BODY
\magnification=\magstep1
\hfuzz=2pt
\headline={\ifnum\pageno=1 \hfil\else\hss{\tenrm\folio}\hss\fi}
\footline={\hfil}
\hoffset=-.2cm
\font\ninerm=cmr9
\font\eightrm=cmr8
\font\titlefont= cmbx10 scaled \magstep3
\def\integer{\mathchoice{\rm I\hskip-1.9pt N}{\rm I\hskip-1.9pt N}
{\rm I\hskip-1.4pt N}{\rm I\hskip-.5pt N}}
\def\real{\mathchoice{\rm I\hskip-1.9pt R}{\rm I\hskip-1.9pt R}
{\rm I\hskip-.8pt R}{\rm I\hskip-1.9pt R}}
\def\Real{\real}
\def\Log{\mathop{\rm Log}}
\def\Romannumeral#1{\uppercase\expandafter{\romannumeral#1}}
\def\date{\line{\number\day/\number\month/\number\year\hfil}}
\def\expectation{\mathchoice{\rm I\hskip-1.9pt E}{\rm I\hskip-1.9pt E}
{\rm I\hskip-.8pt E}{\rm I\hskip-1.9pt E}}
\def\zinteger{{\rm Z\hskip-1.9pt\slash}}
\def\Oun{{\cal O}(1)}
\def\oun{{\hbox{\sevenrm o}(1)}}
\def\proof{\noindent{\bf Proof. }}
\def\longto{\mathop{\longrightarrow}}
\newskip\refskip\refskip=4em
\def\refsize{\advance\leftskip by \refskip}
\def\ref#1#2{\noindent\hskip -\refskip\hbox to
\refskip{[#1]\hfil}{\noindent #2\hfil}\medskip}
\def\proba{\mathchoice{\rm I\hskip-1.9pt P}{\rm I\hskip-1.9pt P}
{\rm I\hskip-.9pt P}{\rm I\hskip-1.9pt P}}
\def\logd{\log\log}
\def\shift{{\cal S}}
\def\bvnorm{|||}
\def\bvn{\bvnorm}
\def\bvnt{\bvnorm_{\theta}}
\def\exo{\par\noindent{\bf Exercise. }}
\font\gbf=cmmib10
\font\gbfs=cmmib8
\textfont12=\gbf
\scriptfont12=\gbfs
\let\mcd=\mathchardef
\mcd\alphag="7C0B \mcd\thetag="7C12 \mcd\varpig="7C24
\mcd\betag="7C0C \mcd\pig="7C19 \mcd\varrhog="7C25
\mcd\gammag="7C0D \mcd\Pig="7C05 \mcd\varsigmag="7C26
\mcd\deltag="7C0E \mcd\sigmag="7C1B \mcd\Omegag="7C0A
\mcd\varepsilong="7C22 \mcd\etag="7C11 \mcd\Thetag="7C02
\mcd\varphig="7C27 \mcd\omegag="7C21 \mcd\Sigmag="7C06
\mcd\psig="7C20 \mcd\kappag="7C14 \mcd\Deltag="7C01
\mcd\zetag="7C10 \mcd\lambdag="7C15 \mcd\Phig="7C08
\mcd\epsilong="7C0F \mcd\mug="7C16 \mcd\Gammag="7C00
\mcd\rhog="7C1A \mcd\xig="7C18 \mcd\Psig="7C09
\mcd\taug="7C1C \mcd\xig="7C18 \mcd\Lambdag="7C03
\mcd\upsilong="7C1D \mcd\nug="7C17 \mcd\Xig="7C04
\mcd\iotag="7C13 \mcd\varthetag="7C23 \mcd\Upsilong="7C07
\mcd\bfiw="7C77
\mcd\bfih="7C68
\mcd\bfix="7C78
\mcd\bfiU="7C55
\mcd\bfic="7C63
\def\E{\expectation}
\centerline{ CONSERVATIVE LANGEVIN DYNAMICS}
\centerline{OF SOLID ON SOLID INTERFACES}
\bigskip
\centerline{Pierre Collet$^1$, Fran\c cois Dunlop$^{1,3}$, and Thierry
Gobron$^2$}
\medskip
\centerline{$^1$ Centre de Physique Th\'eorique {\sl (CNRS -- UPR14)}}
\centerline{ Ecole Polytechnique, 91128 Palaiseau, France}
\medskip
\centerline{$^2$ Laboratoire de Physique de la Mati\`ere Condens\'ee
{\sl (CNRS --URA D1254)}}
\centerline{ Ecole Polytechnique, 91128 Palaiseau, France}
\medskip
\centerline{$^3$ Universit\'e de Cergy-Pontoise}
\centerline{49, avenue des Genottes - BP 8428, 95806 Cergy-Pontoise, France}
\vskip2.truecm
{\narrower\smallskip
\centerline{\bf Abstract}
We study the dynamics of an interface between
two phases in interaction with a wall in the case when the evolution is
dominated by surface diffusion. For this, we use an $S O S$
model governed by a conservative Langevin
equation and suitable boundary conditions. In the partial wetting case,
we study various scaling regimes and show
oscillatory behaviour in the relaxation of the interface
toward its equilibrium shape.
