Content-Type: multipart/mixed; boundary="-------------9911220455952" This is a multi-part message in MIME format. ---------------9911220455952 Content-Type: text/plain; name="99-441.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-441.keywords" Hydrodynamics, $k$-step exclusion, nonconvex or nonconcave flux function, contact discontinuities ---------------9911220455952 Content-Type: application/x-tex; name="GRS.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="GRS.tex" \documentclass{article} \usepackage{amssymb} %\renewcommand{\baselinestretch}{1.7} %espace double \begin{document} \title{Hydrodynamics for totally asymmetric $k$-step exclusion processes} \author{ H. Guiol\thanks{IMECC, Universidade de Campinas, Caixa Postal 6065, CEP 13053-970, Campinas, SP, Brasil}, K.Ravishankar\thanks{Dep. of Mathematics and Computer Science, State University of New York, College at New Paltz, New Paltz, NY, 12561, USA}, E. Saada\thanks{CNRS, UPRES-A 6085, Universit\'e de Rouen, Site Colbert, 76821 Mont Saint Aignan Cedex, France}\\ AMS 1991 {\sl subject classifications}. Primary 60K35;Secondary 82C22.\\ {\sl Key words and phrases.} Hydrodynamics, $k$-step exclusion,\\ nonconvex or nonconcave flux function, contact discontinuities} \maketitle %\textwidth 4.9in %\textheight 7.75in %\renewcommand %\baselinestretch{1.0} %\topmargin -0.55in %\oddsidemargin -0.13in %\evensidemargin -0.13in %\begin{document} %\title{Hydrodynamics for totally asymmetric $k$-step exclusion %processes} %\author{H. Guiol\thanks{IMECC, Universidade de Campinas, %Caixa Postal 6065, CEP 13053-970, Campinas, SP, Brasil}, %K.Ravishankar\thanks{Dep. of Mathematics and Computer Science, %State University of New York, College at New Paltz, New Paltz, NY, %12561, USA}, E. Saada\thanks{CNRS, UPRES-A 6085, Universit\'e de % Rouen, Site Colbert, 76821 Mont Saint Aignan Cedex, France}} % % %\maketitle \begin{abstract} We describe the hydrodynamic behavior of the $k$-step exclusion process. Since the flux appearing in the hydrodynamic equation for this particle system is neither convex nor concave, the set of possible solutions include in addition to entropic shocks and continuous solutions those with contact discontinuities. We finish with a limit theorem for the tagged particle. \end{abstract} \newcommand{\carn}{\hfill\rule{0.25cm}{0.25cm}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \def\E{{\mathbb E}} \def\P{{\mathbb P}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\V{{\mathbb V}} \def\N{{\mathbb N}} \section{Introduction and Notation} In his paper Liggett (1980) introduced a Feller non conservative approximation of the long range exclusion process to study the latter. A conservative version of this dynamics, called $k$-step exclusion process was defined and studied in Guiol (1999). It is described in the following way. Let $k\in\N^*:=\{1,2,...\}$, ${\bf X}:=\{0,1\}^{\Z}$ be the state space, and let $\{X_n\}_{n\in{\N}}$ be a Markov chain on $\Z$ with transition matrix $p(.,.)$ and ${\bf P}^x(X_0=x)=1$. Under the mild hypothesis $\sup _{y\in {\Z}}\sum_{x\in {\Z}}p(x,y)<+\infty$, $L_k$, defined below, is an infinitesimal pregenerator: For all cylinder function $f$, \begin{equation}\label{generateur} L_kf(\eta )=\sum_{\eta (x)=1,\eta (y)=0}q_k(x,y,\eta )\left[ f(\eta^{x,y})-f(\eta )\right], \end{equation} where $q_k(x,y,\eta )={\bf E}^x\left[ \prod_{i=1}^{\sigma _y-1} \eta (X_i),\sigma _y\leq\sigma_x,\sigma _y\leq k\right] $ is the intensity for moving from $x$ to $y$ on configuration $\eta $, $\sigma_y=\inf\left\{ n\geq 1:X_n=y\right\} $ is the first (non zero) arrival