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Planck's law, relaxation times, glasses
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Content-Type: application/x-tex; name="elsart.cls"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="elsart.cls"

%%
%% This is file `elsart.cls',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% esl.dtx  (with options: `package,elsart,ONECOL,DEEPLIST')
%% 
%% IMPORTANT NOTICE:
%% 
%% For the copyright see the source file.
%% 
%% You are *not* allowed to modify this file.
%% 
%% You are *not* allowed to distribute this file.
%% For distribution of the original source see the terms
%% for copying and modification in the file esl.dtx.
%% 
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%% esl.dtx Copyright (C) 1994-1997 Elsevier Science
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 \def\@journal{Elsevier Preprint} \def\@company{Elsevier Science}
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\newtheorem{case}{Case}
\def\@pnumwidth{2.55em}
\def\@tocrmarg{2.55em \@plus 5em}
\def\@dotsep{-2.5}
\setcounter{tocdepth}{2}
\newcommand\listoffigures{%
    \section*{\listfigurename
      \@mkboth{\MakeUppercase\listfigurename}%
              {\MakeUppercase\listfigurename}}%
    \@starttoc{lof}%
    }
\newcommand*\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand\listoftables{%
    \section*{\listtablename
      \@mkboth{%
          \MakeUppercase\listtablename}{\MakeUppercase\listtablename}}%
    \@starttoc{lot}%
    }
\let\l@table\l@figure
\def\tableofcontents{%
  \begin{small}
  \leftline {{\bfseries \contentsname\/}}
  \setcounter{secnumdepth}{4}%
  \setcounter{tocdepth}{2}%
  {\@starttoc{toc}}%
\end{small}
}
\newcommand*\l@section{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand*\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\newcommand*\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\newcommand*\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\newcommand*\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\def\@dotsep{2000}
\def\thebibliography{%
  \@startsection{section}{1}{\z@}{20\p@ \@plus 8\p@ \@minus 4pt}
  {\@bls}{\normalsize\bfseries}*{\refname}%
\addcontentsline{toc}{section}{\refname}%
  \@thebibliography}
\let\endthebibliography=\endlist
\def\@thebibliography#1{\@bibliosize
  \list{\@biblabel{\arabic{enumiv}}}{\settowidth\labelwidth{\@biblabel{#1}}
  \if@nameyear
    \labelwidth\z@ \labelsep\z@ \leftmargin\parindent
    \itemindent-\parindent
  \else
    \labelsep 3\p@ \itemindent\z@
    \leftmargin\labelwidth \advance\leftmargin\labelsep
\fi
     \itemsep 0.3\@bls \@plus 0.1\@bls \@minus 0.1\@bls
    \usecounter{enumiv}\let\p@enumiv\@empty
    \def\theenumiv{\arabic{enumiv}}}%
    \tolerance\@M
   \hyphenpenalty\@M
     \hbadness5000 \sfcode`\.=1000\relax}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
\if@nameyear
  \def\@biblabel#1{}
\else
 \def\@biblabel#1{[#1]\hskip \z@ \@plus 1filll}
\fi
\newcount\@tempcntc
\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
 \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
  {\@ifundefined
   {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries ?}\@warning
   {Citation `\@citeb' on page \thepage \space undefined}}%
  {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
   \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
    \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
   \else
    \advance\@tempcntb\@ne
    \ifnum\@tempcntb=\@tempcntc
    \else\advance\@tempcntb\m@ne\@citeo
    \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else\@citea\def\@citea{,}%
 \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
  {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else \def\@citea{--}\fi
   \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
\@namedef{cv*}{\section*{Curriculum Vitae}\cv}
 \def\cv{\hangindent=7pc \hangafter=-12 \parskip\bigskipamount \small}
\def\footnote{\@ifnextchar[{\@xfootnote}{\refstepcounter
   {\@mpfn}\xdef\@thefnmark{\thempfn}\@footnotemark\@footnotetext}}
\def\footnotemark{\@ifnextchar[{\@xfootnotemark
    }{\refstepcounter{footnote}\xdef\@thefnmark{\thefootnote}\@footnotemark}}
\def\footnoterule{\kern-3\p@
  \hrule \@width 3pc               % The \hrule has default \@height of 0.4pt.
  \kern 2.6\p@}
\def\thempfootnote{\alph{mpfootnote}}
\def\mpfootnotemark{%
  \@ifnextchar[{\@xmpfootnotemark}{\stepcounter{mpfootnote}%
  \begingroup
    \let\protect\noexpand
    \xdef\@thefnmark{\thempfootnote}%
  \endgroup
  \@footnotemark}}
\def\@xmpfootnotemark[#1]{%
  \begingroup
    \c@mpfootnote #1\relax
    \let\protect\noexpand
    \xdef\@thefnmark{\thempfootnote}%
  \endgroup
  \@footnotemark}
\def\@mpmakefnmark{\,\hbox{$^{\mathrm{\@thefnmark}}$}}
\long\def\@mpmakefntext#1{\noindent
                     \hbox{$^{\mathrm{\@thefnmark}}$} #1}
\def\@iiiminipage#1#2[#3]#4{%
  \leavevmode
  \@pboxswfalse
  \setlength\@tempdima{#4}%
  \def\@mpargs{{#1}{#2}[#3]{#4}}%
  \setbox\@tempboxa\vbox\bgroup
    \color@begingroup
      \hsize\@tempdima
      \textwidth\hsize \columnwidth\hsize
      \@parboxrestore
      \def\@mpfn{mpfootnote}\def\thempfn{\thempfootnote}\c@mpfootnote\z@
      \let\@footnotetext\@mpfootnotetext
      \let\@makefntext\@mpmakefntext
      \let\@makefnmark\@mpmakefnmark
      \let\@listdepth\@mplistdepth \@mplistdepth\z@
