Plain TeX, 2 figures autogenerated as blz1.ps and blz2.ps, 50.000 K, 10 pages. Use a postscript printer and compile into postscript with DVIPS. This is a reprint of a 1972 preprint that was never published, but which is occasionally quoted and requested to me. If one does not use a postscript printer the figures cannot be printed: the paper can still be printed provided one changes the parameter \driver on line 3 from 1 to 5. The figures can be found in ref. [11] or [12]. BODY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\driver \newcount\mgnf \newcount\tipi \mgnf=0 \driver=1 \tipi=2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% FORMATO \ifnum\mgnf=0 \magnification=\magstep0\hoffset=0.cm \voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=24.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=24.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 %%%%%%%%%%%%%%%% 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\tempo %\tempo=\number\time\divide\tempo by 60} \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} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} %\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 dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure 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); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure 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 o figure 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\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% %\write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#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% %\write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} % \global\advance\numfig by 1% %\write15{\string\FU (#1){\equ(#1)}}% \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{$\scriptstyle##1$}}}}} \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} } \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{#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} %\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 %\newcount\pinclude \newcount\pcount %\def\include#1{\pinclude=0 %\openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi %\openout15=#1.aux %\input #1 %\pcount=\count0 \advance\pcount by 1 %\write15{\string\ifnum% %\string\pinclude =1 % %\string\pageno = \number\pcount \string\fi} %\closeout15 } % %\def\includeaux#1{\pinclude=1 %\openin13=#1.aux \ifeof13 \immediate\write16{#1.aux does not exist}% %\else \input #1.aux \closein13 \fi } %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \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 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utilizzate solo se il driver % e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o ???? ( \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.ps global} \fi %\ifnum\driver=1 \special{header=ini.ps} \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 ">". 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\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.\ }} 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translate exch 4 1 > \write13 \write13 \write13<1 roll mul add sqrt } def > \write13 \write13 \write13< 222.820 209.594 10.00000 0 360 arc stroke> \write13< 222.820 209.594 2 0 360 arc fill stroke> \write13< 368.473 -5.31179 10.00000 0 360 arc stroke> \write13< 368.473 -5.31179 2 0 360 arc fill stroke> \write13< 41.4465 44.8195 10.00000 0 360 arc stroke> \write13< 41.4465 44.8195 2 0 360 arc fill stroke> \write13< 96.0397 209.182 10.00000 0 360 arc stroke> \write13< 96.0397 209.182 2 0 360 arc fill stroke> \write13< 389.974 69.2845 10.00000 0 360 arc stroke> \write13< 389.974 69.2845 2 0 360 arc fill stroke> \write13< 0. 110.000 40.0000 110.000 tlinea> \write13< 220.000 200.000 257.022 215.145 tlinea> \write13< 360.000 0. 382.938 -32.7693 tlinea> \write13< 50.0000 50.0000 10.51036 56.3693 tlinea> \write13< 100.0000 200.000 112.649 237.947 tlinea> \write13< 380.000 70.0000 416.280 53.1555 tlinea> \write13< 0. 110.000 20.0000 0 22.2490 arc stroke> \write13< 220.000 200.000 20.0000 22.2490 -55.0080 arcn stroke> \write13< 360.000 0. 20.0000 -55.0080 -189.162 arcn stroke> \write13< 50.0000 50.0000 20.0000 -189.162 -288.435 arcn stroke> \write13< 100.0000 200.000 20.0000 -288.435 -384.905 arcn stroke> \write13< 380.000 70.0000 20.0000 -384.905 -523.301 arcn stroke> \write13 \write13<0 0 moveto dup dup 0 exch 2 div lineto 0 > \write13 \write13<0 0 lineto fill stroke } def > \write13 \write13<0 0 moveto 0 2 copy lineto stroke exch 2 > \write13< div exch translate 7 punta0 > \write13 \write13 \write13 \write13 \write13< 0. 