time to site $y$ of the chain starting at site $x$ and $\eta^{x,y}$ is configuration $\eta$ where the states of sites $x$ and $y$ were exchanged. In words if a particle at site $x$ wants to jump it may go to the first empty site encountered before returning to site $x$ following the chain $X_n$ (starting at $x$) provided it takes less than $k$ attempts; otherwise the movement is cancelled. By Hille-Yosida's theorem, the closure of $L_k$ generates a continuous Markov semi-group $S_k(t)$ on $C({\bf X})$ which corresponds to the $k$-step exclusion process $(\eta_t)_{t\geq 0}$. Notice that when $k=1$, $(\eta_t)_{t\geq 0}$ is the simple exclusion process. An important property of $k$-step exclusion is that it is an {\sl attractive process}. Let ${\cal I}_k$ be the set of invariant measures for $(\eta_t)_{t\geq 0}$ and let ${\cal S}$ be the set of translation invariant measures on ${\bf X}$. If $p(x,y)=p(0,y-x)$ for all $x,y\in{\Z}$ and $p(.,.)$ is irreducible then \[ ({\cal I}_k\cap {\cal S})_e=\left\{ \nu _\alpha :\alpha \in \left[ 0,1\right] \right\}, \] where the index $e$ mean extremal and $\nu_\alpha$ is the Bernoulli product measure with constant density $\alpha$, {\it i.e.} the measure with marginal \[ \nu_{\alpha}\{\eta\in{\bf X}:\eta(x)=1\}=\alpha. \] In this paper we prove conservation of local equilibrium for the totally asymmetric process in the Riemann case {\it i.e.}: $p(x,x+1)=1$ for all $x\in\Z$ and the initial distribution is a product measure with densities $\lambda$ to the left of the origin and $\rho$ to its right, we denote it by $\mu_{\lambda,\rho}$. The derived equation involves a flux which is neither concave nor convex and appears for the first time as a hydrodynamic limit of an interacting particle system. Up to now the ``constructive'' proofs for hydrodynamics relied on the concavity of the flux, see Andjel \& Vares (1987) or the papers by Sepp\"al\"ainen ({\it e.g.} Sepp\"al\"ainen (1998)), whose key tool is the Lax-Hopf formula. The entropy solution for the type of equation we consider was first studied in Ballou (1970). Such solutions can have entropy shocks as well as contact discontinuities. Our aim is to take advantage of Ballou's result to deduce conservation of local equilibrium also in a constructive way, in the spirit of Andjel \& Vares (1987). However we explain in the last section how to derive hydrodynamics for general initial profiles, and nearest neighbor dynamics. In section 3 we prove a law of large numbers for a tagged particle in a $k$-step exclusion process. \section{The hydrodynamic equation} \subsection{Heuristic derivation of the equation} Since the process is (totally) asymmetric we take Euler scaling. For every $r\in\R$ define $\eta_t^{\varepsilon}(r):=\eta_{\varepsilon^{-1}t}([\varepsilon^{-1}r])$, where $[\varepsilon^{-1}r]$ is the integer part of $\varepsilon^{-1}r$, for $\varepsilon>0$. Given a continuous function $u^0(x),x\in\R$ (initial density profile), we define a family of Bernoulli product measures $\{\nu_{u^0}^\varepsilon\}_{\varepsilon>0}$ on ${\bf X}$, by: For all $x \in\Z$, \[ \nu_{u^0}^\varepsilon \{\eta\in{\bf X}:\eta (x)=1\}=u^0(\varepsilon x). \] We call $\{\nu_{u^0}^\varepsilon\}$ the family of measures determined by the profile $u^0$. Let $u^\varepsilon(r,t):=\int (S_k(\varepsilon^{-1}t)\eta_0^\varepsilon(r)) d\nu^\varepsilon_{u^0}(\eta_0)$. Then for all $r\in\R$, $u^\varepsilon(r,0)$ converges to $u^0(r)$ when $\varepsilon$ goes to 0; applying the generator (\ref{generateur}) to $\eta_t^\varepsilon(r)$ we have \[ \frac{d}{dt}S_k(t\varepsilon^{-1})\left(\eta_0^{\varepsilon}(r)\right)= \varepsilon^{-1}S_k(t\varepsilon^{-1})\left[ -\sum_{i=0}^{k-1}\prod_{j=0}^i \eta_0^\varepsilon (r+j\varepsilon)\left[1-\eta_0^\varepsilon\left(r+(i+1)\varepsilon \right)\right] \right. \] \[ \left. +\sum_{i=0}^{k-1}\prod_{j=0}^i \eta_0^\varepsilon(r-j\varepsilon)\left[1-\eta_0^\varepsilon(r)\right] \right]. \] Assuming that local equilibrium is preserved (thus expectation of products factor), and taking expectations with respect to $\nu_{u^0}^\varepsilon$, we obtain \[ \frac{\partial u^\varepsilon}{\partial t} (r,t)= \varepsilon^{-1}\left[ -\sum_{i=0}^{k-1}\prod_{j=0}^i u^\varepsilon (r+j\varepsilon,t)\left[1-u^\varepsilon \left(r+(i+1)\varepsilon, t\right) \right] \right. \] \[ \left. +\sum_{i=0}^{k-1}\prod_{j=0}^i u^\varepsilon(r-j\varepsilon,t)\left[1-u^\varepsilon (r,t)\right] \right]. \] If we now let $\varepsilon$ converge to 0, $u(r,t):=\lim_{\varepsilon\to 0}u^\varepsilon(r,t)$ should satisfy \begin{equation}\label{hydro} \left\{ \begin{array}{l} \displaystyle{\frac{\partial u}{\partial t}+\frac{\partial G_k(u)}{\partial x}=0} \\ \\ u(x,0)=u^0(x),\\ \end{array}\right. \end{equation} where $G_k$ represents the flux of particles: \[ G_k(u)=\sum_{j=1}^k ju^j(1-u). \] This is a non standard form because $G_k$ is neither convex nor concave, thus equation (\ref{hydro}) is no longer ``a genuinely nonlinear conservation law'', using the language of Lax (1973). To deal with this equation, we have to use an extended version of non linear Cauchy problems treated by Ballou (1970). \begin{remark} \end{remark} Let $k$ go to infinity and denote by $G_{\infty}$ the limiting flux function: \[ G_{\infty}(u)=\frac u{1-u}. \] That case corresponds to the totally asymmetric long range exclusion process. The resulting equation is simpler because the flux function $G_{\infty}$ is strictly convex. The hydrodynamics in this case should follow from the arguments of Aldous \& Diaconis (1995) for the Hammersley's process.\par \smallskip For notational simplicity, from now on we restrict ourselves to the case $k=2$. However our arguments can be easily extended for all $k$. \subsection{Hydrodynamics in the Riemann case} \subsubsection{Notation and result} Our main theorem characterizes the hydrodynamic (Euler) limit of the 2-step exclusion process at points of continuity, when the family of initial measures is determined by a step function profile. From the heuristic derivation we would expect that in the hydrodynamic limit the density profile would satisfy equation (\ref{hydro}). We show that the limiting density profile at time $t$ is the entropy solution of equation (\ref{hydro}) starting with the initial value $u^0$, a step function profile. We now give a brief summary of results concerning the solution of equation (\ref{hydro}), due to D.P. Ballou (1970), when $u^0$ is a step function. This will motivate the formulation of the theorem as well as some aspects of the proof. Existence of weak solution to the Cauchy problem given by equation (\ref{hydro}) with bounded measurable initial condition was proved in Ballou (1970), under the assumptions: 1. $G_k\in C^2(\R)$. 2. $G_k''$ vanishes at a finite number of points and changes sign at these points. In order to obtain uniqueness further conditions are needed. We require our solutions to satisfy the: \noindent {\bf Condition E:} (O.A. Ole\u{\i}nik) {\sl Let $x(t)$ be any curve of discontinuity of the weak solution $u(t,x)$, and let $v$ be any number lying between $u^-:=u(t,x(t)-0)$ and $u^+:=u(t,x(t)+0)$. Then except possibly for a finite number of $t$,} \[ S[v;u^-]\geq S[u^+;u^-], \] \noindent where \[ S[v;w]:=\frac{G_k(w)-G_k(v)}{w-v}. \] It is known (Ballou (1970)) that the following two conditions are necessary and sufficient for a piecewise smooth function $u(x,t)$ to be a weak solution of equation (\ref{hydro}): 1. $u(x,t)$ solves equation (\ref{hydro}) at points of smoothness. 