We also consider complete wetting
and the structure of the precursor film.
\vskip.7truecm
Keywords: Wetting, Surface Diffusion, Conservative Langevin Dynamics,
Solid on Solid Model.\smallskip}
\bigskip
\bigskip\noindent
{\bf 1. Introduction}
\bigskip\noindent
In recent years, a large amount of work -- experimental as well as
theoretical, has been done to study the physics of wetting phenomena.
Various treatments, mostly phenomenological, have been proposed to
grasp some understanding of the dynamics. Recently, simplified models
have been proposed which can be amenable to the methods of Statistical
Mechanics [ACDD].
The scope of the present work is to extend this kind of approach to the
case of a conservative dynamics. A direct application of such a model
can be found in the study of polycrystalline surfaces which
develop grooves around grain boundaries, and whose evolution is dominated by
surface diffusion [M].
In the next section, we describe our model given by a Langevin equation
for a $S O S$ type model. We derive the dynamics mostly by requiring
that it should converge to the correct equilibrium measure (in the
partial wetting case). As a consequence, we get a connection between
the noise correlations induced by the conservation prescription and the
structure of the drift term.
In Section 3, we study the simple case of a Gaussian interaction between
the layers. In the limit of infinite volume, we show that for time
$t << L^4$, $L$ being the size of the system, the typical profile scales
as $t^{1/4}$ with an oscillating shape, as for the surface diffusion
equation previously devised by Mullins [M ] and mathematically
studied by Baras and al. [BDR].
In Section 4, we extend the results to the case of a general
interaction, which we study under the hypothesis of local equilibrium.
In particular, we show that the surface diffusion
equation that we obtain under this hypothesis is the same as the one derived
from a more phenomenological basis, along the lines proposed by Spohn [S].
In the last section, we investigate the complete wetting case in a more
heuristic fashion and show various new features related to the
conservative character of the dynamics.
\bigskip\noindent
{\bf 2. Conservative Langevin Dynamics }
\medskip
Let us consider an interface between two phases in a tube in two
dimensions and describe it
using some $S O S$ type model. We divide arbitrarily our system into
$L+1$ layers parallel to the walls of the tube and assume that the interface
crosses each layer $i$, $i \in \{0,\dots,L\}$ at a definite position,
$h_i$, thus neglecting possible overhangs. The energy of this interface
can be written as
$$
H(h_0,\dots,h_L) = \sum_{i=1}^L U(h_{i-1} - h_i) - \sum_{i=0}^L \mu_i
h_i \eqno(2.1)
$$
where $U(x)$ is an even function, increasing at least linearly for $x>0$, and
$\{\mu_i\}_{i \in \{0,\dots,L\}}$ is a distribution of chemical
potentials representing the interaction of each layer with the walls.
Obviously enough, the fact that the dynamics we consider is conservative means
that if we take an initial condition such that $\sum_{i=0}^L~h_i~=~0$,
this constraint will be obeyed at all times; this has for direct
consequence that the chemical potentials $\mu_i$ are relevant up to a
constant and for instance can be replaced in (2.1) by another
distribution $\tilde \mu_i$ with zero average:
$$
\tilde \mu_i = \mu_i - {1 \over L+1} \sum_{j=0}^L \mu_j \eqno(2.2)
$$
The finite volume canonical Gibbs measure, to which the system
should converge as time goes to infinity, has a
density with respect to the Lebesgue measure, namely:
$$
\proba(h_0,h_1,...,h_L) = {1 \over \cal Z} \exp \bigl( -\beta
H(h_0,h_1,...,h_L)\bigr) \delta( \sum_{i=0}^L h_i )\eqno(2.3)
$$
The partition
function $\cal Z$ normalising the probability can be finite only if $H$
is bounded from below, which requires a condition such as
$$
\Bigl | \sum_{i=0}^{j}\tilde \mu_{i}\Bigr |<\lim_{x\rightarrow\infty}
{U(x)\over x}=\lim_{x\rightarrow\infty} U'(x) < \infty \quad \forall\, j
\eqno(2.4)
$$
where we assumed for simplicity that both limits in (2.4) exist.
In the case of contact interactions with the walls,
$$
\mu_{i}=\mu_{0}\delta_{i,0}+\mu_{L}\delta_{i,L}\quad ,
$$
the canonical Gibbs measure (2.3) describes an interface with
fluctuations ${\cal O}(L^{1/2})$ around a Wulff shape [DDR].
In order to set up a conservative dynamics which converges to the correct
Gibbs measure (2.3), we start from a general form for the Langevin Equation
and look for sufficient conditions.
Assume that $w_0(t)$, $w_1(t)$, $\dots$, $w_L(t)$ are independent Wiener
processes and consider the system of stochastic differential equations
$$
d h_i(t) = F_i(\bfih (t) ) dt + \sqrt{2 \over \beta}\sum_{j=0}^L
\sigma_{i j} dw_j(t) \eqno(2.4)
$$
where the functions $F_i(\bfix)$ are defined for $\bfix\in\real^{L+1}$
and $\sigmag$ is an $(L+1) \times (L+1)$ matrix with constant coefficients.
