The following is a plain TeX (Unix) file. BODY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%macros start here%%%%%%%%%%%%% \newcount\driver \newcount\mgnf \newcount\tipi \newskip\ttglue %%cm completo \def\TIPITOT{ \font\dodicirm=cmr12 \font\dodicii=cmmi12 \font\dodicisy=cmsy10 scaled\magstep1 \font\dodiciex=cmex10 scaled\magstep1 \font\dodiciit=cmti12 \font\dodicitt=cmtt12 \font\dodicibf=cmbx12 scaled\magstep1 \font\dodicisl=cmsl12 \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\seirm=cmr6 \font\seibf=cmbx6 \font\seii=cmmi6 \font\seisy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\dodicitruecmr=cmr10 scaled\magstep1 \font\dodicitruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\seitruecmr=cmr6 \font\seitruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% definizioni per 10pt %%%%%%%% \textfont\truecmr=\tentruecmr \scriptfont\truecmr=\seventruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\tentruecmsy \scriptfont\truecmsy=\seventruecmsy \scriptscriptfont\truecmr=\fivetruecmr \scriptscriptfont\truecmsy=\fivetruecmsy %%%%% cambio grandezza %%%%%% \def \ottopunti{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\seirm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\seii \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\seisy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\seibf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\seirm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\seitruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\eighttruecmsy \scriptfont\truecmsy=\seitruecmsy }\let\nota=\ottopunti} \newfam\msbfam %per uso in \TIPITOT \newfam\truecmr %per uso in \TIPITOT \newfam\truecmsy %per uso in \TIPITOT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%cm ridotto \def\TIPI{ \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seitruecmr=cmr6 \textfont\truecmr=\tentruecmr \textfont\truecmsy=\tentruecmsy %%%%% cambio grandezza %%%%%% \def \ottopunti{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \textfont1=\eighti \textfont2=\eightsy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\seirm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\seitruecmr %\textfont\truecmsy=\eighttruecmsy }\let\nota=\ottopunti} %%am \def\TIPIO{ \font\setterm=amr7 %\font\settei=ammi7 \font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7 %%%%% cambiamenti di formato %%% \def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti \textfont0=\setterm %\textfont1=\settei \textfont2=\settesy %\textfont3=\setteit %\textfont\itfam=\setteit \def\it{\fam\itfam\setteit} \textfont\bffam=\settebf \def\bf{\fam\bffam\settebf} \normalbaselineskip=9pt\normalbaselines\rm }\let\nota=\settepunti} %%%%%%%%% GRAFICA % % Inizializza le macro postscript e il tipo di driver di stampa. % Attualmente le istruzioni postscript vengono utilizzate solo se il driver % e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o PSPRINT (\driver=2); % qualunque altro valore di \driver produce un output in cui le figure % contengono solo i caratteri inseriti con istruzioni TEX (vedi avanti). % %\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi %\ifnum\driver=1 \special{header=ini.pst} \fi \newdimen\xshift \newdimen\xwidth \newdimen\yshift % % inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro % occupi la posizione (#1,#2) % \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} % % Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in % #4.ps; in questo disegno si possono introdurre delle stringhe usando \ins % e mettendo le istruzioni relative nell'argomento #3. % Il file #4.ps contiene le istruzioni postscript, che devono essere scritte % presupponendo che l'origine sia nell'angolo inferiore sinistro della % scatola, mentre per il resto l'ambiente grafico e' quello standard. % #5 deve essere della forma \eq("nome simbolico"). % % Le istruzioni postscript possono essere inserite nel file che contiene % l'istruzione \insertplot, racchiudendole fra le istruzioni \initfig{#4} % e \endfig; inoltre ogni riga deve cominciare con "write13<" e deve finire % con ">". In questo modo si crea il file #4.ps relativo alla figura. % \def\insertplot#1#2#3#4#5{\par% \xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2% \yshift=#2 \divide\yshift by 2% \line{\hskip\xshift \vbox to #2{\vfil% \ifnum\driver=0 #3 \special{ps::[local,begin] gsave currentpoint translate}% \special{ps: plotfile #4.ps} \special{ps::[end]grestore}\fi \ifnum\driver=1 #3 \special{psfile=#4.ps}\fi \ifnum\driver=2 #3 \special{ \ifnum\mgnf=0 #4.ps 1. 1. scale \fi \ifnum\mgnf=1 #4.ps 1.2 1.2 scale\fi} \special{ini.ps} \fi }\hfill \raise\yshift\hbox{#5}}} \def\initfig#1{% \catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2 \openout13=#1.ps} \def\endfig{% \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12} %%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ps=\psi \let\r=\rho \let\s=\sigma %\let\t=\tau \let\th=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega \let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon % %%%%%%%%%%%%%%%%%%%%% Numerazione pagine % \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year;\,\the\time} % \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} % \def\footnormal{ \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} } \def\footappendix{ \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglioa\hss} } % %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% Per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...); %%% dentro le parentesi e al posto di ... si puo' scrivere qualsiasi commento; %%% per avere i nomi simbolici segnati a sinistra delle formule si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi: \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn). %%% Si possono citare formule seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \global\newcount\numsec \global\newcount\numfor \global\newcount\numtheo \global\advance\numtheo by 1 \gdef\profonditastruttura{\dp\strutbox} % \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} % \def\SIA #1,#2,#3 {\senondefinito{#1#2}% \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} % \def\etichetta(#1){(\veroparagrafo.\veraformula)% \SIA e,#1,(\veroparagrafo.\veraformula) % \global\advance\numfor by 1% \write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} % \def\letichetta(#1){\veroparagrafo.\verotheo \SIA e,#1,{\veroparagrafo.\verotheo} \global\advance\numtheo by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ Sta \equ(#1) == #1 }} % \def\letichettaa(#1){A.\verotheo \SIA e,#1,{A.\verotheo} \global\advance\numtheo by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ Sta \equ(#1) == #1 }} % \def\tetichetta(#1){\veroparagrafo.\veraformula %%%%copy four lines \SIA e,#1,{(\veroparagrafo.\veraformula)} \global\advance\numfor by 1 \write15{\string\FU (#1){\equ(#1)}} \write16{ tag #1 ==> \equ(#1)}} % \def\FU(#1)#2{\SIA fu,#1,#2 } % \def\etichettaa(#1){(A.\veraformula)% \SIA e,#1,(A.\veraformula) % \global\advance\numfor by 1% \write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} % \def\BOZZA{ \def\alato(##1){% {\rlap{\kern-\hsize\kern-1.4truecm{$\scriptstyle##1$}}}}% \def\aolado(##1){% {%\vtop to \profonditastruttura {%\baselineskip %\profonditastruttura\vss \rlap{\kern-1.4truecm{$\scriptstyle##1$}}}}} } % \def\alato(#1){} \def\aolado(#1){} % \def\veroparagrafo{\number\numsec} \def\veraformula{\number\numfor} \def\verotheo{\number\numtheo} \def\verafigura{\number\numfig} % \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\leq(#1){\leqno{\aolado(#1)\etichetta(#1)}}%%%%%this line for \leqno \def\teq(#1){\tag{\aolado(#1)\tetichetta(#1)\alato(#1)}}%%%%%this line for\tag \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1 \write16{#1 non e' (ancora) definito}% \else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} % %%%% next six lines by paf (no responsibilities taken) \def\Lemma(#1){\aolado(#1)Lemma \letichetta(#1)}% \def\Lemmaa(#1){\aolado(#1)Lemma \letichettaa(#1)}% \def\Theorem(#1){{\aolado(#1)Theorem \letichetta(#1)}}% \def\Proposition(#1){\aolado(#1){Proposition \letichetta(#1)}}% \def\Corollary(#1){{\aolado(#1)Corollary \letichetta(#1)}}% \def\Remark(#1){{\noindent\aolado(#1){\bf Remark \letichetta(#1).}}}% \def\Definition(#1){{\noindent\aolado(#1){\bf Definition \letichetta(#1)$\!\!$\hskip-1.6truemm}}} \def\Example(#1){\aolado(#1) Example \letichetta(#1)$\!\!$\hskip-1.6truemm} \let\ppclaim=\plainproclaim % \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} % \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \openout15=\jobname.aux % \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa % %%%%%%%%%%%%%%% DEFINIZIONI LOCALI % \let\ciao=\bye \def\fiat{{}} \def\pagina{{\vfill\eject}} \def\\{\noindent} \def\bra#1{{\langle#1|}} \def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }} \let\ii=\int \let\ig=\int \let\io=\infty \let\i=\infty % \let\dpr=\partial \def\V#1{\vec#1} \def\Dp{\V\dpr} \def\oo{{\V\o}} \def\OO{{\V\O}} \def\uu{{\V\y}} \def\xxi{{\V \xi}} \def\xx{{\V x}} \def\yy{{\V y}} \def\kk{{\V k}} \def\zz{{\V z}} \def\rr{{\V r}} \def\pp{{\V p}} % \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} % \def\guida{\ ...\ } \def\Z{{\bf Z}}\def\R{{\bf R}}\def\tab{}\def\nonumber{} \def\mbox{\hbox}\def\lis#1{{\overline#1}}\def\nn{{\V n}} \def\Tr{{\rm Tr}\,}\def\EE{{\cal E}} \def\Veff{{V_{\rm eff}}}\def\Pdy{{P(d\psi)}}\def\const{{\rm const}} %\def\RR{{\cal R}} \def\NN{{\cal N}}\def\ZZ#1{{1\over Z_{#1}}} \def\OO{{\cal O}} \def\GG{{\cal G}} \def\LL{{\cal L}} \def\DD{{\cal D}} \def\fra#1#2{{#1\over#2}} \def\ap{{\it a priori\ }} \def\rad#1{{\sqrt{#1}\,}} \def\eg{{\it e.g.