2.If $x(t)$ is a curve of discontinuity of the solution then the Rankine-Hugoniot condition ({\sl i.e.} $d (x(t))/dt = S[ u^+;u^-]$) holds along $x(t)$. \noindent Moreover condition E is sufficient to ensure the uniqueness of piecewise smooth solutions, which are the entropy solutions to the equation. Hereafter we only deal with the case $k = 2$. We denote $G(u):=G_2(u)$.\par \smallskip If $G$ were convex (concave) only two types of solutions would be possible. We now describe these two types of solutions. Let $u^0(x)=\lambda 1_{\{x<0\}}+\rho 1_{\{x\geq 0\}}$. If $\lambda>\rho$ ($\rho>\lambda$), then the speed of characteristics which start from $x\leq 0$ (given by $G'$) is greater than speed of characteristics which start from $x>0$. If the intersection of characteristics occurs along a curve $x(t)$, then since \[ S[u^+;u^-]=\frac{G(\lambda)-G(\rho)}{\lambda-\rho} =S[\lambda;\rho] \] Rankine-Hugoniot condition will be satisfied if $x'(t)=S[\lambda;\rho]$. Thus \[ u(x,t)=\left\{ \begin{array}{ll} \lambda, & x\leq S[\lambda;\rho]t;\\ \rho, & x > S[\lambda;\rho]t.\\ \end{array} \right. \] \noindent is a weak solution. The convexity of $G$ implies that condition E is satisfied across $x(t)$. Therefore $u(x,t)$ defined above is the unique entropic solution in this case and will be referred to as a shock in the sense of Lax (1973). \noindent If $\lambda<\rho \quad(\rho<\lambda)$, then the characteristics starting respectively from $x\leq 0$ and from $x>0$ never meet. Moreover they never enter the space-time wedge between lines $x=\lambda t$ and $x=\rho t$. We can choose values in this region to obtain a continuous solution, the so-called {\sl continuous solution with a rarefaction fan}: Let $h$ be the inverse of $H:=G'$, \[ u(x,t)=\left\{ \begin{array}{ll} \lambda, & x\leq H(\lambda)t;\\ h(x/t), & H(\lambda)t< x \leq H(\rho)t;\\ \rho, & H(\rho)t< x.\\ \end{array} \right. \] \noindent It is possible to define piecewise smooth weak solutions with a jump occurring in the wedge satisfying the Rankine-Hugoniot condition. But the convexity of $G$ prevents such solutions to satisfy condition E. Thus the continuous solution with a rarefaction fan is the unique entropic solution in this case.\par \smallskip For the 2-step exclusion process, the flux function $G$ is neither concave nor convex. Instead $G(u)=u+u^2- 2 u^3$ is convex for $u<1/6$ and concave for $u>1/6$. In this case in addition to the shock and continuous solution with a rarefaction fan it is possible to have solutions for which the curve of discontinuities never enters the region of intersecting characteristics. The quotation in boldface is the original number of lemmas, prop... in Ballou (1970), but the notation refers to 2-step exclusion: \begin{definition}{\bf [B def2.1]}\label{Bd2.1} For any $u< 1/6$, define $u^*:=u^*(u)$ as \[ u^*=\sup\{\eta>u: S[u;\eta]>S[v;u]\ \forall v\in(u,\eta)\}. \] For any $u> 1/6$, define $u_*:=u_*(u)$ as \[ u_*=\inf\{\etaS[v;u]\ \forall v\in(\eta,u)\}. \] \end{definition} In other words, for $u< 1/6$, if we consider the upper convex envelope $G^c$ of $G$ on $(u,+\infty)$, then $u^*$ is the first point where $G^c$ coincides with $G$. In the same way when $u> 1/6$, $u_*$ is the first point where the lower convex envelope $G_c$ of $G$ on $(-\infty,u)$ coincides with $G$. For $\eta< 1/6$, $\eta^*=(1-2\eta)/4$, and for $\eta> 1/6$, $\eta_*=(1-2\eta)/4$. Let $h_1$ and $h_2$ be the inverses of $H$ respectively restricted to $(-\infty,1/6)$ and to $(1/6,+\infty)$, {\it i.e.