The solution of Equation (2.4) is a Markov process whose probability density
$\proba(\bfih,t)$ satisfies the Fokker-Planck Equation:
$$
{\partial \proba(\bfih,t) \over \partial t} = -\sum_{i=0}^L {\partial
\over \partial h_i}
\bigl( F_i(\bfih) \proba(\bfih,t) \bigr) +{1 \over \beta} \sum_{i,j,k=0}^L
\sigma_{i k} \sigma_{j k} {\partial^2 \over \partial h_i \partial h_j}
\proba(\bfih,t)\eqno(2.5)
$$
\bigskip
In order to check (2.5), it suffice to take any function $g$ on $\real^{L+1}$
twice continuously
differentiable with compact support, and define the process
$\xi(t)=g(\bfih(t))$ whose stochastic differential is given by the
Ito formula [V]:
$$
\eqalignno{
d\xi(t) &= \bigl\{\sum_{i=0}^L {\partial g(\bfih(t)) \over \partial
h_i} F_i(\bfih)
+ {1 \over \beta} \sum_{i,j,k = 0}^L {\partial^2 g(\bfih(t)) \over
\partial h_i\partial h_j}\sigma_{i k} \sigma_{j k} \bigr\} dt\cr
&+ \sqrt{2 \over \beta}
\sum_{i,k = 0}^L {\partial g(\bfih(t)) \over \partial h_i} \sigma_{i k}
dw_k(t) &(2.6)}
$$
By taking the expectation value in the integral form of (2.6), the third
term in the right hand side vanishes. Using
$$
\expectation\bigl(g(\bfih(t))\bigr) = \int \prod_{i=0}^L dh_i
\proba(\bfih,t) g(\bfih) \eqno(2.7)
$$
and integrating by parts leads to the Fokker-Planck equation (2.5).
\bigskip
The condition that the Gibbs distribution (2.3) be the equilibrium distribution
for the Fokker-Planck equation determines the functions $F_i(\bfih)$ up to
a divergence free vector. We make the following choice:
$$
F_i(\bfih) = -\sum_{j,k=0}^L \sigma_{i k} \sigma_{j k}
{\delta H(\bfih) \over \delta h_j} \eqno(2.8)
$$
The conservation of volume in Equation (2.4) then appears to be a
consequence of taking a conservative noise:
$$
\sum_{i=0}^L \sigma_{i k} =0 \qquad \forall k \in
\{0,\dots,L\}\eqno(2.9)
$$
This choice does not fully prescribe Equation (2.4) yet. Assuming that
the random part in (2.4) is due only to local exchange between
neighboring layers, we get an expression for the matrix $\sigmag$:
$$
\sigmag=\pmatrix{1&&&\cr
-1&\ddots&&\cr
&\ddots&1&0\cr
&&-1&0\cr}\eqno(2.10)
$$
and the conservative Langevin Equation explicitely reads:
$$
\eqalignno{\dot h_0 =& -2 U'(h_0 - h_1) + U'(h_1 - h_2) \cr
&\phantom{...................}+ \mu_0 - \mu_1 +
\sqrt{2 \over \beta} \dot w_{1 \over 2} \cr
\dot h_1 =& 3 U'(h_0 - h_1) - 3 U'(h_1 - h_2) + U'(h_2 - h_3) \cr
&\phantom{...................}- \mu_0 + 2 \mu_1
-\mu_2 + \sqrt{2 \over \beta} (\dot w_{3 \over 2} - \dot w_{1 \over 2})
&(2.11)\cr
\dot h_2 =& - U'(h_0 - h_1)+3 U'(h_1 - h_2) -3 U'(h_2 - h_3) + U'(h_3 - h_4)\cr
&\phantom{...................}- \mu_1 + 2 \mu_2
-\mu_3 + \sqrt{2 \over \beta} (\dot w_{5 \over 2} -\dot w_{3 \over 2}) \cr
\vdots&\cr}
$$
where we have labelled the Wiener processes with half integers to get a more
symmetric expression. Using more compact notations, we can write (2.11) as:
$$d\bfih(t) = - \sigmag \sigmag^\dagger
\bigl( \sigmag\, \bfiU '(\sigmag^\dagger \bfih ) - \mug\bigr) dt
+ \sqrt{2 \over \beta} \sigmag d{\bfiw}(t) \eqno(2.12)$$
where $\bfiU ' (\bfix)$ is the vector whose i'th component is $U'(x_i)$.