      \@minipagerestore\global\@minipagetrue %% \global added 24 May 89
      \everypar{\global\@minipagefalse\everypar{}}}
\def\fn@presym{}
\long\def\@makefntext#1{\noindent\hbox to 1em
  {$^{\fn@presym\mathrm{\@thefnmark}}$\hss}#1}
\def\@makefnmark{\,\hbox{$^{\fn@presym\mathrm{\@thefnmark}}$}\,}
\def\patched@end@dblfloat{%
  \if@twocolumn
      \par\vskip\z@skip      %% \par\vskip\z@ added 15 Dec 87
      \global\@minipagefalse
      \outer@nobreak
    \egroup                  %% end of vbox
    \color@endbox
    \ifnum\@floatpenalty <\z@
      \@largefloatcheck
      \@cons\@dbldeferlist\@currbox
    \fi
    \ifnum \@floatpenalty =-\@Mii \@Esphack\fi
  \else
    \end@float
  \fi
}
\setcounter{topnumber}{5}
\def\topfraction{0.99}
\def\textfraction{0.05}
\def\floatpagefraction{0.9}
\setcounter{bottomnumber}{5}
\def\bottomfraction{0.99}
\setcounter{totalnumber}{10}
\def\dbltopfraction{0.99}
\def\dblfloatpagefraction{0.8}
\setcounter{dbltopnumber}{5}
\long\def\@maketablecaption#1#2{\@tablecaptionsize
    \global \@minipagefalse
    \hbox to \hsize{\parbox[t]{\hsize}{#1 \\ #2}}}
\long\def\@makefigurecaption#1#2{\@figurecaptionsize
  \vskip \@overcaptionskip
  \setbox\@tempboxa\hbox{#1. #2}
  \ifdim \wd\@tempboxa >\hsize              % IF longer than one line THEN
    \unhbox\@tempboxa\par                   %   set as justified paragraph
  \else                                     % ELSE
    \global \@minipagefalse
    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%   center single line.
  \fi}
\def\@makecaption{\@makefigurecaption}
\def\conttablecaption{\par \begingroup \@parboxrestore \normalsize
  \@makecaption{\fnum@table\,---\,continued}{}\par
  \vskip-1pc \endgroup}
\def\contfigurecaption{\vskip-1pc \par \begingroup \@parboxrestore
  \@captionsize
  \@makecaption{\fnum@figure\,---\,continued}{}\par
  \endgroup}
\newcounter{figure}
\def\thefigure{\@arabic\c@figure}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{\figurename~\thefigure}
\def\figure{%
 \let\@makecaption\@makefigurecaption
  \let\contcaption\contfigurecaption \@float{figure}}
\let\endfigure\end@float
\@namedef{figure*}{%
 \let\@makecaption\@makefigurecaption
  \let\contcaption\contfigurecaption \@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}
\newcounter{table}
\def\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{\tablename~\thetable}
\let\old@floatboxreset\@floatboxreset
\def\table{%
\let\@makecaption\@maketablecaption
\def\@floatboxreset{%
 \old@floatboxreset
 \@tablesize
}%
  \let\footnoterule\relax
  \let\contcaption\conttablecaption \@float{table}}
\let\endtable\end@float
\@namedef{table*}{%
\let\@makecaption\@maketablecaption
\def\@floatboxreset{%
 \old@floatboxreset
 \@tablesize
}%
  \let\footnoterule\relax
  \let\contcaption\conttablecaption \@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}
\newtoks\t@glob@notes             % List of all notes
\newtoks\t@loc@notes              % List of notes for one element
\newcount\note@cnt                % Number of notes per element
\newcounter{author}               % Author counter
\newcount\n@author                % Total number of authors
\def\n@author@{1}                  % idem, read from .aux file
\newcounter{collab}               % Collaboration counter
\newcount\n@collab                % Total number of collaborations
\def\n@collab@{}                  % idem, read from .aux file
\newcounter{address}              % Address counter
\def\theHaddress{\arabic{address}}% for hyperref
\newdimen\sv@mathsurround         % Dimen register to save \mathsurround
\newcount\sv@hyphenpenalty        % Count register to save \hyphenpenalty
\newcount\prev@elem \prev@elem=0  % Variables to keep track of
\newcount\cur@elem  \cur@elem=0   % types of elements that are processed
\chardef\e@title=1
\chardef\e@subtitle=1
\chardef\e@author=2
\chardef\e@collab=3
\chardef\e@address=4
\newif\if@newelem                 % Switch to new type of element?
\newif\if@firstauthor             % First author or collaboration?
\newif\if@preface                 % If preface: omit history and abstract
\newif\if@hasabstract             % If abstract / keywords: do not omit rules
\newif\if@haskeywords             % If abstract / keywords: do not omit rules
\newbox\fm@box                    % Box for collected front matter
\newdimen\fm@size                 % Total height of \fm@box
\newbox\t@abstract                % Box for abstract
\newbox\t@keyword                 % Box for keyword abstract
 \let\report@elt\@gobble
\def\add@tok#1#2{\global#1\expandafter{\the#1#2}}
\def\add@xtok#1#2{\begingroup
  \no@harm
  \xdef\@act{\global\noexpand#1{\the#1#2}}\@act
\endgroup}
\def\beg@elem{\global\t@loc@notes={}\global\note@cnt\z@}
\def\@xnamedef#1{\expandafter\xdef\csname #1\endcsname}
\def\no@harm{%
  \let\\=\relax  \let\rm\relax
  \let\ss=\relax \let\ae=\relax \let\oe=\relax
  \let\AE=\relax \let\OE=\relax
  \let\o=\relax  \let\O=\relax
  \let\i=\relax  \let\j=\relax
  \let\aa=\relax \let\AA=\relax
  \let\l=\relax  \let\L=\relax
  \let\d=\relax  \let\b=\relax \let\c=\relax
  \let\bar=\relax
  \def\protect{\noexpand\protect\noexpand}}
\def\proc@elem#1#2{\begingroup
    \no@harm                             % make a few instructions harmless
    \let\thanksref\@gobble               % remove \thanksref from element