110.000 220.000 200.000 freccia > \write13< 220.000 200.000 360.000 0. freccia > \write13< 360.000 0. 50.0000 50.0000 freccia > \write13< 50.0000 50.0000 100.0000 200.000 freccia > \write13< 100.0000 200.000 380.000 70.0000 freccia > \write13< 380.000 70.0000 80.0000 -20.0000 freccia > \write13< grestore> \figfin \def\Sb{} \def\ov{\overline} \def\rot{\hbox{\rm rot}\,} \def\y{\underline y} \def\ga{\underline\gamma} \def\om{\underline\omega} \def\os{\mathop{\omega}\limits_{\sim}} \font\sixrm=cmr6 \font\eightrm=cmr8 %%%%%%%%%%%%% \def\us{\mathop{u}\limits_{\sim}} \def\ps{\mathop{\partial}\limits_{\sim}} \def\u{\underline u} \def\pa{\underline\partial} \def\x{\underline x} \def\A{\underline A} \def\und{\underline} \def\va{\varepsilon} \def\r{\underline r} \font\ftitle=cmbx10 scaled\magstep1 \font\ftitleit=cmti10 scaled\magstep1 \def\d{\hbox{\rm d}\,} \def\k{\underline k} \def\np{\par\noindent} \font\eightrm=cmr8 \def\i{\accent18\char16} \let\0=\noindent \let\V=\und \def\*{\vskip0.3cm} \def\equ{} %%%%%%% \centerline{RIGOROUS THEORY OF THE BOLTZMANN EQUATION} \centerline{IN THE LORENTZ GAS\footnote*{\eightrm The first version appeared as a preprint: Nota Interna n. 358, Istituto di Fisica, Universit\`a di Roma, 10 feb 1972. The reprint has been deposited in the archive {\it mp\_arc@math.utexas.edu}, \# 93--???.}} \bigskip \centerline{{\bf Giovanni Gallavotti}\footnote{**}{\eightrm Permanent address: Dipartimento di Fisica, Universit\`a di Roma.}} \medskip \centerline{\it Istituto di Fisica, Universit\`a di Roma} \centerline{\it C.N.R., Gruppo Nazionale Analisi Funzionale} \vskip1.cm {\it Abstract: The Boltzmann limit conjecture of Grad is discussed in general and proved for the Lorentz gas case (where the Boltzmann equation is linear). This is a reprint of an unpublished preprint of 1972, with one footnote added, one postscript (to quote the Lanford theorem), and improved with language editing. I reprint it in this form to make it accessible, as it has been quoted by other authors in later papers. The original preprint was commissioned for a book that eventually was not published.} \vskip1.cm {\bf 1. --- Introduction} \numsec=1\numfor=1 \vskip0.5cm The Boltzmann equation is an approximation to the ``true'' evolution equation: this is due to the fact that in its derivation the following assumptions are made [1,2]: \item{1)} only binary collisions are considered \item{2)} ``Molecular chaos'' is assumed at all times: i.e. the high order correlation functions can be expressed in terms of the one--particle distribution as: $$f(\r_1\und v_1,\,\r_2\und v_2,\dots,\r_n\und v_n\ ;\ t)= \prod ^n_{i=1}f(\r_1\und v _1\ ;\ t)\Eq(1.1)$$ \item{3)} in the computation of the collision term one disregards the fact that the molecules have a non vanishing dimension. Having realized that the Boltzmann equation is only an approximation it becomes of interest to investigate if there are, at least, limiting cases in which it holds rigorously. If $n$ denotes the particle density and $a$ the radius of interaction (i.e. a parameter proportional to the interaction range or to the square root of the scattering cross--section) then a critical examination of assumptions 1), 2), 3) suggest that the Boltzmann equation should hold rigorously in the limiting case $n\to\infty$, $a\to 0$ in such a way that [3]: $$\eqalignno{ \hbox{i)}\quad &na^2\not= 0&\eq(1.2)\cr \hbox{ii)}\quad &na^3\to 0&\eq(1.3)\cr \hbox{iii)}\quad&\hbox{``Molecular chaos'' \equ(1.1) is assumed at $t=0$}&\eq(1.4)\cr}$$ In fact \equ(1.2) says that the mean free path (m.f.p.) is finite (i.e. there are collisions); on the other hand \equ(1.3) insures that, in the molecular scale, the gas is infinitely dilute (i.e. no particle can be found in a region of dimension $a$; hence the probability that a given particle collides with any other particle fixed vanishes {\it a priori} (although the particle will certainly suffer collisions because of \equ(1.2)). The last facts prevent multiple collisions and the building up of correlations capable of destroying the molecularly chaotic character of the initial state. To discuss rigorous results we need, however, a more precise statement of the conjecture that in the above limiting case the Boltzmann equation is rigorously true. This is done in the following lines. Consider a gas of particles described, at $t=0$, by a molecularly chaotic state (i.e. by a state such that the $n$--point correlation function factorizes as in \equ(1.1)) with a one particle distribution: $$f(\r,\und v\ ;\ 0)={1\over a^2}f_0(\r,\und v)\Eq(1.5)$$ where $f_0(\r,\und v)$ is a given ($a$--independent) function. Suppose that the gas of particles just introduced evolves through