Content-Type: multipart/mixed; boundary="-------------0001130408590" This is a multi-part message in MIME format. ---------------0001130408590 Content-Type: text/plain; name="00-15.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-15.comments" 10 pages ---------------0001130408590 Content-Type: text/plain; name="00-15.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-15.keywords" Chaos, Turbulence, Fluids, Nonequilibrium, Intermittency ---------------0001130408590 Content-Type: application/x-tex; name="mf2000.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mf2000.tex" %%%%% mp_arc \# 00-???, chao-dyn \#0001??? %**start of header %\input fiat \newcount\mgnf\newcount\tipi\newcount\tipoformule \newcount\aux\newcount\piepagina\newcount\xdata % \mgnf=0 \aux=1 %1 produce aux \tipoformule=1 %0 usa aux; 1 no (usa i simboli dati) \piepagina=1 %0 =data e #par.#pag; 1=data e #pag; 2=#pag \xdata=0 %0 data del giorno, 1 data fissa da \Di: \def\Di{} \ifnum\mgnf=1 \aux=0 \tipoformule =1 \piepagina=1 \xdata=1\fi \newcount\bibl %\bibl= ? % 0= rif [XXX], 1= rif. numerici \ifnum\mgnf=0\bibl=0\else\bibl=1\fi \bibl=0 % Per poter cambiare a piacimento il formato dei riferimenti % bibliografici in .tex: % % 1: citare nella forma esemplificata da \ref{B}{2}{20}} % ove XXX e' un simbolo per le iniziali e 2 distingue i lavori con % le stesse iniziali, 7 e' il numero SIMBOLICO del riferimento per XXX2. % Il numero 7 puo' essere ARBITRARIO e viene automaticamente % riaggiustato al momento della compilazione (vedi punto 4) % % 2: Se si sceglie \bibl=0 si cita nella forma [XXX2]; se si sceglie % \bibl=1 si cita nella forma [numero di ordine di prima citazione]. % % 3: La bibliografia va scritta nella forma \def{\qqq}{} % in ordine alfabetico per autore attribuendo un simbolo % qualsiasi al testo che (usando ref.b) produce fin.tex e .tex % con i riferimenti giusti % in \bibl=1 e la si ricompila e stampa. La scheda iniziale .tex % diventa .old. \ifnum\bibl=0 \def\ref#1#2#3{[#1#2]\write8{#1@#2}} \def\rif#1#2#3#4{\item{[#1#2]} #3} \fi %\def\rif#1#2#3#4{\write9{\noexpand\raf{#1}{#2}{\noexpand#3}{#4}}} \ifnum\bibl=1 \openout8=ref.b \def\ref#1#2#3{[#3]\write8{#1@#2}} \def\rif#1#2#3#4{} \def\raf#1#2#3#4{\item{[#4]}#3} \fi \def\9#1{\ifnum\aux=1#1\else\relax\fi} \ifnum\piepagina=0 \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss}\fi \ifnum\piepagina=1 \footline={\rlap{\hbox{\copy200}} \hss\tenrm \folio\hss}\fi \ifnum\piepagina=2\footline{\hss\tenrm\folio\hss}\fi \ifnum\mgnf=0 \magnification=\magstep0 \hsize=13.5truecm\vsize=22.5truecm \parindent=4.pt\fi \ifnum\mgnf=1 \magnification=\magstep1 \hsize=16.0truecm\vsize=22.5truecm\baselineskip14pt\vglue5.0truecm \overfullrule=0pt \parindent=4.pt\fi \let\a=\alpha\let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\ch=\chi \let\ps=\psi \let\o=\omega \let\y=\upsilon \let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda\let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\U=\Upsilon {\count255=\time\divide\count255 by 60 \xdef\oramin{\number\count255} \multiply\count255 by-60\advance\count255 by\time \xdef\oramin{\oramin:\ifnum\count255<10 0\fi\the\count255}} \def\ora{\oramin } %\Di e' definito all' inizio e ridefinito qui \ifnum\xdata=0 \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;\ \ora} \def\Di{\number\day\kern2mm\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\kern0.1mm\number\year} \else \def\data{\Di} \fi \setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 \9{\write15{\string\FU (#1){\equ(#1)}}} \9{ \write16{ EQ \equ(#1) == #1 }}} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1 \9{\write15{\string\FU (#1){\equ(#1)}}} \9{ \write16{ EQ \equ(#1) == #1 }}} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 \9{\write15{\string\FU (#1){\equ(#1)}}} \9{ \write16{ Fig. \equ(#1) ha simbolo #1 }}} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\write16{No translation for #1} \else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13\fi \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \9{\openout15=\jobname.aux} \newskip\ttglue %\font\dodicirm=cmr12\font\dodicibf=cmbx12\font\dodiciit=cmti12 %\font\titolo=cmbx12 scaled \magstep2 %\font\ottorm=cmr8\font\ottoi=cmmi8\font\ottosy=cmsy8 %\font\ottobf=cmbx8\font\ottott=cmtt8\font\ottosl=cmsl8\font\ottoit=cmti8 %\font\sixrm=cmr6\font\sixbf=cmbx6\font\sixi=cmmi6\font\sixsy=cmsy6 \font\titolone=cmbx12 scaled \magstep2 \font\titolo=cmbx10 scaled \magstep1 \font\ottorm=cmr8\font\ottoi=cmmi7\font\ottosy=cmsy7 \font\ottobf=cmbx7\font\ottott=cmtt8\font\ottosl=cmsl8\font\ottoit=cmti7 \font\sixrm=cmr6\font\sixbf=cmbx7\font\sixi=cmmi7\font\sixsy=cmsy7 \font\fiverm=cmr5\font\fivesy=cmsy5\font\fivei=cmmi5\font\fivebf=cmbx5 \def\ottopunti{\def\rm{\fam0\ottorm}\textfont0=\ottorm% \scriptfont0=\sixrm\scriptscriptfont0=\fiverm\textfont1=\ottoi% 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4pt\dimen\footins=30pc\catcode`@=12 \let\nota=\ottopunti %% Grafica \newdimen\xshift \newdimen\xwidth \newdimen\yshift \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} \def\eqfig#1#2#3#4#5{ \par\xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \yshift=#2 \divide\yshift by 2 \line{\hglue\xshift \vbox to #2{\vfil #3 \special{psfile=#4.ps} }\hfill\raise\yshift\hbox{#5}}} \def\8{\write13} \def\figini#1{\catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2\openout13=#1.ps} \def\figfin{\closeout13\catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12} \def\didascalia#1{\vbox{\nota\0#1\hfill}\vskip0.3truecm} %%%%%%%%%%%%%%%%%%%%%% \def\V#1{{\,\underline#1\,}} \def\T#1{#1\kern-4pt\lower9pt\hbox{$\widetilde{}$}\kern4pt{}} \let\dpr=\partial\def\Dpr{{\V\dpr}} \let\io=\infty\let\ig=\int \def\fra#1#2{{#1\over#2}}\def\media#1{\langle{#1}\rangle}\let\0=\noindent \def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} \def\tende#1{\ \vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hglue3.pt${\scriptstyle% #1}$\hglue3.pt\crcr}}\,} \def\otto{\ {\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\ } \def\tto{{\Rightarrow}} \def\pagina{\vfill\eject}\let\ciao=\bye \def\dt{\displaystyle}\def\txt{\textstyle} \def\tst{\textstyle}\def\st{\scriptscriptstyle} \def\*{\vskip0.3truecm} \def\lis#1{{\overline #1}}\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }} \def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }} \def\ie{\hbox{\it i.e.\ }} \def\fiat{{}} \def\\{\hfill\break} \def\={{ \; \equiv \; }} \def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}} \def\sign{{\rm sign\,}}\def\atan{{\,\rm arctg\,}}\let\arctg=\atan \def\annota#1{\footnote{${{}^{\bf#1}}$}} \ifnum\aux=1\BOZZA\else\relax\fi \ifnum\tipoformule=1\let\Eq=\eqno\def\eq{}\let\Eqa=\eqno\def\eqa{} \def\equ{{}}\fi \def\defi{\,{\buildrel def \over =}\,} \def\pallino{{\0$\bullet$}}\def\1{\ifnum\mgnf=0\pagina\else\relax\fi} \def\W#1{#1_{\kern-3pt\lower6.6truept\hbox to 1.1truemm {$\widetilde{}$\hfill}}\kern0pt} \def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,}\def\DD{{\cal D}} \def\nn{{\V n}}\def\NN{{\cal N}}\def\LL{{\cal L}} \def\aa{{\V\a}} \def\EE{{\cal E}}\def\xx{{\V x}}\def\pps{{\V \ps}} \def\cfr{{\it c.f.r.