    \@xnamedef{@#1}{#2}%                 % and store as \@#1
  \endgroup
  \prev@elem=\cur@elem                   % keep track of type of previous
  \cur@elem=\csname e@#1\endcsname       % and current element
  \expandafter\elem@nothanksref#2\thanksref\relax}
\def\elem@nothanksref#1\thanksref{\futurelet\@peektok\elem@thanksref}
\def\elem@thanksref{\ifx\@peektok\relax  % No more \thanksref, so now exit
  \else \expandafter\elem@morethanksref \fi}
\def\elem@morethanksref#1{\add@thanksref{#1}\elem@nothanksref}
\def\add@thanksref#1{\global\advance\note@cnt\@ne
  \ifnum\note@cnt>\@ne \add@xtok\t@loc@notes{\note@sep}\fi
  \add@tok\t@loc@notes{\ref{#1}}}
\def\note@sep{,}
\def\thanks{\@ifnextchar[{\@tempswatrue
  \thanks@optarg}{\@tempswafalse\thanks@optarg[]}}
\def\thanks@optarg[#1]#2{\refstepcounter{footnote}
  \if@tempswa\label{#1}\else\relax\fi
  \add@tok\t@glob@notes{\footnotetext}%
  \add@xtok\t@glob@notes{[\the\c@footnote]}%
  \add@tok\t@glob@notes{{#2}}}
\let\real@refstepcounter\refstepcounter
\def\footnote{\@ifnextchar[{\@xfootnote}{\real@refstepcounter
   {\@mpfn}\xdef\@thefnmark{\thempfn}\@footnotemark\@footnotetext}}
\def\footnotemark{\@ifnextchar[{\@xfootnotemark
    }{\real@refstepcounter{footnote}\xdef\@thefnmark{\thefootnote}\@footnotemark}}
\def\footnoterule{\kern-3\p@
  \hrule \@width 3pc               % The \hrule has default \@height of 0.4pt.
  \kern 2.6\p@}
 \let\report@elt\@gobble
\newenvironment{NoHyper}{}{}
\def\frontmatter{%
  \NoHyper
  \let\@corresp@note\relax
  \global\t@glob@notes={}\global\c@author\z@
  \global\c@collab\z@ \global\c@address\z@
  \sv@mathsurround\mathsurround \m@th
  \global\n@author=0\n@author@\relax
  \global\n@collab=0\n@collab@\relax
  \global\advance\n@author\m@ne   % In comparisons later on we need
  \global\advance\n@collab\m@ne   % n@author-1 and n@collab-1
  \global\@firstauthortrue        % set to false by first \author or \collab
  \global\@hasabstractfalse       % Default:  no abstract or keywords
  \global\@haskeywordsfalse       % Default:  no abstract or keywords
  \global\@prefacefalse           %           not preface
  \ifnum\c@firstpage=\c@lastpage
    \gdef\@pagerange{\@pagenumprefix\ESpagenumber{firstpage}}
  \else
    \gdef\@pagerange{\@pagenumprefix
   \ESpagenumber{firstpage}--\@pagenumprefix\ESpagenumber{lastpage}}%
  \fi
 \parskip 4\p@
  \open@fm \ignorespaces}
\def\preface{\@prefacetrue}
\def\endfrontmatter{%
  \ifx\@runauthor\relax
   \global\let\@runauthor\@runningauthor
  \fi
  \global\n@author=\c@author
  \global\n@collab=\c@collab \@writecount
  \global\@topnum\z@
  \thispagestyle{copyright}%            % Format rest of front matter:
  \if@preface \else                     % IF not preface THEN
  \vskip \@overhistoryskip
  \history@fmt                          % print history (received, ...)
  \newcount\c@sv@footnote
  \global\c@sv@footnote=\c@footnote     % save current footnote number
  \if@hasabstract                       % IF abstract/ keywords THEN
   \vskip \@preabstractskip     % Space above rule
  \hrule height 0.4\p@                 % Rule above abstract/keywords
    \vskip 8\p@
    \unvbox\t@abstract                  % print abstract, if any
  \fi
  \if@haskeywords                       % IF keywords THEN
    \vskip \@overkeywordskip
    \unvbox\t@keyword                   % Keyword abstract, if any
  \fi                                   % FI
  \vskip 10\p@
  \hrule height 0.4\p@                 % rule below abstract/keywords
  \dedicated@fmt                        % print dedication
  \vskip \@belowfmskip                  % Vertical space below frontmatter
\fi                                   % FI
  \close@fm                             % Close front matter material.
   \output@glob@notes  % Put notes at bottom of 1st page
  \global\c@footnote=\c@sv@footnote     % restore footnote number
  \global\@prefacefalse
  \global\leftskip\z@                   % Restore the normal values of
  \global\@rightskip\z@                 % \leftskip,
  \global\rightskip\@rightskip          % \rightskip and
  \global\mathsurround\sv@mathsurround  % \mathsurround.
  \let\title\relax       \let\author\relax
  \let\collab\relax      \let\address\relax
  \let\frontmatter\relax \let\endfrontmatter\relax
  \let\@maketitle\relax  \let\@@maketitle\relax
  \normal@text
}
\let\maketitle\relax
\newdimen\t@xtheight
\t@xtheight\textheight \advance\t@xtheight-\splittopskip
\def\open@fm{\global\setbox\fm@box=\vbox\bgroup
  \hsize=\@frontmatterwidth                 % Front matter is page-wide by default
  \centering                                % and centered
  \sv@hyphenpenalty\hyphenpenalty           % (save \hyphenpenalty)
  \hyphenpenalty\@M}                        % and not hyphenated
\def\close@fm{\egroup                       % close \vbox (\fm@box)
  \fm@size=\dp\fm@box \advance\fm@size by \ht\fm@box
  \@whiledim\fm@size>\t@xtheight \do{%
    \global\setbox\@tempboxa=\vsplit\fm@box to \t@xtheight
    \unvbox\@tempboxa \newpage
    \fm@size=\dp\fm@box \advance\fm@size by \ht\fm@box}
  \if@TwoColumn
    \emergencystretch=1pc \twocolumn[\unvbox\fm@box]
  \else
    \unvbox\fm@box
  \fi}
\def\output@glob@notes{\bgroup
  \the\t@glob@notes
  \egroup}
\def\justify@off{\let\\=\@normalcr
  \leftskip\z@ \@rightskip\@flushglue \rightskip\@rightskip}