the action of a pair potential $\varphi_a(\r)$ having the form: % $$\varphi_a(\r)=\varphi\left({\vert\r\vert\over a}\right)\Eq(1.6)$$ where $\varphi$ is a short range force (without hard core to avoid inessential complications in the notations and definitions) so that the differential scattering cross section in the solid angle $\Omega$ is of the form $$\sigma_a(\Omega)=a^2\sigma(\Omega)\Eq(1.7)$$ Let $f(\r_1,\und v_1,\r_2,\und v_2,\dots,\r_n,\und v_n\ ;\ t)$ be the $m$--particle correlation function describing the state into which the initial state evolves in time $t$. This function is, of course, no longer ``chaotic'' (i.e. of the form \equ(1.1)). The ``Boltzmann limit conjecture'' (BLC) can be now formulated as: \* \0{\bf Conjecture BLC\/: \it\0For all fixed $t>0$ and under ``mild assumption'' on $f_0(\r,\und v)$, the following limit exists: % $$\tilde f(\r_1\und v_1,\r_2\und v_2,\dots,\r_m\und v_m\ ;\ t)=\lim_{a\to 0} a^{2m}f(\r_1\und v_1,\dots, \r_m\und v_m\ ;\ t)\Eq(1.8)$$ % and is chaotic: % $$\tilde f(\r_1\und v_1,\r_2\und v_2,\dots,\r_n\und v_n\ ;\ t)=\prod_{i=1}^n f(\r_i\und v_i\ ;\ t)\Eq(1.9)$$ % and, further, $f(\r,\und v\ ;\ t)$ verifies the Boltzmann equation: % $$\eqalignno{ &{\partial\tilde f\over\partial r}(\r,\und v,t)+\und v\cdot{\partial\tilde f\over \partial r}(\r,\und v;t)=\int\d \und v_1\int\d\Omega\cdot\cr &\quad\cdot\vert\und v-\und v_1\vert\sigma(\Omega)(\tilde f(\r,\und v';t)\tilde f(\r_1,\und v'_1;t)-\tilde f(\r,\und v,t)\tilde f(\r,\und v_1,t))&\eq(1.10)\cr}$$ % with initial condition $$\tilde f(\r,\und v,0)=f_0(\r,\und v)\Eq(1.11)$$ % where $\und v',\und v'_1$ are functions of $\und v$, $\und v_1$, $\Omega$ in such a way to conserve kinetic energy, linear momentum and to have a relative direction parallel to $\Omega$.} \* We refrain to state some examples of ``mild assumptions'' on $f_0(\r,\und v)$ since they should become clear in the course of a hypothetic proof of BLC. We observe that, for the time being, it seems quite difficult to attack the problem of costructing a rigorous proof of the BLC. This is mainly due to the fact that the existence and stability theorems necessary for a proper mathematical definition of time evolutions of large systems are still far from being proved. We stress that this is not a ``technical point'' but reflects our lack of understanding of some basic physical properties of the time evolution of large assemblies of particles (for an example of such problems see [5]; for an example of their applications see [6] and [7]). In this paper we investigate the BLC in the case of simple models introduced by Lorentz [8] and used, for instance, to study the diffusion between gases of very different molecular weight [9] or, in its quantum version, to study the properties of a degenerate gas [10]. The Lorentz models are described in the next section and are such that the mathematical problems concerning existence and stability of the solutions of the microscopic equations of motion are very easily dealt with. This mathematical simplicity, reflected also in the fact that the Boltzmann equation turns out to be linear, will enable us to push to the end the proof of the BLC at least in some cases. \penalty-1000 \vskip1.cm {\bf 2. --- The Lorentz models} \penalty10000 \vskip0.5cm \numsec=2\numfor=1 In the models there are two types of particles: the $W$--particles (wind--particles) and the $T$--particles (tree--particles). The $W$--particles move through the space interacting only with the $T$--particles which, however, are supposed to be infinitely heavy compared to the $W$--particle and are supposed at rest and randomly distributed in space. Each model is completely described by the $W-T$ interaction and by the $T$--particle distribution. >From now on we shall focus our interest to the case in which the $T$--particles are distributed as the space distribution of a perfect gas (Poisson distribution) with density $n$. We shall also assume that the $T$--particles are, with respect to the $W$--particles, hard spheres of radius $a$, reflecting the $W$--particles on their surface. The assumed tree distribution is such that the probability for finding inside a given region $\Lambda$, with volume $V(\Lambda)$ exactly $N$ tree particles, and for finding them in the infinitesimal cubes $d\und c_1,\dots,d\und c_N$ around $\und c_1,\dots,\und c_N$, is: % $$f_\Lambda(\und c_1,\dots,\und c_N)d\und c_1,\dots,d\und c_N=e^{-nV(\Lambda)} {n^N\over N!