} $h_1(x)=(1/6)(1-\sqrt{7-6x})$ and $h_2(x)=(1/6)(1+\sqrt{7-6x})$ for $x\in(-\infty,7/6)$. The following lemmas are taken from Ballou (1970). \begin{lemma}{\bf [B lem2.2]}\label{Bl2.2} Let $\eta<1/6$ be given, and suppose that $\eta^*<\infty$. Then $S[\eta;\eta^*]=H(\eta^*)$. \end{lemma} \begin{lemma}{\bf [B lem2.4]}\label{Bl2.4} Let $\eta< 1/6$ be given, and suppose that $\eta^*<\infty$. Then $\eta^*$ is the only zero of $S[u;\eta]-H(u)$, $u>\eta$. \end{lemma} If $\lambda<\rho<1/6$, the relevant part of the flux function is convex and the unique entropic weak solution is the continuous solution with a rarefaction fan. If $\rho <\lambda<\rho^*$ $(\rho <1/6)$, then $H(\eta)>H(\rho)$ if $\rho<\eta\leq 1/6$, and $H(\eta)>H(\rho^*)>H(\rho)$ if $1/6<\eta<\rho^*$ since $H$ is decreasing in this region. Thus $H(\lambda)>H(\rho)$, which implies an intersection of characteristics: The unique entropic weak solution is the shock. Let $\rho<\rho^*< \lambda$ $(\rho<1/6)$: Lemma \ref{Bl2.4} applied to $\rho$ suggests that a jump from $\rho^*$ to $\rho$ along the line $x=H(\rho^*)t$ will satisfy the Rankine-Hugoniot condition. Since $\rho^*$ is specially defined for this, a solution with such a jump will also satisfy condition E. Therefore if we can construct a solution with the jump described above it will be the unique entropic weak solution in this case. Notice that since $H(\lambda)0$. Then \[ \lim_{t\to\infty} \mu_{\lambda,\rho}\tau_{[vt]}S_2(t)=\nu_{u(v,1)} \] at every continuity point of $u(.,1)$, where $\nu_{u(v,1)}$ denotes the product measure with density $u(v,1)$ defined by: Case 1. $\lambda<\rho< 1/6$: continuous solution, with a rarefaction fan \[ u(x,1)=\left\{ \begin{array}{ll} \lambda, & x\leq H(\lambda);\\ h_1(x), & H(\lambda)< x\leq H(\rho);\\ \rho, & H(\rho)< x.\\ \end{array} \right. \] Case 2. $\rho<\lambda<\rho^*$, ($\rho< 1/6$): entropy shock \[ u(x,1)=\left\{ \begin{array}{ll} \lambda, & x\leq S[\lambda;\rho];\\ \rho, & x> S[\lambda;\rho].\\ \end{array} \right. \] Case 3. $\rho<\rho^*<\lambda$, ($\rho< 1/6$): contact discontinuity \[ u(x,1)=\left\{ \begin{array}{ll} \lambda, & x\leq H(\lambda);\\ h_2(x), & H(\lambda)< x\leq H(\rho^*);\\ \rho, & H(\rho^*)< x.\\ \end{array} \right. \] Case 4. $1/6<\rho<\lambda$: continuous solution, with a rarefaction fan \[ u(x,1)=\left\{ \begin{array}{ll} \lambda, & x\leq H(\lambda);\\ h_2(x), & H(\lambda)< x\leq H(\rho);\\ \rho, & H(\rho)< x.\\ \end{array} \right. \] Case 5. $\rho>\lambda>\rho_*$, ($\rho>1/6$): entropy shock \[ u(x,1)=\left\{ \begin{array}{ll} \lambda, & x\leq S[\lambda;\rho];\\ \rho, & x> S[\lambda;\rho].\\ \end{array} \right. \] Case 6. $\rho>\rho_*>\lambda$, ($\rho> 1/6$): contact discontinuity \[ u(x,1)=\left\{ \begin{array}{ll} \lambda, & x\leq H(\lambda);\\ h_1(x), & H(\lambda)< x\leq H(\rho_*);\\ \rho, & H(\rho_*)< x.\\ \end{array} \right. \] \end{theorem} \begin{remark} \end{remark} For any $k\geq 2$ the profiles will be of the same kind, because $G_k$ has only one inflection point between 0 and 1 and is first convex then concave. \begin{remark} \end{remark} Comparing with hydrodynamics of simple exclusion we observe that $k$-step exclusion ($k\geq 2$) has not only a stable increasing shock (Case 5) and a decreasing continuous solution (Case 1) but also a stable decreasing shock (Case 2), an increasing continuous solution (Case 4) and two contact discontinuities (Cases 3 and 6). \subsubsection{Proof of Theorem \ref{limhydro}} It follows the scheme introduced in Andjel \& Vares (1987), where the authors obtained the hydrodynamic limit for the one-dimensional zero-range process in the Riemann case, i.e. the hydrodynamic equation \[ \left\{ \begin{array}{ll} \displaystyle{\frac{\partial u}{\partial t}+\frac{\partial \phi(u)}{\partial x}=0}\\ \\ u(x,0)=u^0(x)=\lambda 1_{\{x<0\}}+\rho 1_{\{x\geq 0\}}\\ \end{array} \right. \] was derived. There $\phi$, the mean flux of particles through the origin, was a concave function. Therefore, their proof used both the monotonicity of the process, still valid here, and the concavity of the flux, that we have to replace by an {\it ad hoc} use of the properties of the solution of (\ref{hydro}). Informally speaking, they first showed that a weak Ces\'aro limit of (the measure of) the process is an invariant and translation invariant measure. Then they showed that the (Ces\'aro) limiting density inside a macroscopic box is equal to the difference of the edge values of a flux function. These propositions were based on monotonicity, and on the characterization of invariant and translation invariant measures (both valid for k-step as well),thus we can quote them (with appropriate notation for the $k$-step), and take them for granted. \begin{lemma}{\bf [AV 3.1]} Let $\mu$ be a probability measure on $\{0,1\}^{\Z}$ such that (a) $\nu_\rho\leq\mu\leq\nu_\lambda$ for some $0\leq\rho<\lambda\leq 1$,\ \ (b) either $\mu\tau_1\leq\mu$ or $\mu\tau_1\geq\mu$. \noindent Then any sequence $T_n\to\infty$ has a subsequence $T_{n_k}$ for which there exists $D$ dense (countable) subset of $\R$ such that for each $v\in D$, \[ \lim_{k\to\infty}\frac 1 {T_{n_k}} \int_0^{T_{n_k}} \mu\tau_{[vt]}S_2(t)dt=\mu_v \] for some $\mu_v\in{\cal I}_2\cap {\cal S}$. \end{lemma} \begin{lemma}{\bf [AV 3.2]} For $v\in D$, we can write $\mu_v=\int\nu_{\alpha}\gamma_v(d\alpha)$, where $\gamma_v$ is a probability on $[\rho,\lambda]$. Also, if $u S[\lambda;\rho]\\ \end{array} \right. \] then \[ \lim_{T\to\infty}\frac 1 T \int_0^T \mu\tau_{[vt]}S_2(t)dt=\mu_v. \] \end{proposition} \begin{proposition}{\bf [AV 3.5]}\label{AV3.5} If $\mu$ satisfies (a) $\mu\leq\nu_\lambda$, (b) $\mu\tau_1\geq\mu$, (c) there exists $v_0$ finite so that \[ \lim_{T\to\infty}\frac 1 T \int_0^T \mu\tau_{[vt]}S_2(t)dt=\nu_\lambda \] for all $v>v_0$. Then \[ \lim_{t\to\infty}\mu\tau_{[vt]}S_2(t)=\nu_\lambda \quad {\rm for\,all\,\,}v>v_0. \] \end{proposition} \noindent {\bf Proof of Theorem \ref{limhydro} in Cases 2 and 3.} \noindent In case 1, the proof is not different from the one given in Andjel \& Vares (1987) since $G$ is convex in the relevant region; thus we omit it. In case 2, $G$ is not convex in the relevant region, therefore we supply a proof, though it is quite close to the original one. Case 3 uses ideas from cases 1 and 2 and introduces some new ideas to deal with complications arising from the non-convexity of $G$. Cases 4-6 are symmetric to cases 1-3 in the sense that the roles played by convexity and concavity are exchanged. \par \smallskip The proof has 3 steps. The main ingredients are monotonicity, and inequalities relying on the properties of $S[.;.],\,G,\,H$ given before. \noindent {\it First step.}\par Using monotonicity of the 2-step exclusion, we can proceed as in the beginning of the proof of Lemma [AV3.3], and get two finite values $\underbar v$ and $\bar v$ so that: If $v\in D$ and $v>\bar v$, then $\gamma_v=\delta_\rho$, while $\gamma_v=\delta_\lambda$ if $v<\underbar v$. \smallskip \noindent {\it Second step, preliminary.