\def\justify@on{\let\\=\@normalcr
  \leftskip\z@ \@rightskip\z@ \rightskip\@rightskip}
\def\normal@text{\global\let\\=\@normalcr
  \global\leftskip\z@ \global\@rightskip\z@ \global\rightskip\@rightskip
  \global\parfillskip\@flushglue}
\def\@writecount{\write\@mainaux{\string\global
  \string\@namedef{n@author@}{\the\n@author}}%
  \write\@mainaux{\string\global\string
  \@namedef{n@collab@}{\the\n@collab}}}
\def\title#1{%
  \beg@elem
  \title@note@fmt                      % formatting instruction
  \add@tok\t@glob@notes                % for \thanks commands
    {\title@note@fmt}%
  \proc@elem{title}{#1}%
  \def\title@notes{\the\t@loc@notes}%  % store the notes of the title,
  \title@fmt{\@title}{\title@notes}%   % print the title
  \ignorespaces}
\def\subtitle#1{%
  \beg@elem
  \proc@elem{subtitle}{#1}%
  \def\title@notes{\the\t@loc@notes}%  % store the notes of the title,
  \subtitle@fmt{\@subtitle}{\title@notes}% print the title
  \ignorespaces}
\newdimen \@logoheight \@logoheight 5pc
\def\@Lhook{\vrule \@height \@logoheight \@width \z@ \vrule \@height 10\p@ \@width 0.2\p@ \vrule \@height 0.2\p@ \@width 10pt}
\def\@Rhook{\vrule \@height 0.2\p@ \@width 10\p@ \vrule \@height 10\p@ \@width 0.2\p@ \vrule \@height \@logoheight \@width \z@}
\def\title@fmt#1#2{%
\@ifundefined{@runtitle}{\global\def\@runtitle{#1}}{}%
 \vspace*{12pt}             % Vertical space above title
  {\@titlesize #1\,\hbox{$^{#2}$}\par}%
  \vskip\@undertitleskip
\vskip24\p@  % Vertical space below title
  }
\def\subtitle@fmt#1#2{%               % No vertical space above sub-title
  {\@titlesize #1\,\hbox{$^{#2}$}}\par}
\def\title@note@fmt{\def\thefootnote{\fnstar{footnote}}}
\def\author{\@ifnextchar[{\author@optarg}{\author@optarg[]}}
\def\author@optarg[#1]#2{\stepcounter{author}%
  \beg@elem
  \@for\@tempa:=#1\do{\expandafter\add@thanksref\expandafter{\@tempa}}%
  \report@elt{author}\proc@elem{author}{#2}%
  \ifnum0\n@collab@=\z@ \runningauthor@fmt \fi
  \author@fmt{\the\c@author}{\the\t@loc@notes}{\@author}%
}
\def\runningauthor@fmt{%
  \begingroup\no@harm
    \if@firstauthor
      \ifnum0\n@author@ > 2
        \global\edef\@runningauthor{\@author\ et al.}%
      \else
        \global\let\@runningauthor\@author%
      \fi
    \else % \c@author > 1
      \ifnum0\n@author@ = 2
        \global\edef\@runningauthor{\@runningauthor\ \& \noexpand\@author}%
      \fi
    \fi
  \endgroup
}
\def\author@fmt#1#2#3{\@newelemtrue
  \if@firstauthor
  \first@author \global\@firstauthorfalse \fi
  \ifnum\prev@elem=\e@author \global\@newelemfalse \fi
  \if@newelem \author@fmt@init \fi
  \edef\@tempb{#2}\ifx\@tempb\@empty
    \hbox{{\author@font #3}}\else
    \hbox{{\author@font #3}\,$^{\mathrm{#2}}$}%
  \fi}
\def\first@author{\author@note@fmt  % re-define \thefootnote as
                                    % appropriate for author/address
  \add@tok\t@glob@notes
    {\author@note@fmt\@corresp@note}}%
\def\author@fmt@init{%
  \par
  \vskip 8\p@ \@plus 4\p@ \@minus 2\p@
  \@authorsize
  \leavevmode}                        % Vertical space above author list
 \def\and{\unskip~and~}
\def\collab{\@ifstar{\collab@arg}{\collab@arg}}
\let\collaboration=\collab
\def\collab@arg#1{\stepcounter{collab}%
  \if@firstauthor \first@collab \global\@firstauthorfalse \fi
  \gdef\@runningauthor{#1}%
  \beg@elem
  \proc@elem{collab}{#1}%
  \collab@fmt{\the\c@collab}{\the\t@loc@notes}{\@collab}%
  \ignorespaces}
\def\collab@fmt#1#2#3{\@newelemtrue
  \ifnum\prev@elem=\e@collab \global\@newelemfalse \fi
  \if@newelem \collab@fmt@init \fi
  \par                                 % Start new paragraph
  {\large #3\,$^{\mathrm{#2}}$}}
\def\first@collab{
  \collab@note@fmt                     % re-define \thefootnote as
  \add@tok\t@glob@notes                % appropriate for collab/address
    {\collab@note@fmt}}%
\def\collab@fmt@init{\vskip 1em}       % Vertical space above list
\def\author@note@fmt{\setcounter{footnote}{0}%
  \def\thefootnote{\xarabic{footnote}}}
\let\collab@note@fmt=\author@note@fmt
\def\xarabic#1{%
  \expandafter\expandafter\expandafter\ifnum\expandafter\the\@nameuse{c@#1}<0
  *\else\arabic{#1} \fi}
\def\xalph#1{%
  \expandafter\expandafter\expandafter\ifnum\expandafter\the\@nameuse{c@#1}<0
  *\else\alph{#1} \fi}
\def\xfnsymbol#1{%
  \expandafter\expandafter\expandafter\ifnum\expandafter\the\@nameuse{c@#1}<0
  *\else\fnsymbol{#1} \fi}
\def\address{\@ifstar{\address@star}%
  {\@ifnextchar[{\address@optarg}{\address@noptarg}}}
\def\address@optarg[#1]#2{\real@refstepcounter{address}%
  \beg@elem
  \report@elt{address}\proc@elem{address}{#2}%
  \address@fmt{\c@address}{\the\t@loc@notes}{\@address}{#1}%
  \if@Elproofing\else\label{#1}\fi
  \ignorespaces}
\def\address@noptarg#1{\real@refstepcounter{address}%
  \beg@elem
  \proc@elem{address}{#1}%
  \address@fmt{\z@}{\the\t@loc@notes}{\@address}{\theaddress}%
  \ignorespaces}
\def\address@star#1{%
  \beg@elem
  \proc@elem{address}{#1}%
  \address@fmt{\m@ne}{\the\t@loc@notes}{\@address}{*}%
  \ignorespaces}
\def\theaddress{\alph{address}}
\def\address@fmt#1#2#3#4{\@newelemtrue
  \if@Elproofing\def\@eltag{#4}\else\def\@eltag{\theaddress}\fi
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Content-Type: application/x-tex; name="vulpio2.tex"
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Content-Disposition: inline; filename="vulpio2.tex"