\ }} \def\={{\equiv}}\def\ch{{\chi}} % \def\initfiat#1#2#3{ \mgnf=#1 \driver=#2 \tipi=#3 \ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi %\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi %\ifnum\driver=1 \special{header=ini.pst} \fi % %%%%%%%%%%%%%%% FORMATO % \ifnum\mgnf=0 \magnification=\magstep0 \hoffset=0.cm \voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=25.truecm \baselineskip=14pt % plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 \fi \ifnum\mgnf=1 \magnification=\magstep1 \hoffset=0.cm \voffset=-1truecm \hoffset=-.5truecm \hsize=16.5truecm \vsize=25truecm \baselineskip=12pt % plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt\parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 \fi \setbox200\hbox{$\scriptscriptstyle \data $} } %%%%%%%%%%%end of Gallavotti's macros%%%%%%%%% %%%%%%%%%%%inizialization%%%%%%%%%%% %%%%%%put % in front of \BOZZA to remove labels on the left%%%%%%%%%%% \initfiat {1}{1}{2} %\BOZZA %\input amssym.def% %%%%%%%%amssym.def included here%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \expandafter\ifx\csname amssym.def\endcsname\relax \else\endinput\fi % % Store the catcode of the @ in the csname so that it can be restored later. \expandafter\edef\csname amssym.def\endcsname{% \catcode`\noexpand\@=\the\catcode`\@\space} % Set the catcode to 11 for use in private control sequence names. \catcode`\@=11 % % Include all definitions related to the fonts msam, msbm and eufm, so that % when this file is used by itself, the results with respect to those fonts % are equivalent to what they would have been using AMS-TeX. % Most symbols in fonts msam and msbm are defined using \newsymbol; % however, a few symbols that replace composites defined in plain must be % defined with \mathchardef. % \def\undefine#1{\let#1\undefined} \def\newsymbol#1#2#3#4#5{\let\next@\relax \ifnum#2=\@ne\let\next@\msafam@\else \ifnum#2=\tw@\let\next@\msbfam@\fi\fi \mathchardef#1="#3\next@#4#5} \def\mathhexbox@#1#2#3{\relax \ifmmode\mathpalette{}{\m@th\mathchar"#1#2#3}% \else\leavevmode\hbox{$\m@th\mathchar"#1#2#3$}\fi} \def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or 9\or A\or B\or C\or D\or E\or F\fi} % \font\tenmsa=msam10 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \edef\msafam@{\hexnumber@\msafam} \mathchardef\dabar@"0\msafam@39 \def\dashrightarrow{\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}} \def\dashleftarrow{\mathrel{\mathchar"0\msafam@4C\dabar@\dabar@}} \let\dasharrow\dashrightarrow \def\ulcorner{\delimiter"4\msafam@70\msafam@70 } \def\urcorner{\delimiter"5\msafam@71\msafam@71 } \def\llcorner{\delimiter"4\msafam@78\msafam@78 } \def\lrcorner{\delimiter"5\msafam@79\msafam@79 } \def\yen{{\mathhexbox@\msafam@55 }} \def\checkmark{{\mathhexbox@\msafam@58 }} \def\circledR{{\mathhexbox@\msafam@72 }} \def\maltese{{\mathhexbox@\msafam@7A }} % \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \edef\msbfam@{\hexnumber@\msbfam} \def\Bbb#1{{\fam\msbfam\relax#1}} \def\widehat#1{\setbox\z@\hbox{$\m@th#1$}% \ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5B{#1}% \else\mathaccent"0362{#1}\fi} \def\widetilde#1{\setbox\z@\hbox{$\m@th#1$}% \ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5D{#1}% \else\mathaccent"0365{#1}\fi} \font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \newfam\eufmfam \textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm \scriptscriptfont\eufmfam=\fiveeufm \def\frak#1{{\fam\eufmfam\relax#1}} \let\goth\frak % % Restore the catcode value for @ that was previously saved. \csname amssym.def\endcsname % % %%%%%%%%%%%%end of amssym.def%%%%%%%% % %%%%%%%%%%%%%%%%%%extra definitions already in yau's file%%%%%%%% % \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt \hbox{\vrule width.#2pt height#1pt \kern#1pt \vrule width.#2pt}\hrule height.#2pt}}}} \def\qed{ $\mathchoice\sqr64\sqr64\sqr{2.1}3\sqr{1.5}3$} \def\ZZ{Z\!\!\!Z\,} \def\RR{R\!\!\!\!\!I\,\,} \def\II{\ \hbox{I}\!\!\!\hbox{I}\,} \def\11{\hbox{l}\!\!\!1\,} \def\QIF{\quad\hbox{ if }\quad} \font\tenib=cmmib10 \newfam\mitbfam \textfont\mitbfam=\tenib \scriptfont\mitbfam=\seveni \scriptscriptfont\mitbfam=\fivei \def\mitb{\fam\mitbfam} \def\balpha{{\mitb\mathchar"710B}} \def\bbeta{{\mitb\mathchar"710C}} \def\bgamma{{\mitb\mathchar"710D}} \def\bdelta{{\mitb\mathchar"710E}} \def\bepsilon{{\mitb\mathchar"710F}} \def\bzeta{{\mitb\mathchar"7110}} \def\boeta{{\mitb\mathchar"7111}} %bold eta %above ceta because bold eta should not be beta \def\btheta{{\mitb\mathchar"7112}} \def\biota{{\mitb\mathchar"7113}} \def\bkappa{{\mitb\mathchar"7114}} \def\blambda{{\mitb\mathchar"7115}} \def\bmu{{\mitb\mathchar"7116}} \def\bnu{{\mitb\mathchar"7117}} \def\bxi{{\mitb\mathchar"7118}} %\def\bomicron{{\mitb\mathchar"7122}} %above, not omicron, sort of script e lower case \def\bpi{{\mitb\mathchar"7119}} \def\brho{{\mitb\mathchar"701A}} \def\bsigma{{\mitb\mathchar"701B}} \def\btau{{\mitb\mathchar"701C}} \def\bupsilon{{\mitb\mathchar"701D}} \def\bchi{{\mitb\mathchar"701F}} \def\bpsi{{\mitb\mathchar"7120}} \def\bomega{{\mitb\mathchar"7121}} %\baselineskip7mm \def\and{ \hbox{ and } } \def\la {\big\langle} \def\ra {\big\rangle} \def\pa{\parallel} \def\pt{\partial} \def\l{\lambda} \def\L{\Lambda} \def\e{\varepsilon} \def\a{\alpha} \def\be{\beta} \def\d{\delta} \def\g{\gamma} \def\o{\omega} \def\om{\omega} \def\O{\Omega} \def\n{\nabla} \def\s{\sigma} \def\t{\theta} \def\newpage{\vfill\eject} \def\Cal{\cal} \def\to{\rightarrow} \def\if {\hbox { if } } \def\QAND{\quad\hbox{ and }\quad} \def\frac{\over} \def\\{\cr} \def\ref{} \def\nonumber{} \hbox{} \vfill \baselineskip12pt \overfullrule=0in \def\s{\sigma} \def\si{\sigma} \def\eps{\epsilon} \def\ga{\gamma} \def\a{\alpha} \def\ka{\kappa} \def\h{\hat} \def\d{\delta} \def\rh{\rho} \def\ti{\tilde} \def\om{\omega} \def\z{\zeta} \def\I{\Bbb I} \def\ct{\hbox {cost.}} \def\text{\hbox} \def\\\ {\cr} \def\und{\underline} \def\ve{\varepsilon} % %%%%%%%%References macros start here%%%%%%%%%% 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\expandafter\xdef\csname#2check##1\endcsname{}% \expandafter\xdef\csname#2tag##1\endcsname {#1\number\csname#2tagno\endcsnameX}% \write\t@gsout{#2tag ##1 assigned number \csname#2tag##1\endcsname\space on page \number\count0.}% \csname#2tag##1\endcsname}}% \def\m@kecs #1tag #2 assigned number #3 on page #4.% {\expandafter\gdef\csname#1tag#2\endcsname{#3} \expandafter\gdef\csname#1page#2\endcsname{#4}} \def\re@der{\ifeof\t@gsin\let\next=\relax\else \read\t@gsin to\t@gline\ifx\t@gline\v@idline\else \expandafter\m@kecs \t@gline\fi\let \next=\re@der\fi\next} \def\t@gs#1{\def\alltgs{}\m@ketag#1e\m@ketag#1s\m@ketag\t@d@l p \m@ketag\gl@b@l r \n@wread\t@gsin\openin\t@gsin=\jobname.tgs \re@der \closein\t@gsin\n@wwrite\t@gsout\openout\t@gsout=\jobname.tgs } \outer\def\localtags{\t@gs\l@c@l} \outer\def\globaltags{\t@gs\gl@b@l} \outer\def\newlocaltag#1{\m@ketag\l@c@l{#1}} \outer\def\newglobaltag#1{\m@ketag\gl@b@l{#1}} \outer\def\subheadings{\outer\def\newsection##1{\resetall\message{<##1> }\subheading{##1}}} \def\t@gsoff#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsoff \expandafter\gdef\csname#1cite\endcsname{\relax} \expandafter\gdef\csname#1page\endcsname##1{?} \expandafter\gdef\csname#1tag\endcsname{\relax}\fi\next} \def\verbatimtags{\let\ift@gs=\iffalse\ifx\alltgs\relax\else \expandafter\t@gsoff\alltgs,\fi} \catcode`\X=11 %\catcode`\@=\active \localtags %%%%%%%%%%%%references macro end here%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%end of macros%%%%%%%%%%%%%%%%%%%%%% \let\eps=\varepsilon \let\epsilon=\varepsilon \def\ds{\displaystyle} \def\st{\scriptstyle} \def\sst{\scriptscriptstyle} %%%%%%%%%%%%%%%text starts here%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% Local macros %%%%%%%%%%%%%%%%%%%%%%%%%% \font\ninebf=cmbx9 \font\title=cmbx12\def\1{\,\rlap{\ninerm 1}\kern.15em 1} \def\cd{{\cal D}} %\def\E{{I\kern-.25em{E}}} \def\ce{{\cal E}} \def\cm{{\cal M}} %\def\cite#1{[#1]} \def\cg{{\cal G}} \def\ch{{\cal H}} %\def\ci{{\cal I}} %\def\cj{{\cal J}} \def\cl{{\cal L}} \def\cle{\cl^{\rm ex}} \def\clc{\cl^{\rm c}} \def\cq{{\cal Q}} \def\cv{{\cal V}} \def\div{\,{\rm div}} \let\lra=\leftrightarrow %\def\m#1{\langle #1\rangle} %\def\mod#1{\,{\rm mod}\,#1} %\def\text#1{{\rm #1}} \let\n=\eta \let\na=\nabla \let\ov=\overline \let\p=\partial \def\ps#1{\bigl\langle #1\bigr\rangle} \def\r{\rho} \let\un=\underline \let\ve=\varepsilon \let\vp=\varpi \let\wt=\widetilde \def\S{\Omega} \def\N{\Bbb N} \def\H{\Bbb H} \def\R{\Bbb R} \def\T{\Bbb T} \def\Z{\Bbb Z} \def\E{\Bbb E} \def\P{\Bbb P} \def\bl{\bigl} \def\br{\bigr} \def\Bl{\Bigl} \def\Br{\Bigr} \def\lt{\left} \def\rt{\right} \def\D{\Bbb D} %%%%%%%%%%%%%%%%%%%%%%%%%% Local Format %%%%%%%%%%%%%%%%%%%%%%%%%% %\magnification=\magstep1 \hoffset=0.cm \voffset=-.2truecm \hoffset=-.2truecm \hsize=16.5truecm \vsize=24truecm \baselineskip=12pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt\parskip=0.1pt plus1pt %% comment next line to remove the printing of labels %\BOZZA \centerline {\title Equilibrium Fluctuations for Lattice Gases.} \vskip 1truecm \centerline{\baselineskip=10pt O. Benois\footnote{$^1$}{\eightrm Laboratoire Salem, UMR 6085, site Colbert, Universit\'e de Rouen, 76821 Mont Saint Aignan, France}, \hskip.2cm R. Esposito\footnote{$^2$}{\eightrm Dipartimento di Matematica Pura ed Applicata, Universit\`a di L'Aquila, 67100 Coppito, AQ, Italy} \footnote{$ $} {\eightrm and Centro Linceo Interdisciplinare ``Beniamino Segre'', Via della Lungara 10, 00165 Roma, Italy}, \hskip.2cm R. Marra\footnote{$^3$}{\eightrm Dipartimento di Fisica e Unit\`a INFM, Universit\`a di Roma Tor Vergata, Via della Ricerca Scientifica,}\footnote{$ $} {\eightrm00133 Roma, Italy}} \vskip.5cm \centerline{Universit\'e de Rouen, Universit\`a di L'Aquila, Universit\`a di Roma Tor Vergata} \bigskip {\baselineskip = 8pt\rightskip1.4cm\leftskip 1.4cm\ottopunti {\noindent {\bf Abstract:}} \quad The authors in a previous paper proved the hydrodynamic incompressible limit in $d\ge 3$ for a thermal lattice gas, namely a law of large numbers for the density, velocity field and energy. In this paper the equilibrium fluctuations for this model are studied and a central limit theorem is proved for a suitable modification of the vector fluctuation field $\z(t)$, whose components are the density, velocity and energy fluctuations fields. We consider a modified fluctuation field $\xi^\e(t)=\exp \{-\ve^{-1}t E\}\z^\ve$, where $E$ is the linearized Euler operator around the equilibrium and prove that $\xi^\e(t)$ converges to a vector generalized Ornstein-Uhlenbeck process $\xi(t)$, which is formally solution of the stochastic differential equation $d \xi(t)= N\xi(t)dt+ B dW_t$, with $ BB^*=-2 NC$, where $C$ is the compressibility matrix, $N$ is a matrix whose entries are second order differential operators and $B$ is a mean zero Gaussian field. The relation $-2NC=BB^*$ is the fluctuation-dissipation relation. } \bigskip \noindent{\ottopunti {\bf Key Words:} Fluctuations, Stochastic Cellular Automata, Navier-Stokes Equation.} \bigskip \noindent{\ottopunti {\bf MSC2000 Classification:} 60K35, 82C22} \vskip1.5cm \noindent{\bf 1. Introduction.