\bigskip\noindent
{\bf 3. The Gaussian Model.}
\medskip
In this section we investigate the dynamical behaviour of the model when
the function $U(x)$ is a parabola
$$U(x) = {J \over 2} x^2\eqno(3.1)$$
In this particular case, the Langevin equation (2.12) can be cast in the form:
$$d\bfih(t) = \bigl(-J (\sigmag \sigmag^\dagger)^2 \bfih
+\sigmag \sigmag^\dagger \mug\bigr) dt
+\sqrt{2 \over \beta} \sigmag d{\bfiw}(t) \eqno(3.2)$$
The solution is easily found, given a flat initial profile:
$$\bfih(0)= {\bf 0}$$
We get:
$$\bfih(t)= (1-e^{-J (\sigmag \sigmag^\dagger)^2 t}) \bfih^e
+\sqrt{2 \over \beta}
\int_0^t e^{-J (\sigmag \sigmag^\dagger)^2 t} \sigmag d\bfiw(t)
\eqno(3.3)$$
The components of the equilibrium mean profile $ \bfih^e$ are given by:
$$h_k^e =h_0 - {1 \over J} \sum_{l=0}^{k-1} (k-l) \tilde \mu_l
\eqno(3.4)$$
where the $\tilde \mu_l$ are defined as in (2.2) and the value of $h_0$
is determined from the value of the (preserved) volume:
$$h_0= {1 \over J} \sum_{l=0}^{L-1} {(L-l) (L-l+1) \over 2 (L+1)} \,
\tilde \mu_l \eqno(3.5)$$
In order to push the computation a little bit forward, we need to
compute the eigenvalues of the matrix $(\sigmag \sigmag^\dagger)^2$
which are :
$$\lambda_q=16 \sin^4({\pi q \over 2(L+1)})\; , \qquad q \in \{0,\dots,L\}
\eqno(3.6)$$
The components of the associated normalised eigenvectors are:
$$\varphi_k^0={1\over \sqrt{L+1}} \; ; \qquad
\varphi_k^q=\sqrt{2 \over L+1} \cos({\pi q (2k+1)\over 2(L+1)})
\; ,\;\; q \in \{1,\dots,L\}
\eqno(3.7)$$
The mean profile averaged over the Brownians has components:
$$\expectation\bigl(h_k(t)\bigr) = \sum_{q=1}^L
\bigl(\sum_{l=0}^L \varphi^q_l h^e_l\bigr)
\,(1 - e^{- J \lambda_q t}) \,\varphi_k^q \eqno(3.8)$$
while its variance is
$$\expectation\bigl(h_j(t) h_k(t)\bigr) - \expectation\bigl(h_j(t)\bigr)
\expectation\bigl(h_k(t)\bigr) =
\sum_{q=1}^L {1 \over \beta J \sqrt{\lambda_q}} \,(1 - e^{- J \lambda_q t})
\varphi_j^q \varphi_k^q \eqno(3.9)$$
\medskip\noindent
In the case of a contact interaction with the walls of the container,
the chemical
potential can be taken as zero except in the first and last layers:
$$\mu_0 = \mu_L = \mu \, , \qquad\mu_{i,i\not= 0,L} = 0 \eqno(3.10)$$
In that particular case, the shape of the equilibrium mean profile is a
parabola,
$$h_k^e = {\mu \over J} \biggl[ {L (L-1) \over 6 (L+1)} -
{k (L-k) \over L+1} \biggr]\eqno(3.11)$$
and the infinite volume limit of Equations (3.8-3.9) can be cast in an
integral form as:
$$\lim_{L \to \infty} \expectation\bigl(h_k(t)\bigr) = {\mu \over \pi J}
\int_0^{\pi/2} dx \;(1 - e^{-16 J \sin^4(x)t}) {\cos(x) \over \sin^2(x)}
\,\cos((2k+1) x)\eqno(3.12)$$
$$\eqalignno{\lim_{L \to \infty} &\bigl[\E\bigl(h_j(t)h_k(t)\bigr)-
\E\bigl(h_j(t)\bigr)
\E\bigl(h_k(t)\bigr)\bigr]\cr
& = {1 \over \beta \pi J}
\int_0^{\pi/2} dx\; {1 - e^{-16 J \sin^4(x)t} \over \sin^2(x)}\,
\cos((2j+1) x)\cos((2k+1) x)&(3.13)}$$
In order to study the behaviour of the profile for times long enough but
still small with respect to the equilibrium time, we need to scale the
distances by a factor $t^{1/4}$ and study the limit:
$$\lim_{t \to \infty} \lim_{L \to \infty} t^{-1/4}
\E\bigl(h_{[y t^{1/4}]}\bigr)
= - {\mu\over J} {\cal Z} (J^{-1/4} y)\eqno(3.14)$$
where
$$
{\cal Z} (u) = -{2 \over \pi} \int_0^{\infty} dz {1 - e^{-z^4} \over z^2}
\cos(u z)\eqno(3.15)$$
The function $\cal Z $ is the explicit solution of the linearised
surface diffusion equation (34) of reference [M], namely:
$$
{\cal Z}^{(4)}(u) = -{1\over 4} \bigl\{{\cal Z}(u) - u {\cal Z}'(u)\bigr\}
\; , \qquad u > 0\eqno(3.16a)
$$
together with the boundary conditions:
$$
{\cal Z}(0_+) = {- 1 \over \sqrt{2}\, \Gamma ({5\over 4})}\; ,\quad
{\cal Z}'(0_+) = 1\; ,\quad
{\cal Z}''(0_+) = {- 1 \over \sqrt{2}\, \Gamma ({3\over 4})}\; ,\quad
{\cal Z}'''(0_+) = 0\eqno(3.16b)
$$
In order to evaluate ${\cal Z}(u)$, we consider the
real function $f$ defined by $f(\infty)=f'(\infty)=0$ and
$$
f''(x)=g(x)=\int_{-\infty}^{\infty}e^{ixy}e^{-y^{4}}dy\;,\eqno(3.17)
$$
and determine its behaviour for large $x$.