\documentclass{elsart}
\usepackage{graphicx}

%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI (dette macro) %%%%%%%%%%%%%%%%%%%%%
\def\fraz#1#2{ \frac{#1}{#2} }
\def\dif{{\rm d}}
\def\opt{{\rm opt}}
\def\min{{\rm min}}
\def\eps{ { \varepsilon } }
\def\nome{{\eta}}
\def\phi{{\varphi}}
\def\equal{\buildrel {\rm def} \over {=} }
\def\={ \equal } 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{document}
\begin{frontmatter}


\title{PLANCK'S FORMULA AND GLASSY BEHAVIOR IN CLASSICAL
NONEQUILIBRIUM
STATISTICAL  MECHANICS}

\author{A. Carati\thanksref{mail1}}
and
\author{L. Galgani}

\address{Universit\`a di Milano,Dipartimento di Matematica, 
Via Saldini 50, 20133 Milano, ITALY}

\thanks[mail1]{Grant from Fondazione Cariplo per la Ricerca Scientifica, e-mail: 
carati@unimi.it; galgani@unimi.it}
 



\begin{abstract}
In statistical mechanics it is well known that one has to take into
 account the role of the relaxation times to equilibrium. We discuss the 
relevance of this fact  for systems of weakly coupled harmonic oscillators
in a classical framework, 
 showing  how one has a behavior similar  to that of glassy systems
and how one finds an analogue of Planck's 
formula. From this point of view, quantum  equilibrium statistical mechanics
thus appears as a first order approximation in classical nonequilibrium
statistical  mechanics very far from equilibrium.
\end{abstract}

\begin{keyword}
Planck's law, relaxation times, glasses

PACS:    05.20.--y   05.45.--a  05.70.Ln  61.43.Fs  82.20.Mj 
\end{keyword}

\end{frontmatter}




\section{Introduction}

In  thermodynamics one always makes reference to some relaxation time to 
equilibrium; for example, for a gas inside a cilinder  one commonly considers 
the displacement of a piston, and requires that a sufficiently long time 
be elapsed  for the gas to have reached  a new equilibrium state, and in 
such a way one considers a sequence of equilibrium states. The main thesis
we want to discuss here is that, according to classical mechanics,
 there exists a high nonuniformity in the 
relaxation times for the different ``internal'' degrees of freedom,  such times 
increasing  exponentially fast with the  frequencies of the internal motions, 
so that one is
somehow compelled to take such a high nonuniformity into account.
The zero-th order approximation turns out to be just the one in which such 
differences  are altogether neglected 
(i.e.  one considers extremely long times),  and is well known to lead  to
energy equipartition, namely to predict the same mean energy for each degree 
of freedom
(or rather for any quadratic term in the hamiltonian) irrespective of its
frequency. We present here
some recent results (obtained in the paper\cite{carpre}), which  show that in the next order of 
approximation  a
classical analogue of Planck's formula is recovered, so that quantum equilibrium
 statistical 
mechanics 
appears as a first order approximation within the framework of classical
nonequilibriium statistical  mechanics.
In such a way  we are bringing to its extreme consequences a point of view first
proposed by Boltzmann and Jeans (for the references, see \cite{galgnoto}), 
by actually implementing an idea which was proposed
by Einstein, as we will try to illustrate below.
We finally will  point out that  there should exist a kind of experimentum crucis 
showing 
whether the present point of view is correct. Indeed from the present point 
of view
one expects that, as time goes on, equipartition is actually achieved for all 
frequencies, with a kind of equipartition front advancing very slowly with time
 from the low frequencies towards the high frequencies, while according to 
quantum statistical mechanics Planck's law corresponds to a real 
 equilibrium state.

Preliminarly we address here,  however, another point, namely the question of 
how can one 
obtain  in classical mechanics something quantitatively comparable with 
quantum mechanics,
as the latter involves    a quantity, namely Planck's constant $\hbar$, which is 
completely extraneous to classical mechanics. A first answer, to which we limit 
oureselves in the present paper, is that Planck's constant can be introduced in 
classical mechanics simply through the molecular parameters. Indeed,  consider 
for example a system of equal particles  of the same mass $m$  
interacting through
a typical interatomic potential, such as the familiar one of  Lennard-Jones.
Now,  this potential contains two parameters, say $V_0$ and $\sigma$, with the 
dimensions of an energy and a length respectively, and from them and $m$ one 
constructs an action $A$, namely $A=\sigma\sqrt {m V_0}$, which a priori can take 
any  value. But if one takes for the parameters entering a model just the ones
corresponding to actual atoms, as reported in the standard  textbooks, one
 finds that the relation $A\simeq 2Z\hbar$ holds, where $Z$ is the atomic 
number of 
the considered atoms. This is one  way in which Planck's constant can be made to 
enter classical physics at the level of pure mechanics (see \cite{galg71}). A more fundamental way
would require considering the role of the electromagnetic field, but this 
interesting point will not be discussed here.    