} \vskip .5truecm \numsec= 1 \numfor= 1 \numtheo=1 In this paper we study the equilibrium fluctuations for the stochastic lattice gas introduced in [BEM]. It is a model of particles with discrete velocities jumping on the lattice: a particle with a given velocity moves on the 3-d lattice as the asymmetric simple exclusion process with the jump intensity chosen so to have a drift equal to its velocity. In each site particles collide exchanging velocities in such a way to conserve the number of particles, the momentum in each direction and the energy. This model generalizes the one in [EMY2] to include the case of particles with different kinetic energy. In [BEM] it has been proved the law of large number for this model in the following form. We choose as initial state a Bernoulli measure with density, momentum and energy small perturbation (of order $\e$) of constant profiles. Then the empirical fields $\nu_\b^\e(x,t)=\nu_\b(\ve^{-1}x, \ve^{-2}t), \b=0,\dots,4$ of density, momentum and energy converge weakly in probability as $\e$ goes to $0$ to the solution of the hydrodynamic equations for this model, which are the incompressible Navier-Stokes equation for the velocity field and a diffusive equation (including the transport along the velocity field) for the energy. The dissipative terms in these equations are given in terms of a diffusion tensor $D^{\b,\nu}_{\a,\g}, \b,\nu=0,\dots,4,\ \a,\g=1,\dots,3$, which is expressed by the Green-Kubo formulas. The next natural step is to prove the space-time central limit theorem, namely that the fluctuations fields starting from the equilibrium state converge to a stationary multi-dimensional Gaussian process with a given space-time covariance. Since the macroscopic behavior of this model is very close to the real hydrodynamics we face the main difficulty of the hydrodynamic fluctuations: the Euler terms and the Navier-Stokes terms live on different time scales. The same feature is responsible for the impossibility of obtaining the compressible dissipative hydrodynamic equations as scaling limit. In fact, the previous result on the law of large number is true for an initial condition which is a small perturbation of the global equilibrium. This perturbation remains small at later times of order $\ve^{-2}$ and evolves macroscopically according to the incompressible hydrodynamics. The case of the fluctuations is different because a small perturbation of the equilibrium may become very large and be of order $\ve^{-1}$ on times of order $\ve^{-2}$. We go now in some details to explain better this point. The fluctuation fields %change under diffusive scaling are defined by $$ \z^\e _\b(t,G)=\ve^{3\over 2}\sum_x G_\b(\e x) \big[I_\b(\eta_{\ve^{-2}t}(x) -\E[I_\b]\big], %\Eq(0.1) $$ $\b=0,\cdots,4$, where $G_\b$ are suitable test functions, $\eta_t(x)$ is the configuration in $x$ at time $t$ and $\E$ is the equilibrium expectation. $I_\b$ are the quantities conserved by the dynamics, total number of particles, total momentum and total energy in $x$. \bigskip At time zero the limiting fluctuation fields $$ \lim_{\e\to 0}\z^\e _\b(0,G)=\z _\b(0,G) $$ are jointly Gaussian with covariance $$ \E[\z_\b(0,G)\z_\nu(0,H)]= C_{\b,\nu}\int\!d^3 x\,\ G(x)H(x). $$ The matrix $C=(C_{\b,\nu })$ is called the compressibility matrix. The limit is in the in sense of weak convergence of path measures. \medskip It is not hard to show (it is indeed a by-product of the results and estimates in this paper) that the equilibrium fluctuations %change %on a time scale $\tau=\ve^{-1}t$ under Euler time scale are trivial in the sense that they satisfy in the limit a deterministic equation. This is a general feature first showed in [GP], [FF]. More precisely, the limiting field $\z^E=(\z_\b^E), \b=0,\cdots,4$ $$ \z^E(\tau)=\lim_{\ve\to 0}\z_\ve^E (\tau,G)=\lim_{\ve\to 0}\z^\ve \bl(\ve \tau,G\br) $$ is solution of the deterministic equation $$ d \z^E(\tau)=E\z^E(\tau)d\tau, \Eq(0.2) $$ where $E$ is the linearized Euler operator around the global equilibrium. Equations \equ(0.2) are a system of linear hyperbolic equations. The stochastic noise should appear as a correction of order $\ve$ as $$ d \z^E_\ve(\tau)=(A+\ve {\cal D})\z^E_\ve(\tau)d\tau+\sqrt\ve B dW_\tau +O(\ve^2), $$ where ${\cal D}$ is the linearized Navier-Stokes operator around the global equilibrium and $$ B B^*=-2\cd C $$ is the fluctuation-dissipation relation. Hence, to see a finite noise one has to look at longer times $\tau=\ve^{-1}t$. Formally, %change %setting since $\z^\ve(t)=\z^E_\ve(\ve^{-1}t)$, we get $$ d \z^\ve(t)=(\ve^{-1}E+ {\cal D})\z^\ve(t)dt+ B dW_t +O(\ve). $$ Then the limit $\lim_{\ve\to 0}\z^\ve(t)$ does not exist because the Euler modes are too big on this time scale. A similar difficulty is present also in the case of ASEP but the analogous of $E$ is simply an operator of the form $v\cdot\nabla_x$ with $v_i=(p_i-q_i)(1-2\a)$, $p_i$, $q_i$ the rates of jumping to the left and right respectively and $\a=\E[\eta]$. Therefore, a Galilean shift is sufficient to remove the divergence and in fact in [CLO] the central limit theorem is proved for a fluctuation field of the form $$ Y^\ve(t,G)=\ve^{d\over 2}\sum_x G(\ve x- \ve^{-1}vt) \big[(\eta_{\ve^{-2}t}(x) -\E[\eta]\big]. $$ In our case a possible way to subtract the Euler modes is to consider a modified fluctuation field which moves together with the waves solutions of \equ(0.2), traveling with velocity of order $\ve^{-1}$. %change Denoting by $E^*$ the adjoint operator of $E$, we define the fluctuation field as $$ \xi^\ve (t,G)=\z^\ve(t,e^{-{t\over\ve}E^*}G). $$ We prove that the limit $\ve\to 0$ exists and satisfies a suitable stochastic differential equation. Before writing the equation, we consider the same problem in a very simple case: let $A$ and $M$ be $K\times K$ matrix with complex entries such that $A=-A^*$ where the adjoint is relative to the scalar product in $\R^K$. Consider the linear ODE system $$ \dot x_\ve=({\ve^{-1} A}+M) x_\ve ,\quad x_\ve(0)=\bar x. $$ Then, $y_\ve=e^{-{t\over \ve}A}x_\ve$ is solution of $$ \dot y_\ve=e^{-{t\over \ve}A}M e^{{t\over \ve}A}y_\ve, \quad y_\ve(0)=\bar x. $$ Consider the limit $$ U:=\lim_{\ve\to 0}\int _0^1\! ds\, e^{-{s\over \ve}A}M e^{{s\over \ve}A}=\lim_{\ve\to 0}{\ve}\int_0^{1\over \ve}\!ds\, e^{-{s}A}M e^{{s}A} $$ An asymptotic average theorem [EP] states that for any $\d>0$ and $T>0$ there exists $\ve_0>0$ such that the solution $z$ of $$ \dot z= U z, \quad z(0)=\bar x $$ satisfy $$ \sup_{00. %\Eq(3.7b) $$ So, defining for $\b,\nu=0,\ldots,4$ and $\a=1,2,3$, $$ d_\a^{\b,\nu}={\p\E^{\mu_{L,n}}\bigl[w^{(a),\b}_{x,\a}\bigr]\over \p m_\nu(n)}{\big|_{n=n_0},} %\Eq(3.7c) $$ we obtain $$ d_\a^{\b,\nu}=b_0\d_{\b,0}\d_{\a,\nu} +b_4\d_{\b,4}\d_{\a,\nu}+ \1_{\{1,2,3\}}(\b)\d_{\a,\b}\bigl[a_0\d_{\nu,0}+a_4\d_{\nu,4}\bigr], \Eq(3.7d) $$ with $$ b_0 ={\Phi_2\over 3\Phi},\qquad b_4 =2{\Phi_1\over 3\Phi},\qquad a_0 ={\ps{|v|^2h_1}\over\ps{|v|^2h_0}},\qquad a_4 ={\ps{|v|^4h_1}\over\ps{|v|^2h_0}}, \Eq(3.7e) $$ $$ \eqalign{ \Phi_1 &=\ps{h_1|v|^4}\ps{h_0}-\ps{h_1|v|^2}{\ps{h_0|v|^2}},\cr \Phi_2 &=\ps{h_0|v|^4}\ps{h_1|v|^2}-\ps{h_1|v|^4} \ps{h_0|v|^2}.\cr } %\Eq(2.23) $$ Therefore the local function $$ g^\b_{\a}(\n)= w^{(a),\b}_{0,\a}-c^\b_\a-{1\over 2}\sum_{\nu=0}^4d_\a^{\b,\nu}\bl(\tilde I_\nu(\n(0))+\tilde I_\nu(\n({e_\a}))\br) \Eq(3.13a) $$ belongs to $\cg$. \medskip %Denote by $\r(\ve x,t)$, $u(\ve x,t)=(u(\ve x,t))_{\a=1,2,3}$ and %$\ce(\ve x,t)$ respectively the density, momentum and energy at site %$x$ for $\mu_{L,n(\cdot,t)}$ (that is %$\E^\mu_{L,n(\cdot,t)}}[I_\b(\n_x)],\ \b=0,\ldots,4$) and denote by %$\r^{(1)}(\ve x,t)$, $u^{(1)}(\ve x,t)=(u^{(1)}(\ve x,t))_{\a=1,2,3}$ and %$\ce^{(1)}(\ve x,t)$ the first order term in the $\ve$-Taylor %expansion of the previous quantities. Then, % $$ %\sum_{\a=1}^3\nabla_\a \E^\mu_{L,n}][w^0_{x,\a}]= %-\ve^{-1}a_0\div\un\l^{(1)} %-\un\l^{(1)}\cdot\na\Bigl[a_1\l_0^{(1)}+a_2\l_4^{(1)}\Bigr] %-a_3\div\un\l^{(2)} %+\sum_{\a=1}^3\chi\p_\a^2\r^{(1)} % \Eq(2.13) % $$ %where $\div\un\l^{(i)}=\sum_{\a=1}^3\p_\a\l_\a^{(i)}$, if $J$ is a %smooth function on $\R^3$, %$\na J=(\p_1 J,\ldots,\p_3 J)$ and $a_i,\, i=0,\ldots,3$ are positive %constants depending only on $r$, $\theta$ and $\cv$. The equation for %the momentum is % $$ %\sum_{\a=1}^3\nabla_\a \E^\mu_{L,n}][w^\b_{x,\a}]=-\ve^{-1}\p_\b\Bigl[b_0\l_0^{(1)}+b_1\l_4^{(1)}\Bigr] %-\p_\b p-b_2\p_\b\bigl((\l_\a^{(1)})^2\bigr) %-b_3\sum_{\a=1}^3\l_\a^{(1)}\p_\a\l_\b^{(1)} %+\chi \sum_{\a,\gamma=1}^3\p_\a^2 u^{(1)}_\gamma % \Eq(2.14) % $$ %where, as in the previous formula, $b_0,\ldots,b_3$ only depend on %$r,\theta,\cv$ and the pressure $p$ is defined by quadratic terms in %$\l^{(1)}$ and linear terms in $\l_0^{(2)}$, $\l_4^{(2)}$. %The energy equation is % $$ %\sum_{\a=1}^3\nabla_\a \E^\mu}][w^4_{x,\a}]=-\ve^{-1}c_0\div\un\l^{(1)} %-\un\l^{(1)}\cdot\na\Bigl[c_1\l_0^{(1)}+c_2\l_{4}^{(1)}\Bigr] %-c_3\div\un\l^{(2)} %+\chi\sum_{\a=1}^3\p_\a^2\ce^{(1)} % \Eq(2.15) % $$ %In order to get rid of the diverging transport terms in the previous %equations, the first order Taylor expansion of the chemical potentials %have to satisfy % $$ %\div\un\l^{(1)}=0 % \Eq(2.16) % $$ %and the so-called Boussinesq condition % $$ %b_0\l_0^{(1)}+b_1\l_{4}^{(1)}=\text{const} % \Eq(2.17) % $$ %The real $\vp$ in \equ(o2.8) is choose such that the unusual term %$b_2\p_\b\bigl((\l_\a^{(1)})^2\bigr)$ in \equ(2.14) disappears and %the second order expansions $\l^{(2)}$ are taken in a convenient way. % \bigskip \noindent{\it Slow-fast modes decomposition of the currents} We denote by $\ov I^+_\ell=(\ov I_{0,\ell},\ldots,\ov I_{4,\ell})$ the empirical averages of the conserved quantities over the block $\L_\ell$ of length $\ell$: $$ \ov I_{\b,\ell}={1\over (2\ell +1)^3}\sum_{|y|\le\ell}I_\b(\n_y),\quad \b=0,\ldots,4. %\Eq(2.70) $$ The measure $\hat\mu_{\ell,m}$, $m\in\R^5$ is defined as the canonical Gibbs state of $(2\ell+1)^3$ sites with parameters such that $\ov I^+_\ell=m$. It is the uniform probability on the set $\O_{\ell,m}$ of configurations on the block $\L_\ell$ such that $\ov I^+_\ell=m$. We denote by $\a_\ell(g)$ the conditional expectation of $g$ given the averages $\ov I^+_\ell$ $$ \a_\ell(g)=\E^{\mu}[g|\ov I^+_\ell]. %\Eq(2.71) $$ We call $\cl_{s,\ell}$ the symmetric part of the generator $\cl$ restricted to the block $\L_\ell$. Since the measures $\hat\mu_{\ell,m}$ are the only extremal invariant measures for $\cl_{s,\ell}$, we can define $\cl^{-1}_{s,\ell}g$ for any function $g$ such that $\a_\ell(g)=0$. Given any local function $g$ on $\O_\ell$, the finite volume ``variance'' $V_\ell(g,n)$ is $$ V_\ell(g,n)={1\over (2\ell_1+1)^3}\E^{\mu_{\ell,n}}\Bigl[ \Bigl(\sum_{|x|\le\ell_1} (\tau_xg -\a_\ell(g))\Bigr)(-\cl_{s,\ell})^{-1}\Bigl(\sum_{|x|\le\ell_1} (\tau_xg -\a_\ell(g))\Bigr)\Bigr], %\Eq(2.72) $$ where $\tau_\cdot$ is the translation operator on $\O_L$, $\tau_xg(\n)=g(\tau_x\n)$, $\ell_1=\ell-\ell^{1/9}$, $\ell$ large enough. The ``variance'' $V(G,n)$ of $G$ is given by $$ V(G,n)=\limsup_{\ell\to\infty}V_\ell(G,n). \Eq(o2.73) $$ With an abuse of notation, we denote $V_\ell(G,n)$ by $V_\ell(G,r,\theta)$ when $n$ is the chemical potential $n_0=(r,0,0,0,\theta)$. \medskip % For an integer $\ell$, we let $\ov\ell=\ell^{5}$ and % $k=\ell\ve^{-2/3}$. We assume that the box $\L_k$ is divided into % cubes $\L_{\ell,\s}$ of size $(2\ov\ell+1)$ with centers % $\s\in(2\ov\ell+1)\Z^3$, $|\s|\le k$. We consider the sub-cubes % $\L_{\ov\ell_1,\s}$ of size $\ov\ell_1=\ov\ell-\ell^{1/9}$ and their % union is denoted by $\hat\L_k=\cup_{|\s|\le k}\L_{\ell_1,\s}$. The % functions $\o$ and $\hat\o$ are the normalized indicator functions % $$ % \o(x)=(2k+1)^{-3}\1\{x\in\L_k\},\quad % \hat\o(x)=|\hat\L_k|^{-1}\1\{x\in\hat\L_k\} \Eq(3.5.c) % $$ % \medskip \goodbreak We state here the results in [BEM] \bigskip \noindent{\bf \Theorem (so3.1b).