}d\und c_1,\dots,d\und c_N\Eq(2.1)$$ Note that, since the $T$--particles are hard spheres only with respect to the $W$--particles but not with respect to the each other, there are configurations $\und c_1,\dots,\und c_N$ of trees in which the hard spheres overlap, (for some comments on this point see \S6). If $x=(\und p,\und q)$ is the $W$--particles phase space coordinate ($\und p$= velocity, $\und q$= position) the symbol: % $$S_t^{\und c_1,\dots\und c_N}x\Eq(2.2)$$ % will denote the $W$--particle $x'=(\und p',\und q')$ into which $x$ evolves in time $t$ in the presence of $N$ tree--particles located at $\und c_1,\dots,\und c_N$. The symbol $\omega(\und p)$ will denote the direction of $\und p$. The symbol $\hat x$ will denote the pair $(\omega(\und p),q)$ if $x=(\und p,\und q)$. Since the velocity $\vert\und p\vert$ is conserved it is clear that $S_t^{\und c _1,\dots,\und c_N}x$ depends only on the trees located within a distance $(\vert \und p\vert t+a)$ from $\und q$. The symbols: % $$\left(S_t^{\und c_1,\dots,\und c_N}x\right)_1\;,\; \left(S_t^{\und c_1,\dots,\und c_N}x\right)_2\;,\;\omega\left( \left(S_t^{\und c_1,\dots,\und c_N}x\right)_1\right)\Eq(2.3)$$ will, respectively, denote the velocity, position and momentum direction of (2.2). The following symbols will occur frequently: $$S_t^{\und c_1,\dots,\und c_N}\hat x=\left(\omega\left(\left( S_t^{\und c_1,\dots,\und c_N}x\right)_1\right)\;,\;\left( S_t^{\und c_1,\dots,\und c_N}x_m\right)\right)\Eq(2.4)$$ Similarly we can give a natural meaning to the evolution of $m$ $W$--particles: % $$S_t^{\und c_1,\dots,\und c_N}(x_1,\dots,x_m)=\left( S_t^{\und c_1,\dots,\und c_N}x_1,\dots, S_t^{\und c_1,\dots,\und c_N}x_m\right)\Eq(2.5)$$ % which takes into account the fact that there are no $W-W$ interactions. \vskip1.cm {\bf 3. --- The Boltzmann limit for the Lorentz gas} \numsec=3\numfor=1\vskip0.5cm It is easy to derive the Boltzmann equation for $W$--particles in the case of the above described Lorentz gas. It is not difficult to realize that the assumptions to be made in order to derive the Boltzmann equation are essentially the same as conditions 1), 2), 3), of section 1. They are: \item{i)} a $W$--particle never hits twice the same particle; \item{ii)} molecular chaos is assumed; \item{iii)} the size of the $T$--particles is negligible. Here by ``chaotic'' $W$--particle state we again mean a state such that the $W$--particle correlation functions are a product of one $W$--particle distribution which are independent on the $T$--particle distribution: more precisely a chaotic state is such that the probability distribution for finding a certain configuration $C$ of $T$--particles and a set of $W$--particles in $x_1,\dots,x_m$ has the form: $p(C)\prod ^m_{i=1}f(x_i)$, where $p(C)$ denotes the Poisson distribution \equ(2.1) and this is interpreted as $0$ if any wind particle is inside the hard cores of $C$.\footnote{${}^*$} {\nota More explicitly this means the following. Let $p$ be the probability of finding the W particles in a infinitesimal cube $dx_1\ldots dx_M$ around the configuration $X=(x_1,\ldots,x_M)$ in the box $\L_0$, and a tree configuration in the infinitesimal cube $d\V c_1\ldots d\V c_M$ around $C=(\V c_1,\ldots,\V c_N)$ in the box $\L$, assuming it $a$ wider than $\L_0$, at least. Here $x_i=(\V p_i,\V q_i)$. Then $p$ is the product of \equ(2.1) times $M!^{-1}(\prod_{i=1}^M f(x_i)\,dx_i)\, e^{-\ig_C f(\x) d\x}$, where $\x=(\V p,\V q)$ and $\ig_C d\x$ means integration over $\V p$ and over the $\V q\in \L_0$ which are outside the hard spheres centered on $C=(\V c_1,\ldots, \V c_N )$. In other words the W particles also have a Poisson distribution, in the region outside the T particles, with a density function $f$.} Clearly assumption i), ii), iii) can be only approximately true. Let us formulate the BLC for the Lorentz gas. Assume that the initial $W$--particles state has the form: % $$f(x_1,\dots,x_m\ ;\ 0)=\int_{C\ \hbox{\sixrm comp}\,(x_1,\dots,x_m)}p(C)\prod^m_{i=1} f_0(x_i)\Eq(3.1)$$ % where $f_0(x)$ is a given function of $x$ and the ``integral'' is the ``sum'' over all the $T$--particle configurations compatible with $x_1,\dots,x_m$ (i.e. over the $C$'s such that no $W$--particle is located inside the hard core of a $T$--particle). Note that \equ(3.1) is not a product state for the $W$--particles: this difference with respect to section 1 arises because here we have hard core interactions (which, for simplicity, were not considered in section 1). Consider the state obtained by evolving the initial state \equ(3.1): % $$f(x_1,\dots,x_m\ ;\ t)=\int_{C\ \hbox{\sixrm comp }(x_1,\dots,x_m)}p(C)\prod^m_{i=0} f_0(S^C_{-t}x_j)\Eq(3.2)$$ and then let the $T$--particle density $n$ tend to infinity and the hard core $W-T$ radius tend to zero in such a way that $na^3\to 0$ but $na^2\to l\not=0, \infty$. The BLC becomes: \* \0{\bf Conjecture BLC\/: \it\0If $t\ge 0$ and under ``mild assumptions'' on $f_0$, the following limit exists: % $$\lim_{na^3\to 0\atop na^2\to\ \hbox{\sixrm const }\not=0,\infty} f(x_1,\dots,x_m\ ;\ t)=\tilde f(x_1,\dots,x_m\ ;\ t)\Eq(3.3)$$ % and: % $$\tilde f(x_1,\dots,x_m\ ;\ t)=\prod^m_{i=1}\tilde f(x_i\ ;\ t)\Eq(3.4)$$ % and $f(x;t)$ verifies the Boltzmann equation: % $${\partial\tilde f\over\partial t}(x,t)+\und p\cdot{\partial\tilde f\over\partial \und q}(x,t)=\lambda^{-1}\vert\und p\vert\int(f(x',t)-f(x,t)\sigma(\Omega)\d\Omega\Eq(3.5)$$ % where $x'=(\und p\ ;\ \und q)$ and $\und p'$ is a vector with the same length as $\und p$ but forming with it an angle $\Omega$; $a^2\sigma(\Omega)={a^2\over 4\pi}$ is the scattering cross section of a hard sphere with radius $a$ and $\lambda^{-1}=\pi\ na^2$. A similar conjecture can be formulated in a two--dimensional model; here the solid angle $\Omega$ has to be replaced by the deflection angle $\psi$ and $\sigma(\Omega)$ by $\sigma(\psi)={1\over 4}\sin{\psi\over 4}$ and $\lambda^{-1} =2an$. Of course the Boltzmann limit will be, in this case, $na^2\to 0$, $2na\to\lambda^{-1}\not=0,\infty$.} \* In the next sections we construct a proof of the above conjecture in the two--dimensional case. The three--dimensional case could be treated along the same lines as it will become apparent from the proofs. \vskip1.cm \penalty-1000 {\bf 4. --- Results on the BLC} \numsec=4\numfor=1\penalty10000\vskip0.5cm Assume the spatial dimension to be two. The direction $\omega(\und p)$ will be in this case the angle $\theta$ between $\und p$ and a fixed axis. The function $f_0(x)$ will be thought as $f_0(\vert\und p\vert,\omega(\und p),\und q)$ if $x=(\V p,\V q)$ and we can write: % $$f_0(\vert\und p\vert,\omega(\und p),\V q)=\int\d\und q'\d\omega'f_0(\vert\und p \vert,\omega',\und q')\delta(\und q-\und q')\cdot\delta(\omega(\und p)-\omega')\Eq(4.1)$$ % we shall shorten $(\omega',q')$ as $\xi$, $\d\und q'\d\omega'$ as $\d\xi$, $\delta(\und q-q')\delta(\omega(\und p)-\omega')$ as $\delta(x-\xi)$. Hence, by using definition \equ(2.4), the \equ(3.2) becomes for $m=1$: % $$f(x;t)=\int\d\xi f_0(\vert\und p\vert,\xi)\int_{C\ \hbox{\sixrm comp}\,x}\delta (S^C_{-t}\hat x-\xi)p(C)\Eq(4.2)$$ It is therefore useful to consider the Green's function: % $$g(\xi;x;t)=e^{\pi na^2}\int_{C\ \hbox{\sixrm comp}\,x}p(C)\delta (S^C_{-t}\hat x-\xi)\Eq(4.3)$$ % where the factor $e^{\pi na^2}$ has been introduced for normalization purposes (note that it tends to $1$, in the Boltzmann limit). It is easily checked that: % $$g(\xi;x;0)=\delta(\hat x-\xi),\quad \int g(\xi;x;t)\d\xi\equiv1,\quad f(x,t)=\int\d\xi f_0(\vert\und p\vert,\xi)g(\xi;x;t)\Eq(4.4)$$ % we shall show that as $na^2\to 0$, $2na\to\lambda^{-1}\not=0,\infty$ the function $g(\xi;x;t)$ will tend to a limit $\tilde g(\xi;x;t)$ which verifies the two dimensional analogue of equation \equ(3.5) with initial condition $\tilde g(\xi;x;0)=\delta (\hat x-\xi)$ and $\vert\und p\vert$ fixed. The linearity of \equ(3.5), and of the third \equ(4.4), will imply, under suitable assumptions on $f_0$, that also $\tilde f(x,t)$ verifies \equ(3.5). We will not insist in discussing in which sense $g(\xi;x;t)$ converges to $\tilde g(\xi;x;t)$. It will appear from the proofs below that at least $g(\xi;x;t)$ converges to $\tilde g(\xi;x;t)$ pointwise for $t\not=0$, and in the sense of the distributions for all $t\ge 0$. However a close examination of the proof will provide evidence against any uniformity of the convergence in $t$, unless $t$ is restricted to a bounded interval (for further remarks on this point see section 6). Under the above convergence conditions the ``mild assumptions'' in BLC could, for instance, be the continuity and boundedness of $f_0$. \vskip1.cm %\ifnum\mgnf=0\vfill\eject\fi \penalty-200 {\bf 5. --- Proof} \penalty10000 \vskip0.5cm \numsec=5\numfor=1 The proof is based on several simple changes of variables in \equ(4.3). Let $x=(\und p,\und q)$ and let $R(x,t)$ be the sphere with center $\und q$ and radius $(\vert p\vert t+a)$; then $S^c_{-t}x$ depends only on the $T$--particles in $c$ contained in $R(x,t)$. Hence the integral \equ(4.3) can be explicitly written as: $$g(\xi;x;t)=e^{\pi na^2}\sum^\infty_{M=0}\int_{R(x,t)^M}e^{-nV(R(x,t))} {n^M\over M!