For $x>0$ we change the variable in the integral to $y=x^{1/3}z$ and
obtain
$$
g(x)=x^{1/3}\int_{-\infty}^{\infty}e^{x^{4/3}(iz-z^{4})}dz\;.\eqno(3.18)
$$
We are now going to use the stationary phase method to estimate the
above integral. The derivative of the phase has three simple zeroes
which are the three cubic roots of the number $i/4$. Among these three
zeroes, only two give a decreasing behaviour to the integral, namely
$$
z_{\pm}=\eta(\pm \sqrt{3}+i)\quad\hbox{\tenrm with}\quad
\eta=2^{-5/3}\;.
\eqno(3.19)
$$
We choose a contour parallel to the real axis and passing through the two
previous points, namely $z=u+i\eta$. On this line the phase in the
integral is given by
$$
Q(u)\equiv iz-z^{4}=-\eta+\eta^{4}+6u^{2}\eta^{2}-u^{4}+i(u(1+4\eta^{3})-
4\eta u^{3})\;.
\eqno(3.20)
$$
Moreover, at the critical points we have $Q(\pm \eta\sqrt{3})=3i z_{\pm}/4$ and
$Q''(\pm \eta\sqrt{3})=-12 z_{\pm}^{2}$.
By a standard steepest descent argument we obtain for large $x$
$$
g(x)= {2^{5/3} \sqrt{\pi}\over \sqrt{6}}x^{-1/3}\;e^{-3\;x^{4/3}\;2^{-11/3}}
\cos(3^{3/2}\;2^{-11/3}x^{4/3}-\pi/6)+
e^{-3\;x^{4/3}\;2^{-11/3}}{\cal O}(x^{-5/3})\;.
\eqno(3.21)
$$
We now observe that if $\gamma$ is a real number and $\rho$ is a complex
number with negative real part, we have for large $x$ using integration
by parts
$$
\int_{+\infty}^{x}s^{\gamma}e^{\rho s^{4/3}} ds ={3x^{\gamma-1/3}\over
4\rho} e^{\rho x^{4/3}}+e^{\Re(\rho) x^{4/3}}{\cal O}(x^{\gamma-5/3})\;.
\eqno(3.22)
$$
Therefore,
integrating twice and using the boundary conditions at infinity we
obtain for large $x$
$$
f(x)=-{8x^{-1}\over \sqrt{6\pi}}e^{-3\;x^{4/3}\;2^{-11/3}}
\sin(3^{3/2}2^{-11/3}x^{4/3})+
e^{-3\;x^{4/3}\;2^{-11/3}} {\cal O}(x^{-7/3})\;.
\eqno(3.23)
$$
Hence the long time beaviour of the profile in the range $t\ll L^4$
is dominated by an oscillating term with a rapidly decreasing amplitude [R]:
$$\lim_{t \to \infty} \lim_{L \to \infty} t^{-1/4}
\E\bigl(h_{[y t^{1/4}]}\bigr)
\simeq {4\mu y^{-1} \over J^{3/4} \sqrt{6 \pi}} e^{-3\; J^{-1/3}
y^{4/3}\;2^{-11/3}}\sin(3^{3/2}2^{-11/3}J^{-1/3}y^{4/3})
\eqno(3.24)
$$
An analogous reasoning can be applied to Equation (3.13) and one
finds that the mean amplitude of the fluctuations scales as $t^{1/8}$ in
this regime, namely
$$
\lim_{L \to \infty} \bigl[\E\bigl(h_j(t)h_k(t)\bigr)-
\E\bigl(h_j(t)\bigr)\E\bigl(h_k(t)\bigr)\bigr]
\approx t^{1/4} \quad{\rm as}\quad t\rightarrow\infty
$$
\bigskip\noindent
{\bf 4 Local Equilibrium and Surface Diffusion.}
\bigskip\noindent
At large times, we expect the profile to become smooth in average, and
fluctuations to follow a Gibbsian local equilibrium. Assuming that these two
assumptions hold true, we show that the corresponding
hydrodynamical regime is described by surface diffusion, in agreement with
a linear response argument [S]. This is analogous to the heuristic derivation
of motion by curvature in the non conservative Langevin dynamics [DP].