} {\it There exists a %positive diffusion matrix rank 2 tensor $\bar D=\bl(\bar D^{\b,\nu}_{\a,\g}\br)$ ($\bar D^{\b,\nu}$ positive definite matrix) and a sequence of local functions $h^{(q)}=(h_\a^{(q),\b},\,{\a=1,2,3,\,\b=0,\dots,4})$ in ${\cal G}$ such that, setting $$u^{(q),\b}_{\a}(\n)=g^\b_{\a}(\n) -\sum_{\g=1}^3\sum_{\nu=0}^4\bar D^{\b,\nu}_{\a,\g}\nabla_\g \tilde I_\nu(\n(0)) -\cl h^{(q),\b}_\a(\n), \Eq(3.14) $$ where $g^\b_{\a}(\n)$ is defined in \equ(3.13a), it results $$ \limsup_{q\to\infty}\sum_{\a=1}^3\sum_{\b=0}^4V(u_\a^{(q),\b},r,\theta)=0. %\Eq(3.8d) $$} \smallskip \noindent Above Theorem actually holds for any function in ${\cal G}$. \smallskip \noindent{\bf \Lemma (so4.6).} {\it The tensor $\bar D$ satisfies $$ a\cdot(\bar D C) a=\lim_{q\to \infty}\E^\mu\bl[\Gamma(a\cdot h^{(q)}) (-\cl_s)(a\cdot h^{(q)})\br]. \Eq(4.15) $$ } In this formula, $a\cdot b=\sum_{\a=1}^3\sum_{\b=0}^4a_\a^\b b_\a^\b$, $\G (g)=\sum_x\tau_x g$, $\cl_s$ is the symmetric part of $\cl$ in $L^2(\mu)$, $C$ is the $5\times5$ compressibility matrix (see \equ(comp) below for an explicit expression) and $\bar DC$ is the tensor $(\bar DC)^{\b,\nu}_{\a,\g}=(\bar D_{\a,\g}C)^{\b,\nu}$. We define $D=\bar D+\chi\I$ where $\I^{\b,\nu}_{\a,\g}=\d_{\a,\g}\d_{\b,\nu}$. \bigskip %We also define the temperature current as % $$ %\bar w^\b_{x,\a}=\chi\na_\a I'_4(\n_x)+\bar w^{(a),4}_{x,\a} % \Eq(3.7) % $$ %and for notational convenience, we let $\bar w^{(a),\b}_{x,\a} = %w^{(a),\b}_{x,\a}$ for $\a=1,2,3$ \bigskip \noindent{\it Hydrodynamic limit} Given functions $n_\b(x)$, $\b=0,\ldots ,4$, we consider the Gibbs states with chemical potential $n(x)=(n_0(x),\ldots,n_4(x))$ $$ \mu_{L,n}(\n)=Z_{L,n}^{-1}\prod_{x\in\L_L}\exp \Bigl\{\sum_{\b=0}^{4}n_\b(x) I_\b(\n_x)\Bigr\}. %\Eq(o2.10) $$ Now, assume that the initial distribution of the particles is $\mu_{L,n}$ with $n=(n_\b)$ the slowly varying chemical potentials given by $$ n_\b(x)=\l_\b^{(0)}+\ve\l_\b^{(1)}(\ve x)+\ve^2\l_\b^{(2)}(\ve x), \Eq(o2.11) $$ where $\l^{(0)}=(\l_\b^{(0)})=n_0$ and $\l_\b^{i}$ are smooth functions on the 3-d torus $\T_3$. We define the local equilibrium measure as the Gibbs states $\mu_{L,n(\cdot,t)}$ with $n(\cdot,t)$ the chemical potential given by $$ n_\b(x,t)=\l_\b^{(0)}+\ve\l_\b^{(1)}(\ve x,t)+\ve^2\l_\b^{(2)}(\ve x,t). \Eq(o2.12) $$ Furthermore, we assume $$ \div\, \underline{\lambda}^{(1)}=0,\quad \l_0^{(1)}+{1\over 2}\l_4^{(1)}=0. $$ Then in [BEM] (see also [EMY2]) it has been proved that the the law of the process at time $t>0$ is well approximated by the local equilibrium in the sense that the relative entropy per unit volume of the non-equilibrium measure with respect the local equilibrium times $\ve^{-2}$ vanishes in the limit $\ve\to 0$. \medskip We can now state the result proved in [BEM] on the hydrodynamic limit. Let $u(z,t)$, $z\in\T_3$, $t\in[0,t_0]$, $t_0>0$, be the classical smooth solutions of the following Navier-Stokes equation $$ \eqalign{ & \div\, u =0, \cr & \p_t u_\b+\partial_\b p+Ku\cdot\na u_\b = \sum_{\a=1}^3 D_{\a,\b}\p_\a^2 u_\b,\quad \b=1,2,3,\cr } \Eq(2.21) $$ with initial condition $u_\a(z)=\E^{\mu_{L,n(\ve^{-1}z)}}[I_\a(\n_0)]$ and let $\ce(z,t)$ be the solution of the energy equation $$ {\p\over\p t}\ce +Hu\cdot\na \ce =\sum_{\a=1}^3 {\cal K}_\a(\p_\a^2\ce), \Eq(2.21a) $$ with initial condition $\ce(z)=\E^{\mu_{L,n(\ve^{-1}z)}}[I_4(\n_0)]$. The constants appearing in \equ(2.21) and \equ(2.21a) are given by $$ K=18{\ps{v_1^2v_2^2 h_2}\over \ps{h_0|v|^2}^2},$$ with $h_2={1\over 2}h_1(1-6f_0(1-f_0))$ and $$\quad H= {1\over \ps{h_0|v|^2}}{\Psi_1-2C\Psi_2\over \Phi_2+C\Phi_1},\quad C ={1\over 2}{\ps{h_1|v|^4}\over \ps{h_1 |v|^2}}, %\Eq(2.22) $$ where $$ \eqalign{ \Psi_1&=\ps{ h_2|v|^6}\ps{h_1|v|^2}-\ps{ h_2|v|^4}\ps{ h_1|v|^4},\cr \Psi_2&=\ps{ h_2|v|^4}\ps{h_1|v|^2}- \ps{ h_2|v|^2}\ps{ h_1|v|^4}.\cr } $$ \bigskip \goodbreak Let $\P^{\mu_{L,n}}_\ve$ be the law of the process $\n_t(x,v)$ with generator $\ve^{-2}\cl$ starting from the measure $\mu_{L,n}$ defined in \equ(o2.11), with chemical potentials $n_\a(x)$ of the form \equ(o2.12). The density ($\nu_{0}^\ve(t,z)$), the momentum ($(\nu_{\b}^\ve(t,z))_{\b=1,2,3}$) and energy ($\nu_{4}^\ve(t,z)$) empirical fields are defined as $$ \nu_{\b}^\ve(z,t) =\ve^2\sum_{x\in\L_L} \d(z-\ve x)\tilde I_{\b}(\n_t(x)), %\Eq(2.24) $$ where $\tilde I_\b(\n_x)=I_\b(\n_x)-m_\b$, $m_\b=\E^{\mu}[I_\b(\n_0)]$ and $\n_t(x)=\{\n_t(x,v), \, v\in {\cal V}\}$. \bigskip \noindent{\bf \Theorem (s2.1)} \nobreak {\it The density, momentum and energy empirical fields converge, for $t\le t_0$, weakly (in space) in\/ $\P^{\mu_{L,n}}_\ve\!\!$ probability, to $\rho(z,t)dz$, $u(z,t)dz$ and $\ce(z,t)dz$, where $a\rho+b\ce=c$ for suitable $a,b,c$.} \smallskip Note that the transport coefficients $D_{\a,\b}$ and ${\cal K}_\a$ are suitable combinations of the diffusion coefficients $D^{\b,\nu}_{\a,\g}$ in Theorem \equ(so3.1b). The explicit expressions are given in [BEM], but we omit them because they do not play any role in this paper. \bigskip \vskip 1truecm \goodbreak \noindent{\bf 3. Fluctuation field and results}. \vskip .5truecm \numsec= 3 \numfor= 1 \numtheo=1 In this paper, we are interested in the equilibrium fluctuations of the mass, momentum and energy fields. The initial fluctuations, distributed in terms of the measure $\mu$, are finite but they may become infinite at later very long times because of the effect of waves moving with velocity $\ve^{-1}$, which are the solutions of the linearized (around the equilibrium) Euler equations (linear hyperbolic equations) for this model. To remove the diverging terms we have to modify the usual definition of fluctuation fields not simply by a shift but considering fluctuations which move together with the traveling waves We denote by $U^\ve_t$ the operator $\exp(-{t\over\ve} E^*)$ where $E$ is the linearized Euler operator, a $5 \times 5$ matrix whose entries are first order differential operators with constant coefficients, %change $$ E=\pmatrix{0&-a_0\underline{\partial}&0\cr -b_0{\underline{\partial}}&0&-b_4{\underline{\partial}}\cr 0&-a_4\underline{\partial}&0\cr} %\Eq(E) $$ and $*$ is the adjoint with respect the usual scalar product in $L^2(\T_3,\R^5)$ (the constants $a_i$ and $b_i$ are defined in \equ(3.7e) and ${\underline{\partial}}=(\partial_1,\partial_2,\partial_3)$ is the gradient operator). \bigskip For any smooth function $G=(G_\b)_{\b=0,\ldots,4}:\T_3\rightarrow \R^5$ consider the (scalar) fluctuation field $\xi^\ve$ on the state space $(\T_3)^{\otimes 5}$ $$ \xi^\ve\bl(t,G\br)=\ve^{3/2}\sum_{\b=0}^4\sum_x (U^\ve_t G)_{\b}(\ve x)\tilde I_\b(\n_t(x)). \Eq(2.26) $$ It is equivalent to consider the vector fluctuation field $(\xi^\ve_\b)_{\b=0,\ldots 5}$ on $\T^3$ whose components $\xi^\ve_0$, $(\xi^\ve_\b)_{\b=1,\ldots 3}$ and $\xi^\ve_4$ are respectively the density, momentum and energy fluctuation fields, defined as $$ \xi^\ve_\b(t,\varphi)=\xi^\ve(t,G^{(\b)}),\quad \b=0,\dots,4, $$ where $G^{(\b)}$ is the vector function with only the $\b$ component non vanishing and $G^{(\b)}_\b=\varphi$. % -E^*=\left(\matrix{0&b_0{\bf \nabla}&0\cr % a_0{\bf \nabla}&0&a_4{\bf \nabla}\cr % 0&b_4{\bf \nabla}&0\cr}\right) % $$ \bigskip We want to study the evolution of the fluctuation fields in the limit $\ve\to 0$ when the fields are initially distributed with the equilibrium measure $\mu$, given by \equ(o2.9). % We omit %the dependence on the chemical parameters $n$. We notice that the initial covariance of the limiting fields $\lim_{\ve\to 0}\xi^\ve _\b(0,\varphi)=\xi_\b(0,\varphi)$ is $$\E^\mu_\ve\Bigl[ \xi_\b(0,\varphi)\xi_\nu(0,\psi)\Bigr]=C_{\b,\nu} \int_{\T_3}\!dx\, \varphi(x)\psi(x), \Eq(cov)$$ where $C$ is the compressibility matrix ($5 \times 5$) $$ C=\pmatrix{\ps{h_0}&0&\ps{h_0{|v|^2\over 2}}\cr 0&{1\over 3}\ps{|v|^2h_0}\Bbb I_3&0\cr \ps{h_0{|v|^2\over 2}} &0&\ps{h_0{|v|^4\over 4}}\cr}, \Eq(comp)$$ with $\Bbb I_3$ the $3\times 3$ identity matrix, $h_0$ defined in the paragraph before \equ(3.7bis) and $\ps{\,\cdot\,}$ in the paragraph after \equ(o2.9). \smallskip Remark that $E$ is not anti-hermitian in $L^2(\T_3,\R^5)$, since $a_0\ne b_0$ and $a_4\ne b_4$. However a straightforward computation shows that $EC$ satisfies $EC+CE^*=0$. We want to show that the fluctuation field converges to a stationary Gaussian vector process with a given covariance. The equal time covariance is exactly \equ(cov) because of the stationarity of the limiting process. \bigskip To state the results we need some extra notation. %If $S_{-k}$ %is the Sobolev space of order $-k$ on $\T^3$, then we also consider %the Hilbert space $H^C_{-k}$ endowed with the inner product We introduce the Hilbert spaces ${\cal H}_{k}, k\in\Z$ defined by the scalar product $$\ps{G,H}_{k}=\ps{G,L^kH}_0,$$ where $L=I-\Delta$, $\Delta$ the Laplacian operator and $\ps{.,.