{Prob}_{\gamma}$) on the
configuration space
($\text{Prob}_{\gamma}$
can be concentrated on one configuration) such that
when $\gamma \to 0$, the typical microscopic
configuration resembles more and more a smooth
profile $\rho_0$ stretched by $\gamma^{-1}$, where $\rho_0$ is a smooth
function from $T^d$, the $d$-dimensional torus of length $\ell$,
to [0,1]. More precisely, we shall require
that for any continuous
function $\phi({\bold r})$ on $T^d$ and every $\delta > 0$
$$
\lim_{\gamma \rightarrow 0}
\text{Prob}_\gamma \left(
\left \vert
\gamma ^d \sum_{x \in \Lambda_\gamma} \phi (\gamma x) \eta (x)-
\int_{T^d} \phi ({\bold r}) \rho_0({\bold r}) \text{d}{\bold r}
\right\vert >\delta
\right) =0.
\tag6
$$
Condition (6) is clearly satisfied if $\text{Prob}_{\gamma}$
is such that $\langle \eta (x) \rangle =
\rho_0(\gamma { x})$ for all ${ x}$ in $ \Lambda _\gamma$ and
the occupation number of the sites are independent but this is not
required. We could, for example, also have
initial conditions concentrated
on single periodic configurations, etc.
Our Theorem now says that (6)
holds also in the limit $\gamma \to 0$ for $\eta_{t\gamma^{-2}}$,
if we replace $\rho_0$ with $\rho({\bold r},t)$,
the solution of the integrodifferential equation.
$$
\partial _t\rho({\bold r},t)=
\nabla\cdot \left[
\nabla \rho({\bold r},t)-\beta \rho({\bold r},t)(1-\rho({\bold r},t))
\int _{T^d} \nabla J ({\bold r}-{\bold r}^\prime )
\rho ({\bold r}^\prime) \text{d} {\bold r}^\prime
\right]
\tag7
$$
with initial value $\rho({\bold r},0)=\rho_0({\bold r})$.
Eq.(7) is valid for all $\beta$ and
$t \in [0, \infty)$, including the miscibility gap.
Existence and uniqueness
of a suitable weak solution of (7) globally in time
are part of the proof of this Theorem. Moreover it can be shown
that $\rho ({\bold r},t)$ is bounded between 0 and 1 and
that if $\rho_0 ({\bold r})$ is smooth, then $\rho({\bold r},t)$ is smooth.
Further results on the behavior of this particle system on
a macroscopic scale
can be found in [9].
The result stated here
is proven in [8] using the technique introduced in [10], see also [9]. It
is essentially a consequence of the fact that, in a nearest neighbor
exchange,
due to the long range
interactions, the energy
difference is of order $\gamma$. Hence, the dynamics is a
superposition of a Laplacian and a weak $(O(\gamma))$ asymmetric
perturbation which generates, on the macroscopic scale, a {\it force} term
given by the gradient of the (long range) energy density at ${\bold r}$,
$\int J({\bold r} - {\bold r}^{\prime}) \rho({\bold r}^{\prime}) d{\bold
r}^{\prime}$. This force term is multiplied by a conductivity $\beta
\rho(1 - \rho)$: $\beta$
measures the bias introduced in the exchange rates $c_\gamma$,
given in (4), by energy
differences, while $\rho(1 - \rho)$ gives the rate at which exchanges
actually take place when a particular site is picked at random and the
system is {\it locally described} by a product measure with average density
$\rho$.
We note that the product measure is the
correct equilibrium measure for our
{\it reference} system, i.e.\ the system
without the weak long range interaction
$J$.
The analysis and (7) remain unchanged if we replace (4) with a general
rate
$c_\gamma (x,y; \eta)=
\Phi \left(
\beta
\left[H_\gamma (\eta^{x,y})-H_\gamma (\eta )\right]
\right)
$
which satisfies the
detailed balance condition [3]
$
\Phi(E) =\Phi (-E) \exp(-E)
$
as long as we rescale our time by $\Phi(0)$ and $\Phi$ is smooth.