\ }} \def\kk{{\V k}}\def\uu{{\V u}} \def\VV#1{{\underline #1}_{\kern-3pt$\lower7pt\hbox{$\widetilde{}$}}\kern3pt\,} \def\FINE{ \* \0{\it Internet: Authors' preprints downloadable (latest version) at: \centerline{\tt http://ipparco.roma1.infn.it} \centerline{(link) \tt http://www.math.rutgers.edu/$\sim$giovanni} \* \sl e-mail: giovanni.gallavotti@roma1.infn.it }} \fiat \def\AA{{\cal A}}\def\CS{{\cal S}}\def\KK{{\cal K}} %**end of header \fiat \def\CH{chaotic hypothesis\ } \centerline{\titolo Fluctuations and entropy driven space--time } \centerline{\titolo intermittency in Navier--Stokes fluids.} \*\* \centerline{\it Giovanni Gallavotti} \* \centerline{Fisica, Universit\`a di Roma 1} \centerline{P.le Moro 2, 00185 Roma, Italia} \*\* \0{\bf Abstract: \it We analyze the physical meaning of fluctuations of the phase space contraction rate, that we also call entropy creation rate, and its observability in space--time intermittency phenomena. For concreteness we consider a Navier--Stokes fluid.} \*\* \0{\bf\S1. The chaotic hypothesis in turbulence.} \numsec=1\numfor=1\* Consider a Navier--Stokes (NS) fluid in a container $V$ which we take, for simplicity, cubic with periodic boundary conditions and subject to a constant volume force $f \,\V\f(\xx)$ with $\max |\V\f(\xx)|=1$ and with only Fourier harmonics corresponding to large wavelength of the order of the linear size $L$ of $V$. The viscosity will be denoted $\n$, but it is convenient to rescale space, time, velocity and pressure to write the equations in dimensionless form as $$\dot{{\uu}}+ R\,\W u\cdot\W\dpr\,\uu=\D\,\uu-\Dpr p+\V\f, \qquad \Dpr\cdot\uu=\V0,\qquad R=f L^3\n^{-2}\Eq(1.1)$$ % in a container of side $L=1$, where $R$ is the {\it Reynolds number}. We can suppose that $\ig_V\V u\,d\xx=\V0$ (because of translation invariance). We assume the \CH \* \0{\bf Chaotic hypothesis: \it Asymptotic motions of a turbulent flow develop on an attracting set $\AA$ in phase space on which time evolution $\V u\to S_t\,\uu$ can be regarded as a transitive Anosov system for the purposes of computing time averages in stationary states.} \* Here we investigate some assumptions under which the hypothesis acquires some non trivial predictive value with implications that can have experimental relevance. For earlier reviews on the chaotic hypothesis see [Ga98a], [Ga96d], [Ga99a]. A recent one is [Ru99a]. \* \0{\bf\S2. The OK41 cut--off.} \numsec=2\numfor=1\* Anxiety often mars the beginning of any discussion on the NS equations: it is a fact that to date there is {\it no theory} that allows a constructive solution of the equations via a controlled approximation scheme. Nevertheless most people rapidly recover and adopt the viewpoint that ``physically there is an effective ultraviolet cut--off'' and the NS equations can be reduced to ordinary equations: \* \0{\bf The OK41 cut--off hypothesis: \it There exists $\k_0>0$ such that if the NS equation, \equ(1.1), is truncated in momentum space at $K(R)=R^{\k_0}$ (or higher) then the physically relevant predictions are not affected.} \* The OK41 theory, see [LL71], assigns to $\k_0$ the value $3/4$. Therefore the flows of physical interest should be described by \equ(1.1) truncated at $|\kk|\le K(R)$, \ie $$\dot{{\V u}}_\kk+i\,R\,\sum_{\kk_1+\kk_2=\kk\atop |\kk_j|\le K(R)} \W u_{\kk_1}\cdot\W k\,\P_\kk \V u_{\kk_2}\,=\,-\kk^2\V u_\kk+\V\f_\kk \Eq(2.1)$$ % where $\V u(\xx)\defi\sum_{\kk\ne\V0} e^{i\,\kk\cdot\xx}\uu_{\kk}$ and $\P_\kk$ is the projection orthogonal to $\kk$; $\kk=2\p\V n$ with $\V n$ an integer components vector. Equation \equ(2.1) admits an \ap bound on the energy $\dot E/2=-\sum_{\kk}\kk^2|\V u_\kk|^2+\sum_\kk \lis\f_\kk\cdot \V u_\kk$ which implies that asymptotically in time the energy is bounded by $E\le 2||\V\f||^2/(2\p)^4$. We shall call $\m_R$ the ``{\it statistics}'' of the NS equation defined as the probability distribution on {\it phase space} (\ie on the space of the velocity fields $\{\V u_\kk\},\,|\kk|