Before doing so, we recall some facts about one-dimensional interfaces
at equilibrium in SOS models, in the grand canonical ensemble, with
boundary conditions fixing the average slope. The free energy per unit
length of interface is defined as usual by
$$
\sigma (\theta)={1\over\beta}
\lim_{L\rightarrow\infty}{\cos\theta\over L}\log\cal Z_{\theta}
$$
with, similar to (2.3),
$$
{\cal Z}_{\theta}=\int\exp \bigl( -\beta
H(h_0,h_1,...,h_L)\bigr) \delta(h_{0})\delta(h_{L}-L\tan\theta)
\prod_{i=0}^{L} dh_{i}
$$
It is convenient to replace the boundary condition fixing the average
slope by a slope chemical potential. This change of ensemble leads only
to logarithmic corrections in the total free energy and doesn't modify
$\sigma(\theta$). The chemical potential $c(\tan\theta)$ conjugate to the
desired slope can be defined for suitable $U$ by
$$
\int_{-\infty}^{+\infty}dx(x-\tan\theta)\exp -\beta\bigl(U(x)-
c(\tan\theta)x\bigr) =0 \eqno{(4.1)}
$$
Dropping the constraint $\delta(h_{L}-L\tan\theta)$ then leaves a random walk
with independent steps, so that the free energy and local expectation values
can be computed from the step distribution~:
$$
\sigma(\theta)=-{1\over\beta}\cos\theta\log
\int_{-\infty}^{+\infty}dx\exp -\beta\bigl(-U(x)
+c(\tan\theta)(x-\tan\theta)\bigr)\eqno{(4.2)}
$$
$$
\langle U'(h_{i-1}-h_{i})\rangle_{\theta}=c(\tan\theta)=
\sigma\sin\theta+\sigma'\cos\theta\eqno{(4.3)}
$$
where $\sigma'$ is the derivative of $\sigma$ with respect to $\theta$.
Let us now return to the dynamical problem, and
take averages in (2.3)-(2.7) to obtain
$$
\E\dot \bfih=-\sigmag \sigmag^\dagger\,\sigmag
\E \bfiU' (\sigmag^\dagger \bfih ) \eqno{(4.4)}
$$
where $\bfiU'(\bfix)$ is the vector whose i'th component is $U'(x_i)$.
We then get an approximation to $\E U'(h_{i-1}-h_{i})$, using the
equilibrium Gibbs measure for a straight interface of slope
$\E (h_{i-1}-h_{i})$. Indeed convergence to local Gibbs equilibrium
applied to the observable $U'(h_{i-1}-h_{i})$ means, using (4.3),
$$
\E U'(h_{i-1}-h_{i})-c(\E (h_{i-1}-h_{i}))\rightarrow 0 \quad{\rm as}
\quad t\rightarrow\infty\eqno{(4.5)}
$$
More precisely we assume,
$$
\sigmag \sigmag^\dagger\,\sigmag
\E{\bfiU'}(\sigmag^\dagger\bfih)
=\sigmag \sigmag^\dagger\,\sigmag
{\bfic}(\sigmag^\dagger\E{\bfih})+{\cal O}(t^{-1})\eqno{(4.6)}
$$
where ${\bfic(\bfix)}$ is the vector whose i'th component is $c(x_{i})$, and
obtain that the averages $\E{\bfih}$ follow approximately a simple
deterministic equation,
$$
\E\dot {\bfih}=-\sigmag\sigmag^\dagger\,\sigmag
{\bfic}(\sigmag^\dagger\E{\bfih})+{\cal O}(t^{-1})\eqno{(4.7)}
$$
or
$$
\E\dot h_{i}=
-c(\E h_{i-2}-\E h_{i-1})+3c(\E h_{i-1}-\E h_{i})-3c(\E h_{i}-\E h_{i+1})
+c(\E h_{i+1}-\E h_{i+2})+{\cal O}(t^{-1})
$$
A further smoothness assumption gives
$$
\eqalign{
\E\dot h_{i}=&
c'\bigl((\E h_{i-2}-\E h_{i})/2\bigr)
\bigl(-\E h_{i-2}+2\E h_{i-1}-\E h_{i}\bigr)\cr &-2
c'\bigl((\E h_{i-1}-\E h_{i+1})/2\bigr)
\bigl(-\E h_{i-1}+2\E h_{i}-\E h_{i+1}\bigr)\cr &+
c'\bigl((\E h_{i}-\E h_{i+2})/2\bigr)
\bigl(-\E h_{i}+2\E h_{i+1}-\E h_{i+2}\bigr)+{\cal O}(t^{-1})}
$$
or, in sloppy notation,
$$
\E\dot h= (c'(\E h')\E h'')''+{\cal O}(t^{-1})\eqno{(4.8)}
$$
We shall show that this is surface diffusion, or linear response to
a gradient of curvature, modified by anisotropy. The interface is now
taken as a four times differentiable curve $C(l)$ parametrised by arc
length. The associated free energy functional is defined as
$$
F=\int_C \sigma (\theta (l))dl
$$
where $\sigma (\theta)$ is the free energy per unit length of a straight
interface of slope $\tan\theta$, as computed in (4.2).