}_0$ is the usual inner product of $L^2(\T_3,\R^5)$: $$ \ps{G,H}_0=\sum_{\b=0}^4\int_{\T_3}\!dx\,G_\b(x)H_\b(x). \Eq(fps) $$ Denote by $\|\cdot\|_k$ the norm of $\ch_{k}$ and by ${\cal H}_{-k}$ the dual of ${\cal H}_{k}$ with respect to the inner product of $L^2(\T_3,\R^5)$. The fluctuation field $(\xi^\ve(t))_{t\ge 0}$ is a distribution valued stochastic process taking values in the Sobolev space ${\cal H}_{-k_0}$ for some suitable $k_0$. Its path space is $D([0,T], {\cal H}_{-k_0})$ ($T>0$), the space of functions with values in ${\cal H}_{-k_0}$, right continuous with left limits, endowed with the uniform (in time) weak (in space) topology. We call $Q^\ve$ the law of $(\xi^\ve(t))_{t\ge 0}$ when the process is initially distributed according to the equilibrium measure $\mu$. It is therefore a probability measure on the space $D([0,T],{\cal H}_{-k_0})$. %We denote %by $P^\ve_\mu$ the law of the process $\eta_{\ve^{-2}t}$ with initial %distribution $\mu$. By analogy with \equ(fps), we define for local functions $g=(g_0,\ldots,g_4)$ on $\O_L$ %with vanishing equilibrium expectation and smooth functions $G=(G_0,\ldots,G_4)$ on $\T_3$ $$ \ps{G,g}_{0,L}=\ve^{3/2}\sum_{\b=0}^4\sum_{x\in\L_L}G_\b(\ve x)\tau_xg_\b. \Eq(fps2) $$ So the fluctuation field \equ(2.26) can be rewritten as $$ \xi^\ve(t,G)=\ps{U^\ve_tG,\tilde I(\n_t(0))}_{0,L}. $$ \vskip.5cm We introduce the linearized Navier-Stokes operator ${\cal D}$ as $$ {\cal D}G=\sum_{\a,\g=1}^3 D_{\a,\g}\pt_\a\pt_\g G. \Eq(D) $$ Then $N=\pi_E({\cal D})$ is the operator defined as the limit $$ \lim_{\ve\to 0}{1\over T}\int_0^T\!dt\, \exp\bl(-{t\over\ve } E\br) {\cal D}\exp \bl({t\over\ve }E\br)G=\pi_E({\cal D}) G $$ and $\pi_E({\cal D})^*$ the adjoint of $N$ with respect to the inner product $\langle\cdot,\cdot\rangle_0$ in $L^2(\T_3,\R^5)$. The main result of this paper is \bigskip \noindent{\bf \Theorem (so2.2)} {\it The probability measures $(Q^\ve)$ converge weakly in $D([0,T],{\cal H}_{-k_0})$ to the law $Q$ of the stationary generalized Ornstein-Uhlenbeck process $\xi$ with mean $0$ and covariance $$ \E^Q\Bigl[\xi_\b(s,\phi)\xi_\nu(t,\psi)\Bigr]= \int_{\T_3}\!dx\, \bl((CS_{|t-s|})_{\b,\nu}\phi\br)(x)\psi(x), $$ where $(S_t)_{t\ge 0}$ is the semi-group in $L^2(\T_3,\R^5)$ associated to $\pi_E({\cal D})^*$ and $C$ the compressibility matrix. It is formally characterized by the SDE $$\eqalign{ d \xi(t) &= N\xi(t)dt+ B dW_t,\cr BB^* &=-2 NC. } $$ } One of the main ingredient needed while studying the equilibrium fluctuations is the so-called Boltzmann-Gibbs principle which states that the non conserved quantities arising in the conservation laws may be replaced by linear combinations of the conserved ones. In the context of a non gradient system, the usual statement is not valid and some corrections to the fluctuation field have to be introduced (see [C], [Lu]). The situation in the case of an asymmetric system is more delicate since the usual Boltzmann-Gibbs estimate is not sharp enough and one has to prove a stronger result ([CLO]). We need to generalize such a result to the present setup. Indeed we prove the following \bigskip \goodbreak %change \noindent{\bf \Theorem (s2.3) (Boltzmann-Gibbs principle)} {\it Assume that $h\in{\cal G}$ (see \equ(2.28)). Then, for any smooth function $G:\R_+\times\T_3\to \R $, the following estimate holds $$ \limsup_{\ve\to 0}\E^\mu_\ve\left[\sup_{0\le t\le T}\left(\ve^{3/2-1}\int_0^t\!\sum_xG(s,\ve x)\tau_xh(\n, s)\, ds\right)^2\right]\le c\, T\|G \|_0^2\,V(h;r,\theta), %\Eq(2.29) $$ \nobreak where $V$ is the infinite volume variance defined in \equ(o2.73). } \vskip1truecm \noindent{\bf 4. Limiting distribution of the fluctuation field.} \numsec= 4 \numfor= 1 \numtheo=1 \bigskip The theory of Holley-Stroock [HS] characterizes the law $Q$ of the Ornstein-Uhlenbeck process $\xi$ described in Theorem \equ(so2.2) by the following martingale problem: $$ \eqalign{ M_1(t,G) & =\xi(t,G)-\xi(0,G)-\int_0^t\!ds\,\xi(s,\pi_E({\cal D})^*G), \cr M_2(t,G) & =\bigl(M_1(t,G)\bigr)^2+2t\,\ps{G,C\,\pi_E({\cal D})^*G}_0\cr } \Eq(mpb) $$ are martingales under $Q$. In this section, we will prove that any limit law $\bar Q$ of $Q^\ve$ satisfies \equ(mpb). Therefore from the tightness of $(Q^\ve)$ (see Theorem \equ(tight) of section 5), it has to converge to $Q$ and Theorem \equ(so2.2) follows. The processes analogous to \equ(mpb) for $Q^\ve$ are $$ \eqalign{ M^\ve_1(t,G) & =\xi^\ve(t,G)-\xi^\ve(0,G)-\int_0^t\!ds\,\xi^\ve(s,\pi_E({\cal D})^*G), \cr M^\ve_2(t,G) & =\bigl(M^\ve_1(t,G)\bigr)^2+2t\,\ps{G,C\,\pi_E({\cal D})^*G}_0\cr } %\Eq(mpb2) $$ and we want to show that these processes are martingales up to some error terms which vanish as $\ve$ goes to 0. Given local functions $h=(h_\a)_{\a=1,2,3}= (h^{\b}_\a)_{\a=1,2,3;\,\b=0,\ldots ,4}\in\cg$, we introduce the modified % vector fluctuation field as %$$ % \z^\ve_\b(t,G,h)=\xi^\ve_\b(t,G) - \ve^{3/2+1}\sum_{\a=1}^3 % \sum_{\nu=0}^4\sum_x\p_\a (U^\ve_t G^{(\b)})_{\nu}(\ve % x,\ve^{-1}t)\tau_xh^\nu_\a \Eq(3.1) % $$ % and the scalar fluctuation field % $$\eqalign{ \z^\ve(t,G,h)&=\ve^{3/2}\sum_{\b=0}^4\sum_x % (U^\ve_t G)_{\b}(\ve x)\tilde I_\b(\n_x)\cr &\quad - % \ve^{3/2+1}\sum_{\a=1}^3 \sum_{\b=0}^4\sum_x\p_\a (U^\ve_t % G)_{\b}(\ve x)\tau_xh^{\b}_\a } % $$ $$ \z^\ve(t,G,h)=\xi^\ve(t,G)-\ve\sum_{\a=1}^3 \ps{\p_\a(U^\ve_tG),h_\a}_{0,L}, $$ where $\ps{\cdot,\cdot}_{0,L}$ was defined in \equ(fps2). %change Actually we will choose for $h$ the terms of the sequence $h^{(q)}$ defined in Theorem \equ(so3.1b), but we will omit the label $q$ for sake of shortness. It is clear that the difference between $\z^\ve(t,G,h)$ and $\xi^\ve(t,G)$ vanishes in $L^2(\P^\mu_\ve)$ with $\ve$. Moreover, it is well known that the following processes are martingales with respect to the usual filtration related to the process $(\n_t(x,v))$ $$ \eqalign{ M^\ve_1(t,G,h) & =\z^\ve(t,G,h)-\z^\ve(0,G,h)-\int_0^t\g^\ve_1(s,G,h)\, ds,\cr M^\ve_2(t,G,h) & =\bigl(M^\ve_1(t,G,h)\bigr)^2-\int_0^t\g^\ve_2(s,G,h)\, ds,\cr } \Eq(3.2) $$ with $$ \eqalign{ \g^\ve_1(t,G,h) &=\bigl(\p_t+\ve^{-2}\cl\bigr)\z^\ve(t,h,G),\cr \g^\ve_2(t,G,h) & = \bigl(\p_t+\ve^{-2}\cl\bigr)\bigl(\z^\ve(t,h,G)^2\bigr) -2\big(\z^\ve(t,h),G\big) \bigl(\p_t+\ve^{-2}\cl\bigr)\big(\z^\ve(t,h),G\big).\cr } %\Eq(3.3) $$ \bigskip We first compute the compensator $\g^\ve_1$. Let $w^{(a)}_{x,\a}=(w^{(a),\b}_{x,\a})_{\b=0,\ldots ,4}$. Then $$ \eqalign{ \g^\ve_1(t,G,h) = & \ps{\p_t(U^\ve_t G),\tilde I(\n_t(0))}_{0,L} -\ve^{-1}\sum_{\a=1}^3\ps{\p_\a (U^\ve_t G),\cl h_\a(\n_t)}_{0,L}\cr & -\ve^{-2}\sum_{\a=1}^3\ps{\nabla_\a(U^\ve_t G), \chi\nabla_\a \tilde I(\n_t(0))+w^{(a)}_{0,\a}(\n_t)}_{0,L} +R_5(t,G,h),\cr } \Eq(3.8) $$ where, remembering that $\partial_tU^\ve_t=-\ve^{-1}E^*U^\ve_t$, $$ R_5(t,G,h)=\sum_{\a=1}^3\ps{\p_\a (-E^*U^\ve_t) G,h_\a(\n_t)}_{0,L}. $$ Now, given $\bar D_{\a,\g}=(\bar D^{\b,\nu}_{\a,\g})_{\b,\nu=0,\ldots ,4}$, $\a,\g=1,2, 3$, we add and subtract the term $$ \sum_{\a,\g=1}^3\ps{ \bar D_{\a,\g}\p_\a(U^\ve_t G),\nabla_\g \tilde I(\n_t(0))}_{0,L} %\Eq(3.10) $$ in \equ(3.8). Then $\g^\ve_1(t,G,h)$ is equal to $$ \eqalign{ & \ps{\p_t(U^\ve_t G),\tilde I(\n_t(0))}_{0,L} +\ps{ {\cal D}^*(U^\ve_t G), \tilde I(\n_t(0))}_{0,L}\cr & -\ve^{-1}\sum_{\a=1}^3\Bl\langle\p_\a (U^\ve_t G), w^{(a)}_{0,\a}(\n_t)-c_\a -\sum_{\g=1}^3\bar D_{\a,\g}\nabla_\g \tilde I(\n_t(0)) -\cl h_\a(\n_t)\Br\rangle_{0,L}\cr & +R_5(t,G,h)+R_1(t,G)+R_2(t,G)+R(t,G),\cr } \Eq(3.11) $$ with ${\cal D}^*$ the adjoint in $L^2(\T_3,\R^5)$ of the differential operator ${\cal D}$ in \equ(D), $c_\a=(c_\a^\b)_{\b=0,\ldots,4}$ the equilibrium value of the current $w^{(a),\b}_\a$ (see \equ(3.7bis)) and $$ R_1(t,G)=\ve^{-1}\sum_{\a=1}^3 \ps{(\p_\a -\ve^{-1}\nabla_\a) (U^\ve_t G), \chi\nabla_\a \tilde I(\n_t(0))}_{0,L}, $$ $$ R_2(t,G)=\sum_{\a,\g=1}^3\ps{ (\ve^{-1}\nabla^-_\g -\p_{\g}) \p_\a D_{\a,\g}(U^\ve_t G), \tilde I_\nu(\n_t(0))}_{0,L}, $$ $$ R(t,G)=\ve^{-1}\sum_{\a=1}^3\ps{\left(\p_\a -\ve^{-1}\nabla_\a\right) (U^\ve_t G), w^{(a)}_{0,\a}-c_\a}_{0,L}. $$ >From the definition of $U^\ve_t G$, the first term of the sum \equ(3.11) can be written as $$\ve^{-1}\sum_{\a=1}^3 \ps{(-E^* U^\ve_t) G,\tilde I(\n_t(0))}_{0,L} =\ve^{-1}\sum_{\a=1}^3 \ps{\p_\a(U^\ve_t G), d_\a \tilde I(\n_t(0))}_{0,L} %\Eq(3.13) $$ where the coefficients of the matrix $d_\a=(d^{\b,\nu}_\a)_{\b,\nu=0\ldots,4}$ were defined in \equ(3.7d). Recalling the definition of the local functions $g_\a=(g^\b_{\a})_{\b=0\ldots,4}$ and $u_\a=(u^\b_{\a})_{\b=0\ldots,4}$ in \equ(3.13a), \equ(3.14) (we omit the label $q$), we obtain $$ \g^\ve_1(t,G,h) = \ps{{\cal D}^*(U^\ve_t G),\tilde I_\nu(\n_t(0))}_{0,L} +\sum_{i=1}^4R_i(t,G)+\sum_{i=5}^6R_i(t,G,h), $$ where $$ R_3(t,G)= \ve^{-1}\sum_{\a=1}^3\Bl\langle\Bl(\p_\a -\ve^{-1}\nabla_\a+{1\over 2}\nabla^-_{e_\a}\partial_\alpha\Br) (U^\ve_t G),d_\a\tilde I(\n_t(0))\Br\rangle_{0,L}, $$ $$ R_4(t,G)=\ve^{-1}\sum_{\a=1}^3\ps{(\p_\a -\ve^{-1}\nabla_\a) (U^\ve_t G), g_{\a}(\n_t)}_{0,L} $$ and $$ R_6(t,G,h)=-\ve^{-1}\sum_{\a=1}^3\ps{\p_\a (U^\ve_t G),u_{\a}(\n_t)}_{0,L}. $$ To prove that the compensator $\int_0^t\g^\ve_1(t,G,h)ds$ is converging, %$$\limsup_{\ve\to % 0}E^\ve\left[\left(\int_0^t\g^\ve_1(t,G,h)ds\right)^2\right]=0,\quad %\b=0,\dots,4$$ we have to control the remainder terms. The remainders for $i=1,2,3$ are easily controlled by the following \vskip.2cm \noindent{\bf \Lemma (s4444)} {\it Let $h$ be a mean zero local function and $G:\Bbb R_+\times \Bbb T^3\to \Bbb R$ a continuous function. Then there exist a constant $c$ depending only on $h$ such that, for all $t\ge 0$ and all $\ve >0$ $$\Bbb E^\mu_\e\left[\sup_{0\le t\le T}\left(\int_0^t\!ds\, \ve^{3/2}\sum_{x} G(s, \e x)\tau_x h(\eta_s)\right)^2\right]\le c\,T^2\|G\|^2_\infty.$$ } \vskip.2cm The proof is an easy consequence of the Schwartz inequality and the stationarity of $\P_\ve^\mu$. We refer to Lemma 4.1 in [CLO] for details. By Taylor expanding and using Lemma \equ(s4444) we immediately obtain $$ \lim_{\ve\to 0}\E^\mu_\ve\left[\left( \int_0^t\!ds\,R_i(s,G)\right)^2\right]= 0. $$ for $i=1,2,3$. The other terms are estimated by using the refined Boltzmann-Gibbs principle (Theorem \equ(s2.3)) because the functions $g^\b_\a$ and $u^\b_\a$ are in $\cg$ ($h^\b_\a\in \cg$ by hypothesis). We get $$ \lim_{\ve\to 0}\E^\mu_\ve\left[\left( \int_0^t\!ds\,R_4(s,G)\right)^2\right] =\lim_{\ve\to 0}\E^\mu_\ve\left[\left( \int_0^t\!ds\,R_5(s,G,h)\right)^2\right]= 0 %\Eq(3.20) $$ and $$ \limsup_{\ve\to 0}\E^\mu_\ve\left[\left( \int_0^t\!ds\,R_6(s,G,h)\right)^2\right]\le c\,t\max_{\a=1,2,3} \|\partial_\a(U^\ve_\cdot G)\|^2_0 \sum_{\a=1}^3\sum_{\b=0}^4V(u^\b_\a;r,\theta). %\Eq(3.20a) $$ >From the definition of the semi-group $(U^\ve_t)$, it is clear that $\|\partial_\a(U^\ve_\cdot G)\|^2_0$ is uniformly bounded in $\ve$. Moreover the diffusion coefficients $\bar D^{\b,\nu}_{\a,\g}$ are chosen in such a way that, since we take for $h^\b_\a$ the terms of the sequence $(h^{(q),\b}_\a)$ given in Theorem \equ(so3.1b), we have $$ \lim_{q\to \infty}V(u^{(q),\b}_\a;r,\theta)=0. %\Eq(3.21a) $$ We have shown so far that there exists a random variable $R^{q}_\ve$ which converges to $0$ in $L^2(\P^\mu_\ve)$ as $\ve\to 0$ and then $q\to \infty$ such that $$ M^\ve_1(t,G,h) =\xi^\ve\bl(t,G\br)-\xi^\ve\bl(0,G\br)-\int_0^t\! ds\,\ps{{\cal D}^*(U^\ve_s G),\tilde I(\n_s(0))}_{0,L}+R^{q}_\ve.