\section{The  analytical argument for the nonuniformity of the relaxation times}

To go to the heart of the problem, we consider the simplest   example
describing the essence of the dynamics of a diatomic gas, which is the prototype 
physical system presenting ``internal'' degrees of freedom (see \cite{ben87a}).
This is a one--dimensional model involving 
two points $P$ and $Q$ on a line;   $P$ 
is attracted by a linear  spring to a fixed origin $O$,  while $Q$ collides with 
$P$ through an analytic potential. 
Denote By $x$ and 
$y$ the abscissas of the spring $P$ and of the impinging point $Q$ respectively. 
Then one has a system of two coupled equations for the unknowns $x(t)$, $y(t)$, 
namely Newton's
equations with the mutual opposite forces due to the given potential.   
In a first  approximation a decoupling  is obtained by considering  the 
case in which the initial
 energy of the spring is very small. Indeed, in such a case, for
a  high frequency it turns out that  the spring remains 
very close to the origin, i.e. one has $|x (t)|\ll 1$ in suitable units. Thus, 
 in the mutual force, 
which depends 
on the relative distance  $|x-y|$, one can simply replace 
$|x-y|$  by $|y|$,
and so  Newton's equation for the impinging particle   decouples from the 
other one, becoming a closed equation for $y$, and this
 is in principle solved by a certain function $y=y(t)$ depending on the initial
data.  The equation of 
motion for the spring $P$ then becomes
$$
\ddot x+\omega^2 x= f(t) 
$$ 
with a  given  function $f(t)$, and it is well known
that such an equation is solved by
$$
z(t)=e^{i\omega t} \big[ z_0+\int_{t_0}^t e^{-i\omega s} f(s) ds \big]
$$
in terms of the complex variable $z=\dot x+i\omega x$. On the other hand the
energy $E=(1/2)({\dot x}^2+\omega^2 x^2)$ of the  spring is immediately written 
down in terms of $z(t)$, by $2E=|z|^2$. Thus the  energy  
$\delta E= E(+\infty)-E(-\infty)$ exchanged  in a single collision turns  out to 
be given  by  
the simple formula (which we call Benettin's formula -- see \cite{baldan} and 
\cite{sempio} --, 
although it was  well 
known for example to Jeans)
$$
\delta E=\eta^2+2\eta\sqrt {E_0}\cos \varphi_0\ ,
$$
involving the quantity
$$
\eta = \big| \int_{-\infty}^{+\infty} e^{-i\omega s}  f(s) ds\, \big| 
$$
and the initial phase $\varphi_0$ of the oscillator.
Consequently in a first approximation the energy $\delta E$ 
exchanged in a single collision 
between a particle and a spring of frequency $\omega$
is the sum of a drift term $\eta^2$ and of a fluctuating one depending on the
 initial phase $\varphi_0$ of the spring; but in any case the exchanged energy 
is estimated essentially
 by $\eta$, namely by the  Fourier transform (evaluated at 
$\omega$)
of an analytic function, which
 is very well known to decrease exponentially fast
with  $\omega$. It can be shown that the qualitative results of 
such a first
order (or Landau--Teller, see \cite{land36}) approximation  hold rigorously for 
the exact equations (see \cite{car92} and \cite{bencargall}). 

The simplicity
of the argument is  so impressive that  one  could hardly imagine  it 
has  no physical  consequences: we have indeed shown that, 
through collisions, the 
energy exchanges between the low frequencies (the point particle) and the high 
frequencies is 
exponentally small, and this  just because the Fourier transform of an analytical function
decreases exponentially fast with the frequency. So the establishment of 
equipartition between the internal degrees of freedom and the translational ones
requires an exponentially large number of collisions, namely an exponentially 
long time. One can find meaningful examples (see \cite{ben87a}) in which there exists a frequency 
$\bar \omega$ having a relaxation time of the order of 1 second, while
the frequency $\bar \omega/2$ relaxes in  $10^{-8}$ seconds, and the frequency
$2\bar \omega$ in $10^5$ years, which indeed constitutes a case of
high nonuniformity.


\section{Planck's law as an approximate thermodynamic formula in a 
metastable regime, similar to that of  glasses}

Let us consider a model of a large number $N$ of springs of the same 
frequency $\omega$,
each of which suffers collisions with a point particle of velocity $v$ 
as discussed above. One should think of the point particles as mimicking a 
heat reservoir at a temperature $T$ with $mv^2/2=kT$, where $k$ is 
Boltzmann's constant; the frequency $\omega$ of the springs is  a 
parameter that can 
take any positive value.  We address the problem of the statistical
behavior of the internal degrees of freedom (the springs) initially
 at rest, trying to take into
account the high nonuniformity of the exchange of energy at each collision
as a function of frequency. Indeed  for springs of low enough frequency 
one can expect
that after a rather short time they go to equipartition with the reservoir, 
 acquiring an energy $kT$. Thus, for any fixed observation time
$t$ there will exist  a maximal equipartition  frequency
$\omega^*=\omega^*(t)$, such that one will have  equipartition for the lower 
frequencies,  
while the larger frequencies will have an average energy exponentially
decreasing with  frequency. 
Clearly the ``equipartition front'' $\omega^*(t)$ will advance 
towards the high frequencies as time goes on, and this is indeed the main point
where the present classical theory qualitatively differs from the quantum 
theory. 

Indeed the latter predicts that the energy distribution at equilibrium
will be Planck's law, which attributes equipartition  only to the 
zero frequency, while all other strictly positive  frequencies should have a 
lower energy. 
Let us recall that Planck's law for the average energy $U$ of system of $N$
oscillators of frequency $\omega$ at temperature $T=1/k\beta$ is 
$$
U(\omega, T)=N\,\big(\, \fraz \epsilon{e^{\beta\epsilon}-1}\,
\big)\ ,
$$
with $\epsilon=\hbar\omega$; so $U$
 tends to $NkT$ for $\omega\to 0$, and decreases exponentially fast to zero
as frequency increases (in fact one might also take into 
account a contribution $N \hbar\omega/2$ coming from the so called zero--point
energy, but this is a quite delicate point we will briefly discuss below).  
Thus quantum mechanics predicts that at equilibrium only the zero frequencies
will be at equipartition, while the higher frequencies will have lower and lower
energies, decreasing exponentially fast as frequency increases. Classical
mechanics predicts instead that at any fixed time there will be a plateau
of equipartition for the low frequencies, with the equipartition front
advancing towards the high frequencies at an extremely low rate. The higher
frequencies will indeed have an average energy decreasing exponentially with 
frequency; but this should be a kind of metastable situation, because for 
any frequency the average energy should increase with time 
towards equipartition at an extremely low rate, as is by now very familiar
in the so called aging phenomena of glasses or spin glasses. Such an analogy
was  pointed out quite recently (see \cite{carjsp}). 