The variation of $F$ under a small deformation $\delta{\bf r}(l)$ is
$$
\delta F=\int_C(\sigma+\sigma'')\ K\ dl\ \delta\bf r\cdot\bf{\hat n}
$$
where ${\bf \hat n}(l)$ is the unitary vector normal to the interface.
The current is the basic linear response in conservative dynamics,
$$
J(l)=-\mu(\theta){d\over dl}
{\delta F\over\delta (\delta{\bf r}(l)\cdot\bf{\hat n})}=
\mu(\theta){d\over dl}
\bigl((\sigma+\sigma'')K\bigr)\eqno{(4.9)}
$$
where the surface mobility $\mu(\theta)$ gives the time scale.
The speed of the interface measured along the normal is then obtained
from the conservation law:
$$
v_{n}=-{d\over dl}J={d\over dl}
\mu(\theta){d\over dl}\bigl((\sigma+\sigma'')K\bigr)\eqno{(4.10)}
$$
In order to compare to (4.8) we change to $h$ and $x$ variables, using
$v_n=\cos\theta\ \dot h$ to get an equation for $h(t,x)$,
$$
\dot h={\partial\over\partial x}\Bigl(
\mu(\theta)\cos\theta{\partial\over\partial x}\bigl(
(\sigma+\sigma'')K\bigr)\Bigr)
\eqno(4.11)
$$
which agrees with (4.8) if $\mu(\theta)=(\cos\theta)^{-1}$, because
$K=h''(\cos\theta )^3$ and $(\sigma+\sigma'')(\cos\theta )^3=c'(\tan\theta$).
One should also note that Equation (11) in [M] can be recovered from (4.10)
by taking $\mu$ and $\sigma$ independant on the orientation and noting
$\mu\sigma = B $.
\bigskip\noindent
{\bf 5 Complete Wetting.}
\bigskip\noindent
The Hamiltonian is here taken as
$$
H(h_0,h_1,...h_L)=\sum_{i=1}^{L}U(h_{i-1}-h_i)-\mu(h_0+h_{L})
\;,\eqno{(5.1)}
$$
with
$$
J = \lim_{x\rightarrow\infty}{U(x)\over x}=
\lim_{x\rightarrow\infty}U'(x) < \mu <\infty
$$
The quantity $\mu-J$ is essentially what is called the spreading
coefficient in dynamical studies of wetting. $\mu>J$ corresponds to dry
spreading, whereas $\mu=J$ would correspond to spreading at the wetting
transition. We shall discuss only $\mu>J$, in which case the measure
(2.3) cannot be normalised and the asymptotics as time goes to infinity
are unusual.
We shall only discuss the time evolution of quantities
averaged over the Brownians, in a heuristic fashion.
The evolution equations take the form
$$
\eqalign{
\E\dot h_{0}=&\mu-2\E U'(h_{0}-h_{1})+\E U'(h_{1}-h_{2})\cr
\E\dot h_{1}=&-\mu+3\E U'(h_{0}-h_{1})-3\E U'(h_{1}-h_{2})+\E U'(h_{2}-h_{3})
\cr
\E\dot h_{2}=&
-\E U'(h_{0}-h_{1})+3\E U'(h_{1}-h_{2})-3\E U'(h_{2}-h_{3})+\E
U'(h_{3}-h_{4}) \cr
...=&...\cr
\E\dot h_{i}=&
-\E U'(h_{i-2}-h_{i-1})+3\E U'(h_{i-1}-h_{i})-3\E U'(h_{i}-h_{i+1})
+\E U'(h_{i+1}-h_{i+2})\cr
...=&...}
$$
let us first consider the profile near one boundary ($i=0$), while the other
boundary ($i=L$) has been sent to infinity, before letting time go to
infinity. The predictions should in fact be
valid in the range $1\ll t\ll L^{4}$. These predictions will now be
given based on a few reasonable assumptions or ansatz.
For any fixed $i$, local equilibrium is approached as time goes to
infinity. We assume the existence of the following limits~:
$$
v_{i}=\lim_{t\rightarrow\infty}E\dot h_{i}(t)
\qquad ; \qquad
U'_{i(i+1)}=\lim_{t\rightarrow\infty}EU'\bigl(h_{i}(t)-h_{i+1}(t)\bigr)
\eqno{(5.2)}
$$
The first observation is that $v_{i}\not=v_{i+1}$ implies
$U'_{i(i+1)}=J\ {\rm sign}(v_{i}-v_{i+1})$.
Thus, for $i\ge 2$, whenever $v_{i-2}\ne v_{i-1}\not=
v_{i}\not=v_{i+1}\not=v_{i+2}$, we have
$$
v_{i}=J\Bigl(-{\rm sign}(v_{i-2}-v_{i-1})+3\ {\rm sign}(v_{i-1}-v_{i})
-3\ {\rm sign}(v_{i}-v_{i+1})+{\rm sign}(v_{i+1}-v_{i+2})\Bigr)
$$
Therefore $v_{i}$ maximum implies $v_{i}<-4J$ and $v_{i}$ minimum
implies $v_{i}>4J$. Allowing negative maxima and positive minima, while
forbidding positive maxima and negative minima, would be difficult to
match with the boundary condition
requiring $v_{i}\rightarrow 0$ as $i\rightarrow\infty$. This leads to
$$
v_{0}>0>v_{1}\le v_{2}\le v_{3}\le ...\le 0
$$
We next observe that a sequence which would be strictly monotonous over
five layers around a given $i$,
$$
v_{i-2}0>v_{1}=v_{2}=...=v_{k}0>v_{1}=v_{2}=...=v_{k}k+2$, which should have $h_{k+3}(t)\rightarrow-\infty$ as
$t\rightarrow\infty$, and $U'_{(k+3)(k+4)}=-J$, implying $v_{k+2}=0$.