}\delta\left(S^{\und c_1,\dots,\und c_M}_{-t}\hat x-\xi\right) \d\und c_1\dots\d\und c_M\Eq(5.1)$$ % where $V(R(x,t))$= area of $R(x,t)$ and where use has been made of the assumed Poisson distribution of the $T$--particles \equ(2.1). Note that, in general, not all the $T$--particles $\V c_1,\dots,\und c_M$ in \equ(5.1) will be hit by the trajectory $S^{\und c_1,\dots,\und c_M}_{-t}x$ $0\leq\tau\leq t$. Let $A_{x,t,N}$ denote the set of configurations $\V c_1,\dots,\und c_N$ of $N$ $T$--particles such that a $W$--particle with initial coordinate $x$ hits, in the time $t$, all the $N$ particles in $\und c_1,\dots\und c_N$ at least once. We deduce from \equ(5.1): % $$\eqalignno{&g(\xi;x;t)=e^{\pi na^2}\sum^\infty_{N=0} \int_{A_{x,t,N}}n^N{\d\und c_1,\dots, \d\und c_N\over N!}\,\chi_{\und c_1,\dots,\und c_N}(x)\cdot\delta\left(S_{-t} ^{\und c_1,\dots\und c_N}\hat x-\xi\right)\cdot\cr &\cdot\left[\sum^\infty_{M=N}\ig_{ R(x,t)^{M-N};\ \und c'_1,\dots,\und c'_{M-N}\in P(t;\und c_1,\dots,\und c_N)} {\d\und c'_1,\dots,\d\und c'_{M-N}\over (M-N)!}\,n^{M-N}e^{-nV(R(x,t))}\right] &\eq(5.2)\cr}$$ % where $\chi_{\und c_1,\dots,\und c_N}(x)$ is one if $w$ is compatible with the hard cores of $\und c_1,\dots,\und c_N$ and zero otherwise: the region $P(t;\und c_1,\dots,\und c_N)$ is the tube like region (see fig. \equ(5.3)) swept by an ideal $T$--particle when its center is moved along the path $S^{\und c_1,\dots,\und c_N}_{-\tau}x$, $0\leq\tau\leq t$. \insertplot{380pt}{220pt}{}{blz1}{\eq(5.3)} \vskip1.cm \noindent{\nota Fig. \equ(5.3): The set $P(t;\V c_1,\ldots,\V c_N)$ is the dashed region. The circles represent trees $\V c_1,\ldots,\V c_N$, ($N=5$) and the length of the trajectory in the dashed region is $|\V p|t$.} \* \0The sum within square brackets in \equ(5.2) can be performed (since the integrals are trivials) and yields: $$e^{-nV(P(t;\und c_1,\dots,\und c_N))}\Eq(5.4)$$ % so: % $$g(\xi;x;t)=e^{n\pi a^2}\sum^\infty_{N=0}\int_{A_{x,t,N}}n^Ne^{-nV(P(t;\und c_1, \dots,\und c_N))}{\delta\und c_1\dots\d\und c_N\over N!}\cdot\delta\left( S^{\und c_1,\dots,\und c_N}_{-t}\hat x-\xi\right)\ .\Eq(5.5)$$ The reader should remark the very simple probabilistic meaning of this equation which makes it almost self--evident [11]. The $T$--particles in $A^1_{x,t,N}$ can be hit more than once in the time $t$. Divide $A_{x,t,N}$ as $A^1_{x,t,N}\cup A'_{x,t,N}$ where $A^1_{x,t,N}$ is the set of $T$--configurations in $A_{x,t,N}$ such that all their $T$--particles are hit just once by the trajectory $S^{\und c_1,\dots,\und c_N}_{-\tau}x$ $0\leq\tau\leq t$. To this decomposition of $A^1_{x,t,N}$ corresponds a decomposition $g(\xi;x;t)= g_1(\xi;x;t)+g'(\xi;x;t)$ with: % $$g_1(\xi;x;t)=e^{\pi na^2}\sum^\infty_{N=0} \int_{A^1_{x,t,N}}n^N{\d\und c_1,\dots,\d\und c_N\over N!} \chi_{\und c_1,\dots,\und c_N}(x)\cdot \delta\left(S^{\und c_1,\dots,\und c_N}_{-t}\hat x-\xi\right)e^{-nV(P(t;\und c_1,\dots,\und c_N))}\Eq(5.6)$$ We now perform the change of variables, illustrated in fig. \equ(5.8), from the $2N$ variables $\und c_1,\dots,\und c_N$ to the new $2N+1$ variables $l_1,\dots,l_{N+1}$, $\beta_1,\dots,\beta_N$; we get: % $${\d\und c_1,\dots,\d\und c_N\over N!}=a^N\delta\left(\sum^{N+1}_{i=1}l_i- \vert\und p\vert t\right)\left(\prod^{N+1}_{i=1}\d l_1\right)\prod^N_{j=1} \left({\d\beta_j\over 2}\,\sin{\beta_j\over 2}\right)\Eq(5.7)$$ \vskip1.cm \insertplot{380pt}{220pt}{ \ins{30.pt}{120.000pt}{$\th$} \ins{-10.pt}{90.000pt}{$O$} \ins{110.pt}{150.000pt}{$l_1$} % 0. 110.000 %1 \ins{250.000 pt}{200.000pt}{$\b_1$} \ins{220.000 pt}{230.000pt}{$\V c_1$} %\ins{220.000 pt}{200.000pt} %2 \ins{290.00pt}{80.pt}{$l_2$} \ins{385.00pt}{5.pt}{$\V c_2$} \ins{340.00pt}{-20.pt}{$\b_2$} %\ins{360.00pt}{0.pt} %3 % \ins{ 50.0000 pt}{ 50.0000pt} %4 % \ins{ 100.0000pt}{ 200.000pt} %5 \ins{ 230.000 pt}{ 130.0000pt}{$l_{N}$} \ins{ 380.000 pt}{ 90.0000pt}{$\V c_N$} \ins{ 380.000 pt}{ 40.0000pt}{$\b_N$} %\ins{ 380.000 pt}{ 70.0000pt} %6 \ins{ 200.0000 pt}{ 0.0000pt}{$l_{N+1}$} % \ins{ 80.0000 pt}{ -20.0000pt} %7 }{blz2}{\eq(5.8)} \vskip1.cm Hence the $N^{\hbox{th}}$ order contribution to \equ(5.6) is given by (if $x=(p,q)=(\vert p\vert,\omega(\und p),\und q)$, $\xi=(\theta',q')$: $$\eqalignno{ &*\ e^{\pi na^2}(2na)^N\int^\infty_0\prod^{N+1}_{i=1}\d l_1\int^{2\pi}_0 \prod^N_{i=1}\left(\sin{\beta_i\over 2}\, {\d\beta_i\over 4}\right) \delta\left(\sum^{N+1}_{i=1}l_i-\vert\und p\vert t\right)\cdot\cr &\cdot\delta\left(\sum^{N+1}_{i=1}\und l_i-(\und q'-\und q)\right)\delta\left(\sum^N_ {i=1}\beta_i-(\theta'-\omega(\und p))\right)e^{-nV(P(t;\und c_1,\dots,\und c_N))} &\eq(5.9)\cr}$$ % where $\und l_i$ are the vectors represented by arrows in fig. \equ(5.3) $(\vert\und l_1 \vert=l_i)$; the * in \equ(5.9) means that there is an extra condition on the integration region. It is the condition that none of the spheres of radius $a$ around $\und c_1, \dots,\und c_N$ has intersection with the straight segments of the broken line representing the trajectory in fig. \equ(5.8) (i.e. this is the condition that $\und c_1,\dots, \und c_N$ really belongs to $A^1_{x,t,N}$). Of course in \equ(5.9), $\delta\left( \sum^N_{i\equiv1}\beta_i-(\theta-\omega(\und p))\right)$ means $\sum^{+\infty}_ {h=-\infty}\delta(\sum_i\beta_i-(\theta-\omega(\und p))-2\p h)$. In the limit $na^2\to 0$, $2na\to\lambda^{-1}\not=0,\infty$ the restrictions indicated by the * in \equ(5.9) become unimportant and $nV(P(t;\und c_1,\dots,\und c_N))$ simplifies enormously: % $$nV(P(t;\und c_1,\dots,\und c_N))\to 2na\sum^{N+1}_{j=1}l_j=\lambda^{-1}\vert p\vert t\ .