\Eq(?) $$ We would like to have instead of the third term in \equ(?) a term of the form $$ \ps{U^\ve_s(HG),\tilde I(\n_s(0))}_{0,L} $$ for some suitable operator $H$ that we could then rewrite as $\xi^\ve(s,H G)$, so to identify the limiting martingale problem. We proceed in the following way: $$ \ps{{\cal D}^*(U^\ve_sG),\tilde I(\n_s(0))}_{0,L}= \ps{U^\ve_s (U^\ve_s) ^{-1} {\cal D^*}U^\ve_s G,\tilde I(\n_s(0))}_{0,L}. $$ % By Lemma A.1 and Lemma A.2 % $$ % \lim_{\e\to 0}||(U^\ve)_s ^{-1} {\cal D}U^\ve_s G||_2=||\Pi_E({\cal % D})G||_2$$ By Lemma A.2 and Lemma A.3 $$ \lim_{\ve\to 0}\E^\mu_\ve\lt[\lt(\int_0^t\! ds\, \Bl[\xi^\ve\bl(s,(U^\ve_s)^{-1} {\cal D}^*U^\ve_s G\br)-\xi^\ve\bl(s,\pi_{E^*}({\cal D}^*) G\br)\Br] \rt)^2\rt]=0. $$ Hence, noticing that $\pi_{E^*}({\cal D}^*)=\pi_{E}({\cal D})^*$, we have proved that there exists a random variable $C_\ve^q$ which converges to $0$ in $L^2(\P^\mu_\ve)$ when $\ve\to 0$ and then $q\to\infty$ such that $$ M^\ve_1(t,G,h) =\xi^\ve(t,G)-\xi^\ve(0,G)-\int_0^t\!ds\,\xi^\ve\bl(s, \pi_E({\cal D})^* G\br)+C_\ve^q. $$ \vskip .5truecm We now compute the compensator $\g^\ve_2$ in \equ(3.2). We first remark that $$ \g^\ve_2(t,G,h) = \bigl(\ve^{-2}\cl\bigr)\bigl(\z^\ve(t,G,h)^2\bigr) -2\z^\ve(t,G,h) \bigl(\ve^{-2}\cl\bigr)\z^\ve(t,G,h). %\Eq(3.21) $$ We introduce the operator $\cl^{(2)}=\cl^{{\rm ex},(2)}+\cl^{{\rm c},(2)}$ for local functions $f$ and $g$ as $$ \eqalign{ \cl^{{\rm ex},(2)}(f;g) &=\cl^{\rm ex}(fg)-f\cl^{\rm ex} g-g\cl^{\rm ex} f, \cr \cl^{{\rm c},(2)}(f;g) &=\cl^{\rm c}(fg)-f\cl^{\rm c} g-g\cl^{\rm c} f. \cr } %\Eq(3.22) $$ Then we obtain $$ \g^\ve_2(t,G,h) = Y^\ve_1(t,G)+Y^\ve_2(t,G,h)+ Y^\ve_3(t,G,h), %\Eq(3.23) $$ where $$ \eqalign{ Y^\ve_1(t,G) &= \ve\sum_{\b,\nu=0}^4\sum_{x,y}(U^\ve_t G)_{\b}(\ve x)(U^\ve_t G)_{\nu}(\ve y) \cl^{{\rm ex},(2)}\bigl(\tilde I_\b(\n_x);\tilde I_\nu(\n_y)\bigr),\cr Y^\ve_2(t,G,h)&= -2\ve^2\sum_{\a=1}^3\sum_{\b,\nu=0}^4\sum_{x,y}(U^\ve_t G)_{\b} (\ve x)\p_\a (U^\ve_t G)_{\nu}(\ve y)\cl^{{\rm ex},(2)} \bigl(\tilde I_\b(\n_x);\tau_yh^\nu_\a)\bigr),\cr Y^\ve_3(t,G,h)&=\ve^3\sum_{\a,\g=1}^3\sum_{\b,\nu=0}^4\sum_{x,y} \p_\a (U^\ve_t G)_{\b}(\ve x)\p_\g (U^\ve_t G)_{\nu}(\ve y)\cl^{(2)} \bigl(\tau_xh^\b_\a;\tau_yh^\nu_\g)\bigr).\cr } %\Eq(3.24) $$ >From the explicit formulas \equ(3.25), \equ(3.27) and \equ(3.28) that we will get below for $Y_i(t,G)$, $i=1,2,3$ and the use of Lemma \equ(s4444) it is easy to see that $$ \int_0^t\!ds\,\Bl(\g^\ve_2(s,h,G)- \E^\mu_\ve\bl[\g^\ve_2(s,h,G)\br]\Br) $$ is converging to 0 in $L^2(\P^\mu_\ve)$. So, all we need to compute is $\E^\mu_\ve\bl[\g^\ve_2(t,h,G)\br]$. \bigskip Notice that $$ \eqalign{ \cl^{{\rm ex},(2)}(f;g)& =\sum_{x,e,v}b(x,x+e,v) \nabla_{x,x+e,v}f\nabla_{x,x+e,v}g, \cr \cl^{{\rm c},(2)}(f;g)& =\sum_{x,q} \nabla_{x,q}f\nabla_{x,q}g, \cr } \Eq(3.24a) $$ with $\nabla_{x,x+e,v}f=f(\n^{x,x+e,v})-f(\n)$, $\nabla_{x,q}f=f(\n^{x,q})-f(\n)$ and $$ b(x,y,v)=\Bl(\chi+{1\over 2}v\cdot (y-x)\Br) \eta(x,v)(1-\eta(y,v)). $$ So, if we let $\phi_0(v)=1$, $\phi_\b(v)=v_\b$ for $\b=1,2,3$ and $\phi_4(v)={1\over 2}|v|^2$, a straightforward computation leads to the following $$ \eqalign{ Y^\ve_1(t,G) = & \ve^3\sum_{\a=1}^3\sum_{\b,\nu=0}^4\sum_{x}\Bl(\pt_\a (U^\ve_t G)_{\b}\pt_\a (U^\ve_t G)_{\nu}\Br)(\ve x)\times\cr &\qquad\sum_v\Bl[b(x,x+e_\a,v)+b(x+e_\a,x,v)\Br]\phi_\b(v)\phi_\nu(v)+O(\ve).} \Eq(3.25) $$ Therefore $$ \eqalign{ \E^\mu\Bigl[Y^\ve_1(t,G)\Bigr]&=\ve^3 2\chi\sum_{\a=1}^3\sum_{\b,\nu=0}^4\sum_x C_{\b,\nu}\Bigl(\p_\a (U^\ve_t G)_{\b}\p_\a (U^\ve_t G)_{\nu}\Bigr)(\ve x) +O(\e)\cr &=-2\chi\ps{(U^\ve_t G),\Delta C(U^\ve_t G)}_0+O(\e),} %\Eq(3.26) $$ where $\Delta$ is the vectorial Laplacian operator defined as $(\Delta G)_\b=\Delta G_\b$. Observe that $$ \ps{(U^\ve_t G),\Delta C(U^\ve_t G)}_0 =\ps{G,e^{-{t\over \ve}E}Ce^{-{t\over \ve}E^*} \Delta G}_0 =\ps{G,Ce^{{t\over \ve}E^*}e^{-{t\over \ve}E^*}\Delta G}_0, $$ where we have used that $EC=-CE^*$. In conclusion, $Y^\ve_1(t,G)$ converges in $L^2(\P^\mu_\ve)$ to $-2\chi \ps{G,C\Delta G}_0$. % $$ % 2\chi\sum_{\a=1}^3\int_{\T_3}\!dx\, % \pt_\a G(x)\cdot C\pt_\a G(x) % $$ \bigskip We get in the same way $$ \eqalign{ Y^\ve_2(t,G,h) = & 2\ve^3\sum_{\a,\g=1}^3\sum_{\b,\nu=0}^4 \sum_{x}\Bl(\partial_\a (U^\ve_t G)_{\b} \partial_\g (U^\ve_t G)_{\nu}\Br)(\ve x) \times\cr &\qquad\sum_v\Bigl[b(x,x+e_\a,v)-b(x+e_\a,x,v)\Bigr] \phi_\b(v)\nabla_{x,x+e_\a,v}\Gamma(h_\a^\nu)+O(\ve), } \Eq(3.27) $$ where $\Gamma(h_\a^\nu)=\sum_x \tau_xh_\a^\nu$. Since $\mu$ is invariant for the jump generator of particles with a given velocity, it is easy to check that $$ \E^\mu\Bl[\n(x,v)(1-\n(x+e_\a,v))\nabla_{x,x+e_\a,v}\Gamma(h_\a^\nu)\Br] =0, $$ which implies that the time integral of $Y^\ve_2(t,G,h)$ converges to $0$ in $L^2(\P^\mu_\ve)$ by Lemma \equ(s4444). \bigskip The last term $Y^\ve_3(t,G,h)$ is given by $$ \eqalign{ & \ve^3\sum_{\a,\g=1}^3\sum_{\b,\nu=0}^4\sum_{x}\Bl(\partial_\a (U^\ve_t G)_{\d} \partial_\g (U^\ve_t G)_{\nu}\Br)(\ve x)\times\cr &\qquad \left[ \sum_{v;|e|=1}b(x,x+e,v) \nabla_{x,x+e,v}\Gamma(h_\a^\b)\nabla_{x,x+e,v}\Gamma(h_\g^\nu) +\sum_{q}\nabla_{x,q}\Gamma(h_\a^\b)\nabla_{x,q}\Gamma(h_\g^\nu)\right]. } \Eq(3.28) $$ By using again Lemma \equ(s4444) it is immediate to show that the time integral of $Y^\ve_3$ converges in $L^2(\P_\ve^\mu)$ to its average that we are going to compute. Let $\cl_s^{\rm ex}$ be respectively the symmetric part of $\cl^{\rm ex}$ in $L^2(\mu)$. It is easy to check that for any local function $f$ and $g$ $$ \sum_{v;|e|=1}\E^\mu\Bigl[\eta(0,v)(1-\eta(e,v))\nabla_{0,e,v}\Gamma(f) \nabla_{0,e,v}\Gamma(g)\Br]= 2\E^\mu\Bigl[\G (f)(-\cl_s^{\rm ex})g\Br] $$ and $$ \sum_{q}\E^\mu\Bigl[\nabla_{0,q}\Gamma(f) \nabla_{0,q}\Gamma(g)\Br]= 2\E^\mu\Bigl[\G (f)(-\cl^{\rm c})g\Br]. $$ Therefore $$ \E^\mu\bl[Y^\ve_3(t,G)\br]=2\ve^3\sum_x\E^\mu\Bl[\G\bl(\partial(U^\ve_t G)(\ve x)\cdot h\br)(-\cl_s)(\partial(U^\ve_t G)(\ve x)\cdot h)\Br], $$ where $\cl_s$ and $a\cdot b$ were defined after \equ(4.15). Remember that the functions $h=(h_\a^\b)$ are chosen as the terms of the sequence $(h_\a^{(q),\b})$ in Theorem \equ(so3.1b). Lemma \equ(so4.6) asserts that $$ \lim_{q\to\infty}\E^\mu\Bl[\G\bl(a\cdot h^{(q)}\br)(-\cl_s)( a\cdot h^{(q)})\Br] =2a\cdot(\bar DC)a. $$ Hence, $$ \eqalign{ \E^\mu\bl[Y^\ve_3(t,G,h)\br]&= 2\ve^3\sum_x\underline{\partial}(U^\ve_t G)\cdot (\bar DC) \underline{\partial}(U^\ve_t G)+o_q(1)\cr &=-2\ps{U^\ve_t G,(\bar{\cal D}C)(U^\ve_t G)}_0 +o_q(1)+O(\ve), } $$ where, denoting by $\bar D_{\a,\g}$ the matrix $(\bar D_{\a,\g}^{\b,\nu})_{\b,\nu=0\ldots,4}$, $$ \bar{\cal D}G=\sum_{\a,\g=1}^3 \bar D_{\a,\g}\pt_\a\pt_\g G. $$ With the property $\bar{\cal D}C=C\bar{\cal D}^*$, we get $$ \ps{U^\ve_t G,(\bar{\cal D}C)(U^\ve_t G)}_0 =\ps{e^{-{t\over \ve}E^*} G,(C\bar{\cal D}^*)e^{-{t\over \ve}E^*}G}_0 =\ps{ G,Ce^{-{t\over \ve}E^*}\bar{\cal D}^*e^{-{t\over \ve}E^*}G}_0 $$ and by Lemma A.2 $$ \lim_{\ve\to 0}\int_0^t\!ds\,\exp\bl({s\over\ve }E^*\br) \bar{\cal D}^*\exp \bl(-{s\over\ve }E^*\br)=t\,\pi_{E^*}(\bar{\cal D}^*)=t\, \pi_{E}(\bar{\cal D})^*, $$ so $$ \lim_{\ve\to 0}\E^\mu\Bl[\int_0^t\!ds\,Y^\ve_3(s,G,h)\Br] =-2t\,\ps{G, C\,\pi_{E}(\bar{\cal D})^* G}_0. $$ To summarize, we have proved that there exists a random variable $R^q_\ve$ vanishing in $L^2(\P^\mu_\ve)$ in the limits $\ve\to 0$ and then $q\to\infty$ such that $$ \eqalign{ M_2^\ve(t,G,h)&=\bl(M_1^\ve(t,G,h)\br)^2 +2t\ps{G, C\, \pi_{E}(\bar{\cal D})^* G}_0+2t\ps{G, C\,\Delta G}_0+R^q_\ve\cr &=\bl(M_1^\ve(t,G,h)\br)^2 +2t\ps{G, C\,\pi_{E}({\cal D})^*G}_0+R^q_\ve. } $$ This completes the proof of Theorem \equ(so2.2), once the Boltzmann-Gibbs principle and Lemmas A.2 and A.3 are proved. \vskip 1truecm \noindent{\bf 5. The Boltzmann-Gibbs principle.} \vskip .5truecm \numsec= 5 \numfor= 1 \numtheo=1 Since we closely follow the strategy proposed in [CLO] to prove Theorem \equ(s2.3), we will only focus our attention to the points where non trivial changes are necessary. One of the ingredients in the proof is the equivalence of ensembles, which is classical for Bernoulli product measures but, as far as we know, is not in our case. We state below a weaker statement which will suffice to our purpose. For a given chemical potential $n\in\R^5$, let $M(n)=(M_0(n),\ldots,M_4(n))$ be defined as $M_\b(n)=\E^{\mu_{L,n}}[I_\b(\n_0)]$. If we put $A=M(\R^5)$, it is easy to verify that $n\mapsto M(n)$ is a $C^1$ diffeomorphism from $\R^5$ onto $A$, in particular the inverse function $M\mapsto n(M)$ is continuous on $A$. Given $a>0$, we introduce the set $A^a$ of $M\in A$ such that, $|n(M)-n_0|\le a$, with $n_0=(r,0,0,0,\t)$ the equilibrium chemical potentials. We denote by $\bar\mu_{L,M}$ the grand canonical measure $\mu_{L,n(M)}$ which satisfies therefore $E^{\bar \mu_{L,M}}[I_\b(\n_0)]=M_\b$ for $\b=0,\ldots,4$. Recall that $\bar I_L^+(\n)=\bl(\bar I_{0,L}^+(\n),\ldots, \bar I_{4,L}^+(\n)\br)$ are the empirical averages of the conserved quantities in $\L_L$. For any particle configuration $\n$ in $\O_L$, we call $\bar N^v_L(\n)$, $v\in\cv$, the average number of particles with velocity $v$ in $\L_L$. Also recall the definition of $\phi_\b(v)$ before \equ(3.24a). Given $k=(k_v)_{v\in\cv}$, we set $I_\b(k)=\sum_v\phi_\b(v)k_v$ and $I^+(k)=\bl(I_0(k),\ldots,I_4(k)\br)$. \medskip \goodbreak \noindent{\bf \Lemma (lbg0) (Equivalence of ensembles)} {\it Let $h$ be a local function. Then there exists a constant $c=c(h,a)$ such that $$ \Bigl|\E^\mu\bigl[h\big|\bar I^+_L=M\bigr]-E^{\bar\mu_{L,M}}[h]\Bigr|\le c\,\ve^{3} $$ \nobreak uniformly in $M\in A^a$. } \smallskip \noindent{\it Proof.