The problem then remains of understanding whether  in classical mechanics
one can go a little further, in the direction of finding a thermodynamic
universal description for such a metastable situation. We will now describe
how it was found that this is indeed the case, and that one thus obtains
a classical analogue of Planck's formula. This was obtained in \cite{carpre} 
by showing how
the  Benettin--Jeans formula recalled above for the energy exchange 
$\delta E$ in a single  collision is transformed, through statistics (i.e.
through suitable averages), into what we call Einsten's dynamical fluctuation 
formula,
which in turn is essentially equivalent to Planck's formula.
But this requires recalling first the original interpretation of Planck's 
formula given by Einstein in the years 1909, 1912 (see \cite{eins09bis} and 
\cite{solvay}), which apparently is not 
familiar in the scientific community.

What Einstein did, was to provide  a physical substantiation for the 
original deduction Planck gave of his law on October 19, 1900 
(see \cite{planck1}). Indeed it was 
only in his second communication (december 1900) that Planck gave his familiar
deduction involving the standard statistical arguments and the
revolutionary   discretization of energy. In his first communication he
was  instead proceeding at a phenomenological heuristic and  formal level, 
without invoking any discretization at all. In fact 
he had  remarked  (we are using here a contamination
of the notations of Einstein and of Planck, using furthemore the inverse
temperature $\beta=1/kT$ instead of temperature $T$) that the differential
equation
$$
\fraz {\dif U}{\dif \beta}=-\fraz {U^2}N
$$ leads by integration (with a 
suitable choice of the integration constant) to the 
 equipartition law $U=N kT$, which is valid for low frequencies, 
while the
differential equation
$$
\fraz {\dif U}{\dif \beta}=- \epsilon U
$$
 leads analogously to a law which was known to be valid for high frequencies,
namely  Wien's law 
$U=C\exp(-\beta\epsilon)$, with a suitable constant $C$.
 Just by virtue of imagination, through an 
interpolation  he was then led to the differential equation
$$
 \fraz{\dif U}{\dif \beta}=- (\epsilon U+\fraz{U^2}N)\ ,
$$
which by integration gives  exactly 
 Planck's formula. Such a firmula  was found to fit very well the experimental data,
with an appropriate choice for $\epsilon$; in fact   it was already 
known, by  a general argument of Wien,  that $\epsilon$ had
to be taken proportional to frequency (namely one should put 
$\epsilon=\hbar \omega$) with a suitable constant action $\hbar$, 
so that the constant to be fitted to the data was the action $\hbar$, namely 
Planck's constant.

So much for what concerns Planck's first communication. The contribution of
Einstein was the  following one. Already in the year 1903 (see \cite{eins03}) 
he had remarked 
that in the canonical ensemble the fluctuations of energy, described by the
corresponding variance
$$
\sigma^2_E=<E^2>-U^2
$$
where $< \cdots >$ denotes the canonical expectation  and $U := <E>$,
are expressed in terms of the mean energy $U$ through a relation
having a kind of universal thermodynamic character, namely 
$\fraz {\dif U}{\dif \beta}=-\sigma^2_E$. 
Thus Einstein  was  led to
split Planck's differential equation into two relations, namely
$$
\fraz{\dif U}{\dif \beta}=-\sigma^2_E
$$
and 
$$
\sigma^2_E= \epsilon U+U^2/N\, ,
$$
the first of which  was conceived  to be just a kind of  general thermodynamic
relation, while the second one should rather  have a dynamical character,
being in principle deducible from a microscopic dynamics. 
In his very words (see \cite{solvay}): these two relations 
 ``{\it exhaust the thermodynamic  content of Planck's}'' formula; and:
 ``{\it a mechanics compatible with the energy
fluctuation $\sigma^2_E= \epsilon U+U^2/N$ must then necessasily lead to 
Planck's}'' formula.
It was  pointed out in \cite{carpre}  that the second of the above 
equations  has indeed a mechanical character, coinciding essentially with
the Benettin--Jeans formula. So we might say that the ``mechanics'' conceived
by Einstein as leading to Planck's formula is nothing but the dear old classical 
mechanics of Newton. 

This is seen as follows.  The energy
$e_k$ that a single oscillator possesses  after $k$ collisions
according to the Benettin--Jeans formula 
(we  denote now by $e$ the energy of a single 
oscillator, reserving the symbol $E$ for the energy of the  system of $N$ 
oscillators, to be considered below)  can be  conveniently
written in the form
 $$
e_k=e_0+k\eta^2+2{\eta}\sum_{j=1}^k \sqrt{e_{j-1}}\, \, \cos\varphi_{j-1}\ ,
$$
which depends parametrically on  
the phases $\varphi_0,\dots ,\varphi_{k-1}$ and on the initial energy $e_0$.
 The phases can be  assumed to be
 independent and uniformly distributed;  thus,  averaging over them in the
familiar way of  random walk theory, one gets for the mean energy
$u_k:=<{e_k}>$ after $k$ collisions ($< \cdots >$ denoting now 
average over the phases) the expression
$$
u_k=e_0+k\eta^2\ . 
$$
Analogously, with $< {(\cos \varphi_j)^2}>=1/2$, one finds
$<{e_k^2}>=u_k^2+2\eta^2\sum_{j=1}^ku_{j-1}$.    The variance 
$\sigma^2_{e_k}:= <{e_k^2}>
-u_k^2$ then
takes the form $\sigma^2_{e_k}=2e_0k\eta^2+k(k-1)\eta^4$ or also, in the
approximation of large $k$ so that  we can  identify  $k(k-1)$ with $k^2$,
$$
\sigma^2_{e_k}=2e_0k\eta^2+(k\eta^2)^2 \ . 
$$
The relevant point is that in the  latter formula the  ``time'' $k$ (we are 
assuming here for simplicity's sake that the ``time  of flight'' between two 
collisions is a constant) enters
only in the combination $k\eta^2$, so that it can be eliminated through the 
expression of $u_k$;
  this leads to an analogue of Einstein's dynamical fluctuation formula, 
namely
$\sigma^2_{e_k}=2e_0(u_k-e_o) +(u_k-e_0)^2$. 
The analogy becomes even stronger if  one introduces the ``exchanged
energy'' after  $k$ collisions, ${\tilde e}_k=e_k-e_0$, because the
corresponding expectation ${\tilde u}_k$ and variance $\sigma^2_{{\tilde e}_k}$
are then related by
$$
\sigma^2_{{\tilde e}_k}=2e_0{\tilde u}_k+{\tilde u}_k^2 \ . 
$$