It is now an observation, which at first sight appears surprising, that (7)
can be rewritten in the form
$$
{\partial \rho({\bold r}, t) \over \partial t} = \nabla \cdot
\left[\sigma^0
(\rho)\nabla \left( {\delta {\Cal F}^0\over \delta \rho }
\right)\right]
\tag8
$$
where $\sigma^0(\rho) = \beta \rho(1 - \rho)$ and
$$
{\Cal F}^0(
\{ \rho\}) = \int f^0_c (\rho({\bold r})) d{\bold r} + {1 \over 4} \int
\int J({\bold r} -
{\bold r}^{\prime})\left[\rho({\bold r}) - \rho({\bold r}^{\prime})
\right]^2
d{\bold r} d{\bold r}^{\prime}
\tag9
$$
with $f^0_c(\rho) = f^0_{eq}(\rho) - {1 \over 2} \alpha
\rho^2$ and $f^0_{eq}(\rho) = -\beta^{-1} s(\rho); s(\rho) = -\rho \log
\rho - (1
- \rho) \log(1 - \rho)$.
$f^0_{eq}(\rho)$ is clearly just the free energy density of
the reference system while ${\Cal F}^0 (\{\rho\})$, which is a non-local
functional of the density, is just the total constrained free energy of this
system in the limit $\gamma \to 0$, see [2, 11]. For $\rho$ constant
the second term in (9) vanishes, and $f^0_c(\rho)$ is in fact the correct
equilibrium free energy density as long as $\beta \leq \beta_c = 1/T_c =
4\alpha^{-1}$. If $\beta > \beta_c$, $f^0_c(\rho)$ has a double minimum at
$\rho = \rho^{\pm}_{\beta}$, the two nontrivial solutions of $\log(\rho/(1
- \rho)) = \beta \alpha(\rho - 1/2)$, and the correct free energy is then
obtained by the double tangent construction [2, 11].
The mobility $\sigma^0 (\rho)$, as in (1), is just $\chi D$,
with $\chi=\chi^0 = \beta \rho(1 - \rho)$ and $D=D^0=1$ for the
reference
system [3].
It is
this, at first sight fortuitous, coincidence of $\chi^0$ with the
{\it conductivity} appearing in front of the long range force term on the right
site of (7) which makes it possible to transform (7) into the physically
more revealing form (8). Further thought shows however that the form (8)
is really forced by the requirement that the Gibbs measure $\exp\{-\beta
H_{\gamma}\}/ Z_N$ be stationary for the dynamics. This is most easily
seen by adding
to the reference system not $H_{\gamma}$ but a one body external potential,
$U_{\gamma} (\eta) = \sum_x U(\gamma x) \eta(x)$, in which case the
integral in (7) is replaced by $\nabla U({\bold r})$.
The coincidence is now simply
the Einstein relation (or linear response theory) between the conductivity
for the current induced by the weak perturbation $U_{\gamma}$ and the
diffusion coefficient $D^0$ of the reference system, c.f.\ ref.\ [2,
p. 233--234] and [12].
The superscript$\phantom{a}^0$ in (8) and (9) is to remind us that our reference
system is one in which there are no short range interactions (except for
the hard core exclusion preventing multiple occupancy at a site). We
consider now the macroscopic dynamics of a system in which there is also a
short
range interaction $H_s ({\bold \eta})$, making the total
energy $H = H_s + H_{\gamma}$.
The system evolves
microscopically according to a Kawasaki exchange rate $c_{\gamma}
(x,y;\eta)$ which satisfies detailed balance with respect to the Gibbs
measure, $\exp\{-\beta[H_s + H_{\gamma}]\}/Z_N$
($Z_N$ is the corresponding canonical partition function). The dynamics of this system,
at a temperature $\beta^{-1}$ above the critical temperature
of the reference system,
will again be a
superposition of diffusive dynamics for the reference system and a current
linear in the force coming from the weak long-range interaction. Using the
same reasoning as before, the conductivity coefficient in front of this
force, given by the integral in (7), must, by the stationarity of equilibrium measure
(Einstein relation), be equal to the mobility $\sigma$ of the reference system.
It is known [2, 11] that the constrained free energy of such a system are
obtained from those with $H_s = 0$ by replacing in the integrand of (9)
$f^0_{eq}(\rho)$ by $f_{eq}(\rho)$, the equilibrium free energy density of
the reference system with short range Hamiltonian $H_s(\bold \eta)$. Hence we expect
that the macroscopic evolution of such a system will again be given by an equation of
the form (8) and (9) without the
superscript$\phantom{a}^0$, i.e.\ $f^0_{c} \to f_{eq}-(1/2) \alpha
\rho^2$
(which may have two minima, so that there will be phase segregation)
and $\sigma^0 \to \sigma = \chi D$, where $D$ is the diffusion coefficient
(generally $\rho$ dependent) and $\chi$ the susceptibility of the reference
system. This has already been proven explicitly in some cases [13] and it is
surely quite general.