}\par Let $uS[\lambda;\rho]$ ($u\in D$) and proceeding in a similar manner we can show that $\gamma_u = \delta_\rho$.\par \smallskip \noindent {\bf Proof for case 3:} $\rho<\rho^*<\lambda,\,\rho< 1/6$.\par \noindent {\it Second step, part 1.}\par Let $u\bar v$, and $H(\rho^*)0$, this implies \[ \int_{[\rho,\lambda]}(\alpha-\rho)\gamma_u(d\alpha)= 0 \] \noindent Therefore we conclude $\gamma_u=\delta_\rho$. \bigskip \noindent {\it Second step, conclusion.}\par Using the two preceding parts, attractivity, Propositions \ref{AV3.4} and \ref{AV3.5}, we conclude in case 3 \begin{equation}\label{4.2.2} \lim_{t\to\infty}\mu_{\lambda,\rho}\tau_{[vt]}S_2(t)=\left\{ \begin{array}{ll} \nu_\lambda, & {\it if\,\,} v< H(\lambda)\\ \nu_\rho, & {\it if\,\,} v> H(\rho^*)\\ \end{array} \right. \end{equation} \noindent and in case 2 \begin{equation}\label{4.2.2a} \lim_{t\to\infty}\mu_{\lambda,\rho}\tau_{[vt]}S_2(t)=\left\{ \begin{array}{ll} \nu_\lambda, & {\it if\,\,} v< S[\lambda;\rho]\\ \nu_\rho, & {\it if\,\,} v> S[\lambda;\rho]\\ \end{array} \right. \end{equation} \bigskip \noindent {\it Third step, first part.}\par Let $u_1,v,v_1\in D$, $u_1 G(\rho) - v \rho \] Because $G(\alpha)-v\alpha$ is increasing in $(\rho^*,\theta)$ (recall that $H(\theta) = v$) we have: For all $\alpha\in (\rho^*,\theta)$ \[ G(\alpha)-v\alpha>G(\rho^*)-v\rho^*>G(\rho)-v\rho \] We conclude that $\theta$ is a global maximum: \[ \max_{\rho\leq\alpha\leq\lambda}[G(\alpha)-v\alpha]=G(\theta)-v\theta \] thus $\gamma_v=\delta_\theta=\delta_{h_2(v)}$. \bigskip \noindent {\it Third step, second part.}\par This part follows closely the argument in Andjel \& Vares (1987), but we detail it for the sake of completeness. Since the measures $\frac 1 T \int_0^T \mu_{\lambda,\rho}\tau_{[vt]}S_2(t)\ dt$ depend monotonically on $v$ and form a relatively compact set, we have for all $v$, \[ \lim_{T\to\infty} \frac 1 T \int_0^T \mu_{\lambda,\rho}\tau_{[vt]}S_2(t)\ dt=\nu_{h_2(v)}. \] It remains to prove that \[ \lim_{t\to\infty}\mu_{\lambda,\rho}\tau_{[vt]}S_2(t)=\nu_{h_2(v)} \] when $H(\lambda)q$, we obtain \[ \begin{array}{ll} G_5^{p,q}(u)=&u(1-u)\left[(p-q)\left\{(1+2u)+3u^2(1-pq)\right.\right.\\ &\left.\left.+4u^3(1-2pq)+5u^4(1-3pq+p^2q^2)\right\}+3u^2 2p^4q\right].\\ \end{array} \] Indeed the last term corresponds to a ``cycle'' jump: To go from $x$ to $x+3$ when sites $x+1$ and $x+2$ are occupied, the particle follows the path $(x,x+1,x+2,x+1,x+2,x+3)$.\par The extension to an initial product measure with a general profile is still possible, using Rezakhanlou (1991) and Landim (1993), but with the help of Young measures.\par \bigskip \noindent {\bf Aknowledgments:} We thank E. Andjel, C. Bahadoran, P. Ferrari, J. Krug and G. Sch\"utz for fruitful discussions. This work was supported by the agreement USP/COFECUB n$^o$ UC 45/97, and FAPESP project n$^o$ 1999/04874-6. The three authors thank IME at S\~ao Paulo for their kind hospitality, H.G. \& K.R. thank Universit\'e de Rouen where part of this work was realized. \begin{thebibliography}{10} \bibitem{ad} Aldous, D.J. \& Diaconis, P., Hammersley interacting particle process and longest increasing subsequences, {\sl Probab. Theory Relat. Fields}, {\bf 103} (1995), 199-213. \bibitem{andkip} Andjel, E.D. \& Kipnis, C., Derivation of the hydrodynamical equation for the zero range interaction process, {\sl Ann. Probab.}, {\bf 12} (1984), 325-334. \bibitem{andvar} Andjel, E.D. \& Vares M.E., Hydrodynamic equations for Attractive Particle Systems on ${\bf Z}$, {\sl J. Stat. 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