} Let $\ell$ be the number of velocities in $\cv$ and denote by $\nu_{\a}$, $\a=(\a_v)_{v\in\cv}$, the product measure on $\O_L$ of Bernoulli measures with parameters $\a=(\a_v)_v$, i.e. $E^{\nu_{\a}}[\n(x,v)]=\a_v$. A straightforward extension of the classical strong equivalence of ensembles asserts that for any local function $g$, $$ \Bigl|E^{\nu_\a}\bigl[g\big|\bar N^v_L=k_v,v\in\cv\bigr]-E^{\nu_k}[h]\Bigr|\le C(h)\ve^3 \Eq(clt) $$ uniformly in $k=(k_v)_{v\in\cv}\in B_L=\{0,L^{-3},\ldots,1\}^\ell$. We first compute the term $\E^\mu\bigl[h\big|\bar I^+_L=M\bigr]$. Since this expectation does not depend on the chemical potential (here $n_0$), it is equal to $E^{\nu_{1/2}}\bigl[h\big|\bar I^+_L=M\bigr]$ with the obvious abuse of notation $1/2=(1/2,\ldots,1/2)$. Therefore, from \equ(clt), $$ \E^\mu\bigl[h\big|\bar I^+_L=M\bigr]= { \sum_{k\in B_L, \bar I^+_L(k)=M}\nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br) E^{\nu_k}[h] \over \sum_{k\in B_L,\bar I^+_L(k)=M}\nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br) }+O(\ve^3). \Eq(f1) $$ Since the particles with different velocities are independent $$ \nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br)=\prod_{v\in\cv} \nu_{1/2}\bl(\bar N^v_L=k_v\br) $$ and the asymptotics of a single term in the product above is given by the Stirling formula $$ \nu_{1/2}\bl(\bar N^v_L=k\br)= {1\over \sqrt{2\pi \ve^{-3}k(1-k)}}\exp\Bl[-\ve^{-3}(s(k)+\log 2)\Br] \Bl(1+O\Bl({\ve^3\over k(1-k)}\Br)\Br), \Eq(f2) $$ where $s(k)=k\log k+(1-k)\log(1-k)$ is the entropy. In particular, if $(k_v)_v$ belongs to $B_L^\d:=B_L\cap[\d,1-\d]^\ell$ for some small $\d>0$, then $$ \nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br)= {1\over \sqrt{(2\pi \ve^{-3})^\ell\prod_vk_v(1-k_v)}} \exp\bl[-\ve^{-3}\sum_v(s(k_v)+\log 2)\br] \bl(1+O(\ve^3)\br). $$ The fact that the entropy is convex suggests to use the Laplace method to derive the asymptotics of both terms in the ratio \equ(f1). This is the aim of Lemma \equ(lbg0a) below which is stated in the $\ell=1$ case without any constraint on $k$, nevertheless the generalization to higher dimension with constrains is easy because, up to a linear change of variables $k\mapsto k'$, the sums over $k$ in \equ(f1) with constraints can be written as a sum without constraint over $k'$ in a cube of dimension $\ell-5$ (5 is the number of linear conditions $I^+(k)=M$). Therefore, we have $$ \eqalign{ &\sum_{k\in B^\d_L, \bar I^+(k)=M}\nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br)E^{\nu_k}[h]\cr &\qquad={T_\ve\over \sqrt{(2\pi \ve^{-3})^\ell\prod_vk^*_v(1-k^*_v)}} \exp\Bl[-\ve^{-3}\sum_v(s(k^*_v)+\log 2)\Br]E^{\nu_{k^*}}[h]\bl(1+O(\ve^3)\br),\cr } $$ where $k^*$ is the minimizer of $\sum_v(s(k_v)+\log 2)$ under the constraints $k\in [\d,1-\d]^n$ and $I^+(k)=M$, $$ T_\ve=\sum_{\scriptstyle k\in B^\d_L,\bar I^+(k)=M\atop\scriptstyle |k-k^*|\le \ve^{3\a}}\exp\Bl[-{\sum_vs''(k^*_v)\over 2}\ve^{-3}(k-k^*)^2\Br]. $$ with $0<\a<1/2$. Notice that this result holds provided that the minimizer $k^*$ satisfies $k^*\in ]\d,1-\d[^\ell$, that will be shown below. As a consequence, the ratio \equ(f1) is equal to $$ E^{\nu_{k^*}}[h]\bl(1+O(\ve^3)\br), $$ provided that the contributions from ``bad'' configurations are negligible. Let $\k$ be the minimizer of $\sum_v(s(k_v)+\log 2)$ under the constraints $k\in [0,1]^\ell$, $I^+(k)=M$. >From Lagrange optimization theorem, $\k$ has to minimize the function $$ \sum_v(s(k_v)+\log 2)+\sum_{\b=0}^4\g_\b \sum_v\br(\phi_\b(v)k_v-M_\b\bl), $$ where $\phi_\b$ have been defined in the line before \equ(3.25) and $(\g_\b)$ are Lagrangian multipliers. So the minimizer satisfies $$ s'(\k_v)=\sum_{\b=0}^4\g_\b\phi_\b(v),\quad v\in\cv. $$ Since the derivative of the entropy $s'(\a)$ is equal to the associated chemical potential $\l=\log{\a\over 1-\a}$, we have $\nu_{\k}=\mu_{L,\g}$, $\g=(\g_0,\ldots,\g_4)$ but the constraint $I^+(\k)=M$ implies that $\g=n(M)$ that is to say $\nu_{\k}=\bar\mu_{L,M}$ and in particular $E^{\nu_{\k}}[h]=E^{\bar\mu_{L,M}}[h]$. Moreover, if $\l_v=\log{\k_v\over 1-\k_v}$ is the chemical potential related to $\k_v$, then we have $$ \l_v=\sum_{\b=0}^4\phi_\b(v)n_\b(M). $$ >From the assumption $M\in A^a$, the previous equality implies that we can choose $\d>0$ small enough such that $\k\in]2\d,1-2\d[^\ell$ uniformly in $M\in A^a$. Such a choice of $\d$ implies that $k^*=\k$. So the lemma will be proved if we finally show that the contribution of the ``bad'' $k$ ($k\in B_L\setminus [\d,1-\d]^\ell$) inside the sums in the numerator and denominator of the ration \equ(f1) is irrelevant with respect to the leading term. From Stirling formula \equ(f2), there is $c>0$ such that $$ \sum_{k\notin B^\d_L, \bar I^+(k)=M}\nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br) \le \exp\bl[-\ve^{-3}\sum_v(s(k_v)+\log 2)-c\log\ve\br]. $$ >From the discussion above, there exists $b>0$ such that $\sum_vs(k_v)\ge \sum_vs(\k_v)+b$, therefore $$ \sum_{k\notin B^\d_L, \bar I^+(k)=M}\nu_{1/2}\bl(\bar N^v_L=k_v,v\in\cv\br) \le c\exp\bl[-\ve^{-3}\sum_v(s(\k_v)+\log 2)\br]\exp[-b\ve^{-3}/2]. $$ \qed \bigskip \noindent{\bf \Lemma (lbg0a)} {\it Let $\psi$ and $\phi$ be smooth functions on $[0,1]$, $\psi$ concave, $\phi$ non negative. Assume that the maximizer $\t$ of $\psi$ is in $]0,1[$, then $$ \sum_{i=0}^N\phi\bl({i\over N}\br)\exp \bl[N\psi\bl({i\over N}\br)\br] =S_N\bl(\a,{\psi''(\t)\over 2}\br)\phi(\t)\exp [N\psi(\t)] \Bigl(1+O\bl({1\over N}\br)\Bigr), $$ with $\t$ the maximizer of $\psi$ and $$ S_N(\a,a)=\sum_{|i-N\t|\le N^{1-\a}}\exp\Bl[a{(i-N\t)^2\over N}\Br], \ 0<\a<{1\over 2}. %\ ,\qquad\bl|O_\phi\bl({1\over N}\br)\br|\le {c(\phi)\over N} $$ } \medskip \noindent{\it Proof.} We start by factorizing the leading term $\exp[N\psi(\t)]$ in the sum. For simplicity call $$ U_N(i)=\phi\bl({i\over N}\br)\exp N\bl[\psi\bl({i\over N}\br)-\psi(\t)\br]. %\Eq(le1) $$ >From the assumption on $\t$, if $\d>0$ there exists a constant $c(\d)>0$ such that $$ \sum_{|i-N\t|> N\d}U_N(i)\le \exp[-Nc(\d)]. $$ Moreover, choosing $\d$ small enough ensures that $\psi(x)-\psi(\t)\le -c(x-\t)^2$ provided that $|x-\t|\le\d$, where $c>0$ is a constant which will change from line to line. Then, given $0<\a<{1\over 2}$, $$ \sum_{N^{1-\a}<|i-N\t|\le N\d}U_N(i)\le \exp[-N^{1-2\a}c(\d)]. \Eq(le2) $$ So the main contribution is coming from $\sum_{|i-N\t|\le N^{1-\a}}U_N(i)$. Using Taylor expansion, we see that in this range of $i$'s, $$ U_N(i)=\exp\Bl[b_0{j^2\over N}\Br]\lt( a_0+a_1\Bl({j\over N}\Br)+a_0b_1N\Bl({j\over N}\Br)^3 +O\Bl(\Bl({j\over N}\Br)^2\Br)+O\Bl(N\Bl({j\over N}\Br)^4\Br)\rt), $$ where $j={i\over N}-\t$, $a_0=\phi(\t)$, $a_1=\phi'(\t)$, $b_0={\psi''(\t)\over 2}<0$ and $b_1={\psi^{(3)}(\t)\over 6}$. By the ``almost oddness'' of $j$, $$ \sum_{|i-N\t|\le N^{1-\a}}\exp\Bl[b_0{j^2\over N}\Br]\lt( a_1\Bl({j\over N}\Br)+a_0b_1N\Bl({j\over N}\Br)^3\rt) =O\bl({1\over N}\br)S_N(\a,b_0). \Eq(le3) $$ We also remark $$ \sum_{|i-N\t|\le N^{1-\a}}\Bl({j\over N}\Br)^2\exp\Bl[b_0{j^2\over N}\Br] \le {c\over N}\sum_{|i-N\t|\le N^{1-\a}}\exp\Bl[b_0{j^2\over 2N}\Br], $$ so that $$ \sum_{|i-N\t|\le N^{1-\a}}\Bl({j\over N}\Br)^2\exp\Bl[b_0{j^2\over N}\Br]=O\bl({1\over N}\br)S_N\bl(\a,{b_0\over 2}\br), \Eq(le4) $$ We get in the same way $$ \sum_{|i-N\t|\le N^{1-\a}}N\Bl({j\over N}\Br)^4\exp\Bl[b_0{j^2\over N}\Br]=O\bl({1\over N}\br)S_N\bl(\a,{b_0\over 2}\br). \Eq(le5) $$ Finally, comparing $N^{-1/2}S_N(\a,a)$ ($a<0$) with the integral of a Gaussian, it is easy to check that $S_N(\a,a)=c(a)\sqrt{N}+O(1)$, therefore $$ S_N\bl(\a,{b_0\over 2}\br)=O(1)S_N(\a,b_0). \Eq(le6) $$ Putting together formulas \equ(le2) to \equ(le6), the Lemma is proved. \noindent\qed \bigskip Even if the equivalence of ensembles that we stated in Lemma \equ(lbg0) is weaker than the classical one, it is enough to prove the following result which is actually the only estimate needed in the proof of Boltzmann-Gibbs principle. \bigskip \noindent{\bf \Corollary (lbg0b)} {\it If $h\in\cg$ is a local function then $$ \E^\mu\Bl[\bl(\E^\mu\bigl[h\big|\bar I^+_L=M\bigr]\br)^2\Br]\le c\,\ve^6. $$ } \smallskip \noindent{\it Proof.} Let $\hat h=\E^\mu\bigl[h\big|\bar I^+_L\bigr]$, $\tilde h(M)=E^{\bar\mu_{L,M}}[h]$ and consider the decomposition $$ \E^\mu\bl[\hat h^2\br]\le 2\E^\mu\Bl[\bl(\hat h-\tilde h(\bar I^+_L)\br)^2\Br]+ 2\E^\mu\Bl[\bl(\tilde h(\bar I^+_L)\br)^2\Br]. $$ Since $h$ is in $\cg$, we have $$\tilde h(m)=0 \quad\hbox{and } \quad{\pt \tilde h\over \pt M_\b}\Big|_{M=m}=0,$$ with $m_\b$ the equilibrium values of $I_\b(\n_0)$. Therefore $$ \bl|\tilde h(\bar I_L^+)\br|\le c\sum_{\b,\nu=0}^4\bl|\bl(\bar I_{\b,L}(\n)-m_\b\br) \bl(\bar I_{\nu,L}(\n)-m_\nu\br)\br|. $$ Hence $\E^\mu_\ve\bl[(\tilde h(\bar I_L^+))^2\br]\le c\, \ve^6$. On the other hand, it results from Lemma \equ(lbg0) that for any $a>0$ $$ \E^\mu\Bl[\bl(\hat h-\tilde h(\bar I^+_L)\br)^2\Br] \le c(a)\ve^6+cP^\mu\bl[I^+_L\notin A^a\br]. $$ >From the continuity of the function $M\mapsto n(M)$, there exists $b>0$ such that $$ P^\mu\bl[I^+_L\notin A^a\br]\le P^\mu\bl[|I^+_L-m|>b\br]. $$ Finally, since $I^+_L={1\over |\L_L|}\sum_xI^+(\n_x)$ with $I^+(\n_x)$ i.i.d. random vectors with finite exponential moments and expectation $m$ under $\mu$, a large deviation estimate provides $$ P^\mu\bl[|I^+_L-m|>b\br]\le\exp(-c\ve^3). $$ \qed \bigskip The first result used in [CLO] (Lemma 4.3) before establishing Boltzmann-Gibbs is a general estimate bounding the equilibrium expectation of the squared time integral of zero mean functions of Markov processes by their $\|\cdot\|_{-1}$ norm. More precisely, if $X$ is a Markov process on the finite state space $\ce$ with generator $L$ and ergodic invariant measure $\pi$, then there exists a (universal) constant $c>0$ such that for any function $f\,:\,[0,T]\times\ce\rightarrow\R$ satisfying $\E^\pi[f(t,X_t)]=0$ for any $t\in[0,T]$, we have $$ \E^\pi\left[\sup_{0\le t\le T}\left( \int_0^t\!ds\,f(s,X_s)\right)^2\right] \le c\int_0^T\!ds\,\|f(s,\cdot)\|_{-1}^2, \Eq(bg1) $$ where $$ \|f\|_{-1}^2=\sup_g\{\ps{f,g}+\ps{f,L^sg}\} $$ and $L^s$ is the symmetric part of $L$ in $L^2(\pi)$. \bigskip The next lemma (Lemma 4.4 in [CLO]) is needed to control remainder terms in the proof of the Boltzmann-Gibbs principle. \medskip \noindent{\bf \Lemma (lbg1)} {\it For any local function $h\in\cg$, there exists a constant $c(h)>0$ such that for any subset $B$ of $\L_L$, any smooth function $G\,:\,[0,T]\times\T_3\rightarrow\R$ and $\ve$ small enough, $$ \E^\mu_\ve\left[\sup_{0\le t\le T}\left( \int_0^t\!ds\,\ve^{3/2-1}\sum_{x\in B}G(s,\ve x)) \tau_xh(\n_s)\,ds\right)^2\right] \le c(h)(1+T)\int_0^T\!\!ds\,\ve^3\sum_{x\in B}G^2(s,\ve x). \Eq(apriori) $$ } \noindent{\it Proof.