An independent argument in the same direction is to compute
$$
\sum_{i=0}^{k+1}E\dot h_{i}=
EU'(h_{k}-h_{k+1})-2EU'(h_{k+1}-h_{k+2})+EU'(h_{k+2}-h_{k+3})
$$
which tends to zero as $t\rightarrow\infty$ in both cases of the above
alternative, indicating that $E\dot h_{k+2}$ should also go to
zero because of volume conservation.
Our ansatz therefore takes the form
$$
x=v_{0}>y=v_{1}=v_{2}=...=v_{k}J$, and leaving aside a discrete set of exceptional
values, this gives
$$
k=\biggl[{5J+\mu\over\mu-J}\biggr]\approx6J(\mu-J)^{-1}\qquad{\rm as}
\qquad\mu\searrow J
$$
while
$$
x\simeq{(\mu-J)^{2}\over3J}\ ,\quad y\simeq{(\mu-J)^{3}\over18J^{2}}\
,\quad
{\rm and}\quad z={\cal O}((\mu-J)^{3})
\qquad{\rm as}
\qquad\mu\searrow J
$$
The profile of these $k$ layers satisfies
$$
-U'_{(i-2)(i-1)}+3U'_{(i-1)i}-3U'_{i(i+1)}+U'_{(i+1)(i+2)}=y
$$
If the constant $y$ would be zero, this would be a Wulff shape, but
$y<0$, which is related to the flux going from the top layers downto the
zero wetting layer $h_{0}$. However, since $y\sim k^{-3}$ as $\mu\searrow J$,
the Wulff shape will be approached in this limit, the effect of the flux
on $\E (h_{i}-h_{j})$ being ${\cal O}((i-j)/k)$ compared to fluctuations
${\cal O}((i-j)^{1/2})$.
Let us now return to the tube $i=0,...L$, and let the
time go to infinity with $L$ fixed. The same reasoning as before leads to
$$
v_{i}=\lim_{t\rightarrow\infty}E\dot h_{i}(t)=v
\qquad {\rm for} \qquad i=k+2,...,L-k-2
$$
and then
$$
U'_{i(i+1)}=\lim_{t\rightarrow\infty}EU'\bigl(h_{i}(t)-h_{i+1}(t)\bigr)
=f({i\over L})+{\cal O}({1\over L})
$$
giving in turn $v\sim L^{-3}$, and a shape differing from a Wulff shape
only at order ${\cal O}(1)$ compared to fluctuations ${\cal O}(L^{1/2})$
\bigskip
{\bf References}
\medskip
\item{[ACDD]} D.B. Abraham, P. Collet, J. De Coninck, F. Dunlop : {\sl
Langevin Dynamics of Spreading and Wetting}, Phys. Rev. Lett. {\bf 65},
195-198 (1990)~; {\sl Langevin
Dynamics of an Interface near a Wall}, J. Stat. Phys. {\bf 61}, 509 (1990).
\item{[BDR]} P. Baras, J. Duchon, R. Robert : {\sl Evolution d'une
interface par diffusion de surface}, Com. Part. Diff. Eq. {\bf 9} 313 (1984) .
\item{[DDR]} J. De Coninck, F. Dunlop, V. Rivasseau : {\sl On the Microscopic
Validity of the Wulff Construction and of the Generalized Young Equation},
Commun. Math. Phys {\bf 121}, 401-419 (1989).
\item{[DP]} F. Dunlop, M. Plapp : {\sl Scaling profiles of a spreading drop
from Langevin or Monte-Carlo dynamics}, pp.~303--308 in ``On Three
Levels~: Micro-, Meso-, and Macro-Approaches in Physics'', Edited by
M.~Fannes et al., Plenum Press, New-York (1994).
\item{[M]} W.W. Mullins : {\sl Theory of thermal grooving} , J. Appl. Phys.
{\bf 28} , 333 (1957).
\item{[R]} W.M. Robertson : {\sl Grain-boundary grooving by surface
diffusion for finite surface slope} , J. Appl. Phys. {\bf 42 } 463 (1971).
\item{[S]} H. Spohn : {\sl Interface motion in models with stochastic
dynamics}, J. Stat. Phys. {\bf 71}, 1081 (1993).
\item{[V]} S.R.S. Varadhan : {\sl Diffusion problems and partial differential
equations} , Tata Institute, Bombay (1980) .
\bye
ENDBODY