\Eq(5.10)$$ Hence: % $$\eqalignno{ &\tilde g(\xi;x;t)=\lim_{ na^2\to 0\atop 2na\to\lambda^{-1}}g_1(\xi;x;t)=\sum^\infty_{N=0}\lambda^{-N}\int^\io_0 \left(\prod^N_{i=1}\d l_i\right)\int^{2\pi}_0\prod^N_{j=1}\left(\sin{\beta_j\over 2}{\d\beta_j\over 4}\right)\cdot\cr &\quad\cdot\delta\left(\sum_i\und l_i-(\und q'-\und q)\right)\cdot\delta\left(\sum_i\beta_i- (\theta'-\omega(\und p))\right)\cdot e^{-\lambda^{-1}\vert p\vert t}\ &\eq(5.11)\cr}$$ In the derivation of equation \equ(5.11) we have systematically disregarded convergence problems connected with the summation over $N$, $M$ etc. since they are trivial as a consequence of the presence of the factorials and of the boundedness of the integration regions. The limit \equ(5.11) is pointwise for $t\not=0$ and it could be checked that it holds also in the sense of the distributions for $t\ge 0$. Furthermore it could be checked that for $t>0$ the function $g(\xi,x,t)\ge g_1 (\xi,x,t)$ is bounded above by a $L_1(d\xi)$ function; hence the limit \equ(5.11) holds also in the $L_1(d\xi)$ sense. Finally, by direct computation, it follows from \equ(5.11) that: % $$\int \tilde g(\xi;x;t)\d\xi\equiv 1\Eq(5.12)$$ % and this fact, together with the above convergence properties and \equ(4.4), implies the validity of the limit relation: $\lim_{ na^2\to 0\atop 2na\to\lambda^{-1}} g(\xi;x;t)=\tilde g(\xi;x;t)$ in $L_1(\d\xi)$ for $t>0$; furthermore it could be proved that this limit holds, for $t\ge 0$, in the sense of the distributions. It is known [12] that \equ(5.11) is a solution of the Boltzmann equation (and this can be checked directly by substituting $\tilde g$ into \equ(3.5)), with initial condition $\tilde g(\xi;x;0)=\delta(\hat x-\xi)$ and $\vert p\vert$ fixed. To complete the proof of the BLC it remains to deal with the $m$--particle distributions. However we skip this point since it involves straightforward calculations based on changes of variable of the type illustrated in fig. \equ(5.8). \vskip1.cm {\bf 6. --- Concluding remarks} \numsec=6\numfor=1\vskip0.5cm In the preceding sections we have described a proof of the Boltzmann limit conjecture in the case of a two--dimensional Lorentz gas with hard core $W-T$ interactions and free gas distribution of the $T$--particles. The generalization to three dimensions would be trivial. A less trivial generalization would be obtained by keeping the hard core $W-T$ interaction but assuming that the $T$--particles are spatially distributed as if they were a gas of hard spheres with hard core size being proportional to the the $W-T$ radius. Other generalizations are conceivable in the direction of allowing soft $W-T$ particle interactions and more general $T$--particle distributions. So far none of these generalizations have been attempted. Much more difficult and interesting would be the treatment of the {\it Knudsen model}, in which the $T$--particles are allowed to move without suffering changes in their momentum in the collisions in the $W$--particles. Another remark is that had we done the calculations associated with the proof of equation \equ(3.4); we would have also found evidence of a lack of uniformity of the Boltzmann limit in the number $m$ of $W$--particles even at fixed $t$: the larger $m$ is the closer one has to get near the Boltzmann limit in order to see factorization of the $W$--particle correlations. We also wish to remark that even when the Boltzmann limit conjecture is true, one cannot expect that the function $\tilde f(\r,\und v,t)/ a^2$ (see section 1) is a good approximation to $f(\r,\und v,t)$ for large $t$: in fact one intuitively expects that for times of the order of $tm\r p/na^3$ some non trivial correlations will start building up thus destroying the molecular chaos and spoiling the validity of the Boltzmann equation. This last remark is quite deceiving since it tells us that we cannot use, {\it without further assumptions}, the Boltzmann equation to investigate the long time behaviour and, in particular, to compute the transport coefficients. From a rigorous point of view we cannot even be sure that the lowest order in $na$ of the transport coefficients is