We shall now compare solutions of the evolution (7) with those of the
CHE ((2) and (3)) in the late stage of the domain
coarsening process [1]: at this stage
there are well defined
interfaces between domains of linear sizes $L$, large
compared to
that of the interface width, in which the densities
are close to those of the pure
phases, $\rho^{\pm}_\beta$. In this {\it sharp} interface
limit,
the CHE has been studied
analytically, mostly by means of formal matched asymptotic expansions
[7,14,15] and this type of analysis is extended in [8] to eq.(7).
Since in this regime the mean curvature
of the boundary of the domains is $O(1/L)$, the
local structure of the interface is given in first approximation by
the one dimensional
{\sl instantonic} solutions of (7). These are stationary solutions
$U_{\beta}(z)$ of (7) which depend only on one coordinate of
${\bold r}$ (in this case ${\bold r}$ is in $\real ^d$), say $z
= r_1$, which connect in a monotone increasing way
the low
($\rho_\beta ^-$)
and high
($\rho_\beta ^+$) density phases, i.e. $U_{\beta}(z) \to
\rho^{\pm}_\beta$ as $z \to \pm \infty$. Once we fix
$U_\beta (0)=1/2$, there is
only one (smooth) {\it instantonic} profile [16], given by the
solution of
$$
m_{\beta}(z) = \text{tanh} \left\{ {\beta \over 2}
\int \tilde J(z-z^{\prime})
m_{\beta}(z^{\prime}) dz^{\prime}\right\}
\tag10
$$where $m_{\beta}(z) = 2U_{\beta}(z) - 1$ and $\tilde J
(r_1)=\int_{\real^{d-1}}
J({\bold r}) \text{d} r_2 \ldots \text{d} r_d$.
Let us now consider the
evolution eq.(7) on
a torus
$L \Omega$, where $\Omega$ is a fixed $d-$dimensional torus.
Setting
${\bold x}=L^{-1}{\bold r}$ and $\tau_q = L^{-q} t$,
$
\rhe ({\bold x}, \tau _q )=
\rho (L {\bold x}, L^q \tau _q)
$,
the equation becomes
$$
L^{2-q} \partial _{\tau _q}\rhe ({\bold x}, \tau _q)
=\nabla \left[
\sigma (\rhe ({\bold x} ,\tau _q)) \nabla
\mu ^{L}({\bold x}, \tau _q)
\right]
\tag11
$$
in which
$
\mu^L(\rhe)({\bold x}, \tau) =
-s^\prime(\rhe ({\bold x}, \tau ))
-\beta \int
J_L( {\bold x}- {\bold x}^\prime ) \rhe ({\bold x}^\prime, \tau )
d{\bold x}^\prime
$
and $J_L (\bold x) = L^{d} J (L {\bold x})$.
We are interested in cases in which
$\rhe ({\bold x},0) =\rho_\beta^\pm +O(1/L)$ unless ${\bold x}$ is
at a distance
$O(1/L)$
from a (hyper)surface $\Gamma_0 \subset \Omega$,
which is the interface. In the thin ($O(1/L)$)
layer around the interface, $\rhe$ will be
approximated by a
stationary instanton.
The relevant time scale for such an initial condition
is given by $q=3$, as for the CHE [7,14]. A formal asymptotic
expansion, based mainly on the assumption that the interface
is stable, can be used (as in [7] for the CHE) to
study the evolution of the interface $\Gamma_{\tau_3}$
for $\tau_3$ positive. One obtains [8] that the
normal velocity of a point $x$ on $ \Gamma _{\tau_3}$
is given by
$$
V_3({\bold x})={ \sigma_\beta \over 2 \rho^+_\beta-1}
[\nu \cdot \nabla \mu_1]^+_- ({\bold x})
\tag12
$$
where $\nu({\bold x})$ is the
unit normal at the point ${\bold x}$ of the interface, pointing
in the direction of the denser phase;
$\sigma_\beta=\beta \rho^+_\beta (1-\rho^+_\beta)=
\beta \rho^-_\beta (1-\rho^-_\beta)$
is the mobility on the homogeneous
phase.