} Following [EMY2] (section 4), we introduce an alternative representation for the particle configuration $\n_x=\bigl(\n(x,v)\bigr)_{v\in\cv}$ at site $x$: one can find $n(n-5)$ numbers $c_\b(v)$, $\b=-n+5,\ldots,-1$ and $v\in\cv$, such that if we put $$ I_\b(\n_x)=\sum_{v\in\cv}c_\b(v)\n(x,v), %\Eq(bg2) $$ then the map $\n_x\mapsto (I_\b(\n_x))_{\b=-n+5,\ldots,4}$ is one to one. Moreover the coefficients $c_\b(v)$ can be chosen in such a way that the covariances $\E^\mu[I_\b(\n_x);I_\nu(\n_x)]$, $\b\not=\nu$, vanish (except $\b,\nu\in \{0,4\}$). We also introduce the variables $\tilde I_\b(\n_x)=I_\b(\n_x)-\E^\mu[I_\b(\n_x)]$. \medskip Let $\cg^{ex}$ be the space of functions $h$ such that $\E^\mu[h]=0$ and $\sum_x\E^\mu[h;I_\b(\n_x)]=0$ for any $\b=-n+5,\ldots,4$. The integration by parts lemma valid for ASEP (Lemma 6.1 in [EMY1]) easily generalizes to a superposition of ASEP. % and, in our setting, it has the % following form: for $h\in\cg^{ex}$, there exist functions % $\Phi_{x,e,v}\in L^2(\mu)$ (depending on $h$) such that for % any $u\in L^2(\mu)$, % $$ % \E^\mu[gu]=\sum_x\sum_{|e|=1}\sum_{v\in\cv}\E^\mu[\Phi_{x,e,v}\nabla_{x,e,v}u] % \Eq(bg3) % $$ % with $\nabla_{x,e,v}u=u(\n^{x,x+e,v})-u(\n)$. Furthermore % $$ % \sum_x\sum_{|e|=1}\sum_{v\in\cv}|x|^{3+1/2}\E^\mu[\Phi^2_{x,e,v}]<+\infty % \Eq(bg4) % $$ \smallskip We now turn to the proof. Fix $h\in\cg$, we can find coefficients $(a_\b)_{\b<0}$ such that $h-\sum_{\b<0}a_\b\bar I_{\b,L}$ is in $\cg^{ex}$, where $\bar I_{\b,L}(\n)=|\L_L|^{-1}\sum_x I_\b(\n_x)$. Therefore, it is enough to prove the lemma in the case where $h\in\cg^{ex}$ and in the case where $h=\bar I_{\b,L}(\n)$. The first case is a straightforward generalization of Lemma 4.4 in [CLO] since the integration by parts formula is valid in $\cg^{ex}$. In the second case, denote by $\hat I_\b(\n_x)$ ($\b<0$ fixed) the conditional expectation of $I_\b(\n_x)$ with respect to the empirical averages of the conserved quantities $\bar I^+_L$. %and let $\hat I^\ve_{\b,L}=|\L_L|^{-1}\sum_x\hat I_\b(\n_x)$. Then the left hand side of \equ(apriori) is bounded above (up to a factor 2) by the sum of the two terms $$ \E^\mu_\ve\left[\sup_{0\le t\le T}\left( \int_0^t\!ds\,\ve^{3/2-1}\bar G^B_s \sum_x\bl(I_\b(\n_x)-\hat I_\b(\n_x)\br)\,ds\right)^2\right], \Eq(bg5) $$ where $\bar G^B_s:=\ve^3 \sum_{x\in B}G(s,\ve x)$, and $$ \E^\mu_\ve\left[\sup_{0\le t\le T}\left( \int_0^t\!ds\,\ve^{3/2-1}\sum_{x\in B}G(s,\ve x) \hat I_\b(\n_0)\,ds\right)^2\right], \Eq(bg6) $$ >From the inequality \equ(bg1), \equ(bg5) is less or equal to $$ c\,V_L\bl(\tilde I_\b(\n_0),r,\t\br)\int_0^T\!ds\,\bl(\bar G^B_s\br)^2 $$ and by corollary 4.6 of [EMY2], $V\bl(\tilde I_\b(\n_0),r,\t\br)=\limsup_L V_L\bl(\tilde I_\b(\n_0),r,\t\br)<+\infty$. So \equ(bg5) is bounded above by $$ c\int_0^T\!ds\,\ve^3\sum_{x\in B}G^2(s,\ve x) $$ ($c$ a positive constant). Finally, by stationarity of $\mu$, the term \equ(bg6) is less than $$ \ve^{-5}\E^\mu\bl[(\hat I_\b(\n_0))^2\br]T\int_0^T\!ds\,\bl(\bar G^B_s\br)^2. $$ >From Corollary \equ(lbg0b), $\E^\mu\bl[(\hat I_\b(\n_0))^2\br]\le c\,\ve^6$ and \equ(bg6) is going to zero as $\ve\to 0$. \noindent\qed \bigskip Finally, Corollary \equ(lbg0b) and Lemma \equ(lbg1) allows to extend straightforwardly the proof of the Boltzmann-Gibbs principle given in section 4 of [CLO] and then to obtain Theorem \equ(s2.3). \bigskip We conclude this section by pointing out that the arguments for the proof of tightness (section 5 of [CLO]) can be easily adapted to our case. Notice that, up to now, we did not need to have the supremum over time inside the expectation in the Boltzmann-Gibbs statement, however it is used in this part to control some terms arising in martingale compensators. So, we can state \medskip \noindent{\bf \Theorem (tight)} {\it The family of probability $(Q^\ve)_{\ve>0}$ on $D([0,T],\ch_{-k_0})$ is tight since $$ \lim_{M\to\infty}\lim_{\ve\to 0}\P^\mu_\ve\Bl(\sup_{0\le t\le T} \|\xi_t^\ve\|_{-k_0}>M\Br)=0\Eq(t1) $$ and for any $a>0$ $$ \lim_{\d\to 0}\lim_{\ve\to 0}\P^\mu_\ve\Bl( \sup_{\scriptstyle |s-t|<\d\atop 0\le s,t\le T} \|\xi^\ve_t-\xi^\ve_s\|_{-k_0}>a\Br)=0.\Eq(t2) $$ } \bigskip \vskip 1truecm \noindent{\bf Appendix} \vskip .5truecm \numfor=1 Let ${\cal A}$ be the space of $n\times n$ matrices with complex entries. ${\cal A}$ is a Hilbert space under the scalar product $$ (X,Y)=\sum_{1\le k,\ell\le n}\bar X_{k,\ell}Y_{k,\ell}. $$ Given a matrix $A$ in ${\cal A}$ the projector $\Pi_A $ is defined as the orthogonal projection onto ${\cal C}(A)$, the commutator space of $A$ $$ {\cal C}(A) =\{M\in A: [M,A]=0\},\quad [M,A]:=MA-AM. $$ \vskip.2cm \goodbreak \noindent{\bf Lemma A.1} \nobreak {\it Let $A$ be a diagonalizable matrix, ${\rm Sp}(A)\in i\R$. Then, for any matrix $M$ and $t>0$ $$\lim_{\ve\to 0}{1\over t}\int_0^t\!ds\,\exp\bl({s\over\ve }A\br) M\exp \bl(-{s\over\ve }A\br)=\Pi_A(M). %\Eqa(A.1) $$ } \medskip \noindent{\it Proof.} We follow the proof in [EP]. Let $P$ be a non-singular matrix and $R$ a real diagonal matrix such that $A=P^{-1}i RP$. Let $\{S_j,j=1,\dots,m\}$ be a partition of the integers $\{j=1,\dots,n\}$ such that $$ \eqalign{R_k&=R_\ell \quad \hbox{if} \quad k,\ell\in S_j \quad \hbox{for some}\ j,\cr R_k&\ne R_\ell\quad \hbox{otherwise,}} $$ where $R_j, j=1,\cdots,n$ are the eigenvalues of $R$. We define the bar operation in the following way: let $K=(K_{k,\ell})\in {\cal A}$ be $$K_{k,\ell}=\cases{1\quad \hbox{if} \quad k,\ell\in S_j \quad \hbox{for some}\ j,\cr 0 \quad \hbox{otherwise.}} $$ Then $\ov M$, $M\in {\cal A}$, is defined as $$\ov M_{k,\ell}=K_{k,\ell}M_{k,\ell}.$$ Observe that $\ov M$ is the diagonal part of $ M$ in the simple case of $R$ with distinct eigenvalues. \medskip We have that $$ \exp\bl({s\over\ve }A\br) M\exp \bl(-{s\over\ve }A\br)= P^{-1}\exp\bl({i s\over\ve }R\br)PMP^{-1}\exp\bl(-{i s\over\ve }R\br)P. $$ It is proved in [EP] that $$ P^{-1}\overline{ P MP^{-1}} P $$ is a projection onto ${\cal C}(A)$. Hence $\ov M=\Pi_R(M)$ because $R$ is diagonal. Moreover, $\Pi_R(M)=\Pi_A(M)$ because $R$ is diagonal. So it is enough to prove that for any $M$ $$ \lim_{\ve\to 0}{1\over t}\int_0^t\!ds\, \exp\bl({i s\over\ve }R\br) M\exp\bl(-{i s\over\ve }R\br)=\ov M. $$ In [EP] it is also shown that for any matrix $M$ there exists a matrix $S$ such that $M$ can be decomposed as $$ M=\ov M +[S,R]. $$ Since $\ov M$ commutes with $R$ $$ \exp\bl({i s\over\ve }R\br) M\exp\bl(-{i s\over\ve }R\br) =\ov M +\exp\bl({i s\over\ve }R\br)[S,R]\exp\bl(-{i s\over\ve }R\br). $$ The second term on the r.h.s gives $$ \Bl(\exp\bl({i s\over\ve }R\br)[S,R]\exp\bl(-{i s\over\ve }R\br)\Br)_{k,\ell}= S_{k,\ell}(R_\ell-R_k)\exp\bl({is\over\ve}(R_k-R_\ell)\br), $$ where $S=(S_{k,\ell})$ and $R=(R_{k,\ell})=(R_k\d_{k,\ell})$. As a consequence, $$ \lim_{\ve\to 0}{1\over t}\int_0^t\!ds\exp\bl({i s\over\ve }R\br)[S,R]\exp\bl(-{i s\over\ve }R\br)=0. $$ \vskip.4cm \noindent{\bf Lemma A.2} {\it Let $\E$ be a first order differential operator such that its Fourier transform $\hat\E(k)$ satisfies ${\rm Sp}(\hat\E(k))\in i\R$ for any $k$ and let $\D=\sum_{\a,\g=1}^3D_{\a,\g}\partial_\a\partial_\g$ be a second order differential operator, where $D=(D_{\a,\g})=(D^{\b,\nu}_{\a,\g})$ is a definite positive rank 2 tensor. Then there exists a definite positive second order differential operator $\pi_\E(\D)$ such that for any $G$ smooth $$ \lim_{\ve\to 0}\Bigl\|\int_0^t\!ds\,\Big[\exp\bl({s\over\ve }\E\br) \D\exp \bl(-{s\over\ve }\E\br)-\pi_\E(\D)\Big]G\Bigr\|_0=0. $$ } \medskip \noindent{\it Proof.} Let $\hat \D(k)$ be the Fourier transform of $\D$ $$ \hat\D(k)=-\sum_{\a,\g=1}^3 D_{\a,\g}^{\b,\nu}k_\a k_\g \hat G(k) $$ It is enough to prove that for any $t>0$ and for any $G$ smooth there exist a matrix $\hat\pi_E(\D)$ such that $$ \lim_{\ve\to 0}\Bigl\|\int_0^t\!ds\,\Bigl[\exp\bl({s\over\ve }\hat \E\br) \hat\D\exp\bl(-{s\over\ve }\hat \E\br)-\hat\pi_\E(\D)\Bigr] G\Bigr\|_0=0 $$ where $\|\cdot\|_0$ is the usual norm in $L^2(\T_3,\R^5)$. Choosing $\hat\pi_\E(\D)=\pi_{\hat\E}(\hat\D)$, that is an easy consequence of Lemma A.1 via dominated convergence theorem since, by assumption, $\hat\E$ is diagonalizable with pure complex eigenvalues which implies $\bigl\|\exp \bigl({s\over\ve }\E\bigr)\bigr\|_0\le {\rm const}$. Finally, since $\hat\D$ is definite positive, the same is true for $\pi_{\hat\E}(\hat\D)$. \noindent\qed \smallskip Notice that Lemma A.2 implies that for any $0\le s\le t$, $$ \lim_{\ve\to 0}\Bigl\|\int_s^t\!du\,\Big[\exp\bl({u\over\ve }\E\br) \D\exp \bl(-{u\over\ve }\E\br)-\pi_\E(\D)\Big]G\Bigr\|_0=0 $$ \vskip.4cm \goodbreak \noindent{\bf Lemma A.3} \nobreak {\it Let $A^\ve(s)$, $A$ be linear operators from $\ch_{k_0+2}$ to $\ch_{k_0}$ such that $$ \sup_{\ve,0\le s\le t}\bl\|A^\ve(s)\br\|_{k_0+2\to k_0}<\infty $$ \nobreak and for any $G\in \ch_{k+2}$ and $0\le s< t$ $$\lim_{\ve\to 0}\lt\|\int_s^t\!du\, [A^\e(u)-A]G\rt\|_{k_0}=0.$$ Then, for any $G\in \ch_{k_0+2}$ $$\lim_{\ve\to 0}E_\ve^{\mu}\lt[\lt(\int_0^t\!ds\,\xi^\ve \bl(s,[A^\ve(s)-A]G\br)\rt)^2\rt]=0, $$ where $\xi^\ve_s$ is the fluctuation field.} \bigskip \noindent{\it Proof.} We set $\ps{\xi^\ve_t,G}=\xi^\ve(t,G)$. Let $0=t_00$. Then $$ \int_0^t\!ds\,\xi^\ve(s,A^\ve(s)G) =\sum_{i=0}^{\ell-1}\Bl\langle\xi^\ve_{t_i},\int_{t_i}^{t_i+1}\!ds\,A^\ve(s)G \Br\rangle+R_1^\ve, $$ with $$R_1^\ve =\sum_{i=0}^{\ell-1}\int_{t_i}^{t_i+1}\!ds\,\ps{\xi_s^\ve -\xi_{t_i}^\e,A^\ve(s)G}. $$ Since $$ |R_1^\ve|\le t\sup_{\scriptstyle |s_1-s_2|\le\delta\atop\scriptstyle 0\le s_1,s_2\le t} \|\xi_{s_1}^\ve-\zeta_{s_2}^\e\|_{-k_0} \sup_{0\le s\le t}\|A^\e(s)\|_{k_0+2\to k_0}\|G\|_{k_0+2}, $$ it results from tightness \equ(t2) that for any $\d>0$, $$ \lim_{\d\to 0}\lim_{\ve\to 0}P^\ve_{\mu}\bl(|R_1^\ve|>\d\br)=0. $$ Moreover $$\int_0^t\!ds\,\ps{\xi^\ve_s,A^\ve(s)G}= \sum_{i=0}^{\ell-1}\ps{\xi_{t_i}^\ve,AG}(t_{i+1}-t_i)+R_1^\ve+R_2^\e,$$ with $$ \eqalign{ |R_2^\e|& =\left|\sum_{i=0}^{\ell-1}\int_{t_i}^{t_{i+1}} \!ds\,\ps{\xi^\ve_{t_i}(A^\ve(s)-A)G} \right|\cr &\le t\sup_{0\le s\le t}\|\zeta_s^\ve\|_{-k_0}\max_i\left\|\int_{t_i}^{t_{i+1}}ds[A^\ve(s)-A]G \right\|_{k_0}. } $$ >From assumption $$ \lim_{\ve\to 0}\max_i\lt\|\int_{t_i}^{t_{i+1}}\!ds\, [A^\ve(s)-A]G\rt\|_{k_0}=0. $$ So, using tightness \equ(t1), we get for $M>0$ and $\ve$ small enough $$ P^\ve_{\mu}\bl(|R_2^\ve|>\d\br)\le P^\ve_{\mu}\Bl( \sup_{0\le s\le t}\|\xi_s^\e\|_{-k_0}>{\d\over M}\Br) $$ which vanishes in the limit ${M\to0}$ after ${\ve\to 0}$. With the same kind of arguments (using tightness again), we get $$ \sum_{i=0}^{\ell-1}\ps{\xi_{t_i}^\ve,AG}(t_{i+1}-t_i) =\int_0^t\!ds\,\ps{\xi_s^\e,AG} +R_3^\ve, $$ where $$ \lim_{\d\to 0}\lim_{\e\to 0}P^\ve_{\mu}\bl(|R_4^\e|>\d\br)=0. $$ We have proved so far that $\int_0^t\!\,ds\bl(\xi^\ve(s,A^\ve(s)G)- \xi^\ve(s,A G)\br)$ converges to 0 in $\P^\mu_\ve$ probability. To assert that the convergence occurs in $L^2(P^\mu)$ it suffices e.g. to check that $$ \sup_\ve \E^\mu_\ve\lt[\lt(\int_0^t\!ds\,\ps{\xi^\ve_s,(A^\ve(s)-A)G}\rt)^4\rt] <\infty, $$ which is clear from the assumptions on the operator $A^\ve(s)$ and $A$. \vfill\eject {\centerline{\bf REFERENCES}} \vskip .4cm \item {[BEM]} O. Benois, R. Esposito and R. Marra {\it Navier-Stokes limit for a thermal stochastic lattice gas.} J. Stat Phys. {\bf 90} 653--713 (1999). \item{[BR]} T. Brox and H. Rost, {\it Equilibrium fluctuations of stochastic particle systems: the role of conserved quantities}. Ann. 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Ikeda eds, Pitman Research Notes % in Math., {\bf 283}, 75--128 J. Wiley \& Sons, New York (1994). \bigskip \noindent{\bf Acknowledgments}: Two of us (R.E. and R.M.) wish to thank the University of Rouen and the IHES in Bures-sur-Yvette, where part of this work was done, for the very kind hospitality. Work partially supported by GNFM-INDAM, MURST and CEE-TMR's on Hyperbolic Systems and Kinetic Models. O.B. acknowledges very much the University of Roma Tor Vergata for hospitality, C. Landim for fruitful discussions and is infinitely grateful to his two co-authors for them patience and understanding about the difficulty to become a father. \end ENDBODY