correctly given by the value obtained in the Boltzmann limit. However it seems reasonable that this is, indeed, the case at least if the dimension of the space is larger than two (in one--dimension a simple counter example can be found by using soluble models [4]; in this case, however, the Boltzmann equation is a priori expected not to be a good approximation). For further readings on the Lorentz gas see ref. [13]. The idea of the Boltzmann limit is clearly stated in [3]; the present proof in the case of the Lorentz gas is done in ref. [12] (for the case of $\tilde g(\x; x;t)$ only) and was inspired by discussions and suggestions from J.L. Lebowitz. \vskip1.cm {\bf Postscript: \* \it\0The Boltzmann limit conjecture has been proved a few years after the present paper was written. It is due to O. Lanford, [14] (1974), and it holds under some very reasonable restrictions on the initial data and for a hard spheres system, but with a still standing limitation on the time interval of validity. The time interval is strictly positive, but it is a small fraction $\e$ of the mean free flight time $t_0$ (which is defined as the ratio of the mean free path over a mean velocity computed in the initial state): $\e\sim\fra15$. This clearly did put an end (or at least it should have) to the diatribes on whether the irreversibility can or cannot be deduced from a microscopically reversible mechanical model ({\rm Boltzmann vindicatus est}); but it left open the question of the mathematical justification of the validity of the Boltzmann equation over the time scales on which it is usually used, going beyond the mean free time by several orders of magnitude. There has been, since, one case in which the proof of the BLC has been pushed to infinite time, [15].} \vskip2.cm {\bf References} \vskip0.5cm \item{[1]} S. Chapman, T. Cowling, {\it The Mathematical Theory of Non--Uniform Gases}, Cambridge University Press, 1953, p. 46. \item{[2]} Cohen, E.G.D.: {\it The kinetic theory of dilute gases}, in "Transport Phenomena in Fluids", H. Hanley ed., Ch. VI, 119--156; Dekker, New York, 1969. \item{[3]} H. Grad, {\it Principles of the kinetic theory of gases}, in {\it Handbuch der Physik}, vol. XII, p. 205--294, see p. 214, ed. S. Fl\"ugge, Springer--Verlag, 1958. \item{[4]} J. Lebowitz, J. Percus, {\it Kinetic equations and exactly solvable one dimensional systems}, Physical Review, {\bf 155}, 122--138, 1966; see also J. Lebowitz, J. Percus, J. Sykes, {\it Time evolution of the total distribution function of a one dimensional systems of hard rods}, Physical Review, {\bf 171}, 224--235, 1968. \item{[5]} O. Lanford, {\it The classical mechanics of one dimensional systems of infinitely many particles.I. An existence theorem}, Communications in Mathematical Physics, {\bf 9}, 176--191, 1968. \item{[6]} O. Lanford, {\it The classical mechanics of one dimensional systems of infinitely many particles.II. Kinetic theory}, Communications in Mathematical Physics, {\bf 11}, 257--292, 1969. \item{[7]} G. Gallavotti, O. Lanford, J. Lebowitz: {\it Thermodynamic limit of time--dependent correlation functions for one--dimensional systems}: Journal of Mathematical Physics: {\bf 11}, 2898--2905, 1970 \item{[8]} S. Chapman, T. Cowling, {\it loc. cit.}, p. 187. \item{[9]} S. Chapman, T. Cowling, {\it loc. cit.}, p. 256. \item{[10]} S. Chapman, T. Cowling, {\it loc. cit.}, p. 309. \item{[11]} G. Gallavotti, {\it Time evolution problems in Classical Statistical Mechanics and the Wind--tree--model}: in "Cargese Lectures in Physics", vol. IV, ed. D. Kastler, Gordon Breach, Paris, 1970, pp.257--275; see p. 271--272, formula (6.2). \item{[12]} G. Gallavotti, {\it Divergences and approach to equilibrium in the Lorentz and the Wind--tree models}, Physical Review, {\bf 185}, 308--322, 1969. \item{[13]} A. Weijland, J. Van Leeuwen, {\it Non analytic behaviour of the diffusion coefficient of a Lorentz gas}, Physica, {\bf 36}, 457--490, 1967; and {\bf 38}, 35--47, 1968; E. Hauge, E. Cohen, {\it Normal and abnormal diffusion in Ehrenfests's wind tree model}, Journal of Mathematical Physics, {\bf 8}, 397--414, 1969; W., Hogey, {\it Convergent generalizations of the Boltzmann equation for a hard sphere gas}, Physical Review, {\bf 185}, 210--218, 1969. \item{[14]} O. Lanford, {\it Time evolution of large classical systems}, in ``Dynamical systems, theory and applications'', p. 1--111, ed. J. Moser, Lecture Notes in Physics, vol. 38, Springer Verlag, 1974. \item{[15]} M. Pulvirenti, {\it Global validity of the Boltzmann equation for two and three dimensional rare gas in vacuum}, Communications in Mathematical Physics, {\bf 113}, 79--85, 1987. \ciao