The expression between square brackets is the jump
of the normal derivative of the first order correction
to the chemical potential ($\mu^L=\mu_0+L^{-1} \mu_1+O(L^{-2})$)
across the interface, in the direction $\nu$.
Such a correction can be computed
by solving the following static problem
$$
\cases
\Delta \mu _1 ({\bold x}) =0 &\text{ for } {\bold x} \in \Omega_{\tau_3} ^+
\cup \Omega_{\tau _3}^- \\
\mu _1({\bold x})= -S_\beta K({\bold x})/( 2 \rho^+_\beta-1)
&\text{ for } {\bold x} \in \Gamma_{\tau _3}
\endcases
\tag13
$$
in which we have used the natural decomposition of
$\Omega$ into an interface and the domains $\Omega _{\tau_3}^\pm$
of the opposite
phases ($\Omega= \Omega _{\tau_3}^+ \cup \Omega _{\tau_3}^-
\cup \Gamma _{\tau_3}$). Moreover in (13) $\mu_1$
is a continuous function on $\Omega$ which is
at least twice differentiable in
$\Omega _{\tau_3}^+ \cup \Omega _{\tau_3}^-$,
$K({\bold x})$ is $(d-1)$ times the mean curvature of the interface at
${\bold x}$
and $S_\beta$ is the surface tension for the model
(i.e. the free energy density associated with the instantonic
solution $U_\beta$, see [17]) which can be written as
$$
S_\beta
= {1\over 2}\int _{\real \times \real}
\partial _z U _\beta (z) U_\beta (z^\prime)
(z^\prime-z)
\tilde J( z-z^\prime)
\text{d} z
\text{d} z^\prime
\tag14
$$
The geometric motion generated by $V_3({\bold x}) \nu
({\bold x})$
is known as the {\sl symmetric solidification
model} or {\sl Mullins--Sekerka} motion
[18].
A trivial extension
of [7], to include
a mobility term, shows that in the same scaling
the CHE gives rise to the same interface evolution
(12) and (13). The only change is in the
surface tension, which in the CHE case is given by,
$S_\beta =\zeta\int (\partial _z U^{CH}_{\beta} (z))^2 \text{d} z$,
where $U_\beta ^{\text{CH}}(z)$
is the instantonic solution for the CHE with $\beta $ dependence.
A way of introducing the temperature in the CHE is
given in [15] and, for $\beta$ finite, $U_\beta$ and
$U_\beta ^{\text{CH}}$ are qualitatively similar. However (see [8])
$U_\beta$ approaches a step function as $\beta$ approaches
infinity, which is not the case for $U_\beta ^{\text{CH}}$,
for fixed $\zeta$.
In [8] we argue that also on the diffusive time scale
($q=2$ in (11)), that is the case in which the density
in the bulk is not yet relaxed to the equilibrium one,
the CHE equation and (7) show a very similar behavior.
And the same is true in the limit $\beta \rightarrow \infty$, in which
$V_3$ (given in (12)) vanishes and the motion (surface diffusion)
has to be
observed on the time scale $q=4$: the analogous result
for CHE is shown in [15].
We conclude by noting that we have derived rigorously a macroscopic
equation describing phase segregation in alloys with long range
interactions $\gamma ^d J(\gamma x)$, whose only inputs are the equilibrium free energy density and
diffusion constant of the reference system. The late stage coarsening
described by this equation, in which the details of $J$
enter only in determining the surface tension,
is in the same universality class as the CHE
[1].
\vskip 0.4 cm
{\it
Acknowledgements:}
We would like to thank A.~Asselah, P.~Butt\`a, R.~Kohn, J.~Percus,
S.~Puri, J.~Quastel, H.~Spohn, L.~Xu,
H.T.~Yau and particularly E.~Presutti
for fruitful discussions.
Supported in part by NSF Grant 92-13424 4-20946, NASA Grant NAG3-1414
and the Swiss National Science Foundation.
\vskip 0.8 cm
\noindent
e-mail addresses:
\item{}gbg\@amath.unizh.ch
\item{}Lebowitz\@math.rutgers.edu
\vfill
\eject
\vskip 1 cm
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\vskip 1 cm
\baselineskip=15pt
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