A similar relation also holds for the global system of $N$ independent identical
oscillators. Indeed, the quantities  of interest  are the   total energy 
$E_k =\sum_{i=1}^N e_k^{(i)}$ (where $e_k^{(i)}$  denotes the energy of the
$i$--th  oscillator after $k$ collisions) and the corresponding
exchanged energy $\tilde E_k=E_k-E_0$, where $E_0$ is the initial energy. By the
central limit theorem,  ${\tilde E}_k$ is normally
distributed with a mean ${\tilde U}_k$ and a variance $\sigma^2_{{\tilde E}_k}$ 
which are
obtained by adding up the corresponding quantities for each
oscillator, namely are given by ${\tilde U}_k=N{\tilde u}_k$ and 
$\sigma^2_{{\tilde E}_k}= N \sigma^2_{{\tilde e}_k}$.
So,  denoting
by $\tilde U$ and $\sigma^2_{\tilde E}$ expectation and variance of the exchanged
energy at any
``time'' $k$,  one gets between $\sigma^2_{\tilde E}$ and
$\tilde U$ a functional relation which is independent of ``time'' $k$, namely
$$
\sigma^2_{{\tilde E}}=2a_0\omega {\tilde U}+{\tilde U}^2/N\ ,
$$
where $a_0$  denotes the initial action  per oscillator, 
$a_0:= E_0/(\omega N)$. Notice that the quantity $\eta$, which contains 
the molecular parameters characterizing the considered system,  has now 
completely disappeared, and the formula just found has a kind of thermodynamic 
universality. 
This  is our analogue of Einstein's functional relation between variance and 
expectation of energy, which we call Einstein's dynamical fluctuation formula. 

An important point to be stressed is that in our formula there appears as a 
parameter the quantity $2a_0$,   twice
the initial action per oscillator,   which takes  the place of Planck's 
constant $\hbar$. At the moment we are unable to say anything rigorous  concerning 
this very  relevant point. We only present  here some heuristic considerations,
on which we hope to come back in the future. The point is that, as was 
recalled above, the Benettin--Jeans dynamical formula  was established for 
small initial energies of the oscillators; think for example of the fact that
if the oscillator had an initial energy greater than that of the particle, then
 in order to go towards equipartition
one should have  a formula for the exchanged energy with a drift term of 
opposite sign with respect to the one given above. So the formula used above
should hold only for low enough energy or action per oscillator, say for 
$a_0<a_*$, with a  certain
critical or threshold action $a_*$. This naturally leads to think of a situation
with the $N$ oscillators having random initial actions all smaller than $a_*$,
so that, on averaging over the initial actions, uniformly distributed over  the 
interval $(0,a_*)$, one would get a formula as above with $2a_*/2$, namely 
$a_*$. in place of  Planck's constant. The actual mean energy would 
correspondingly be given by Planck's formula with the addition of an analogue
of the zero--point energy, namely $a_*\omega/2$, playing here the role
of the initial energy. Naturally, in situations of metaequilibrium of a 
glassy  type as the one described here, the energy distribution would depend 
on the initial distribution, as was particularly stressed in the paper
\cite{carjsp}.

Notice that it is exactly through the critical action $a_*$ that the molecular
parameters (namely $m$,  $V_0$ and $\sigma$), and thus Planck's constant,  are introduced into our classical 
problems. Indeed by any analytical or numerical estimate one would find
$$
a_*=\alpha\,  \sigma\sqrt{mV_0}\ ,
$$
i.e.
$$
a_*\simeq \alpha\,  2Z\hbar
$$
with a suitable pure number $\alpha$, possibly of order of magnitude 1.
In fact it is just in a way analogous to this that Planck's constant was first 
introduced in a problem of this type in the paper \cite{galg71}, where the 
pure number 
$\alpha$ was estimated to be something as 0.01 (see also \cite{rnc} 
\cite{galg71b}).

\section{Conclusions}

So we have illustrated how it can be conceived that in classical statistical
mechanics one gets a formula of Planck's type as a kind of universal
thermodynamic formula  describing, in a first approximation,   a situation of 
metaequilibrium very far from equilibrium for a system of harmonic oscillators
interacting with  a reservoir.  We also stressed how  the present point
of view leads to  a qualitative difference with respect to quantum
mechanics, because Planck's formula should hold only for limited times, and 
one should observe, as time goes on,  an advancing of the  equipartition front
towards the high frequencies. However this advancing should occur at an 
extremely slow rate, as is familiar in the aging phenomenology of glasses
and spin glasses. An important point would be to establish whether such an
advancement of the equipartition front is really observed or not. This  is
indeed  the  experimentum crucis mentioned in the introduction.

Finally, we  would like  to conclude by recalling  some sentences by Einstein which seem to be appropriate in the present context.
It is very well known that Einstein was critical towards the 
``orthodox'' interpretation of quantum mechanics: although having greatly
contributed to creating the new mechanics, he did not like the interpretation
most physicists attributed to it. 
But much less known is the fact that this general attitude  was explicitly
expounded by Einstein
in connection with the photon, namely in the framework of the  statistical
mechanics  of a system of harmonic 
oscillators discussed here. Indeed,  in his scientific autibiography (see 
\cite{schilpp}),
when recalling the way he had invented the photon by  substantiating
Planck's formal approach, he said: ``This way of considering the problem
showed in a definitive and direct way that it is necessary to attribute a certain
immediate concreteness to Planck's quanta and that, under the energetic aspect,
radiation possesses a sort of molecular structure'', 
but a few lines below he also added: 
``This interpretation, that almost all contemporary physicists consider 
as essentially  definitive,  to me appears instead  as a simple 
provisional way out''.

Now, what did he have in mind when  writing  down such words? In our opinion
he had exactly in mind his ``mechanical'' interpretation of Planck's formula,
recalled above, that he had illustrated  in his contribution to the 
Solvay conference (see \cite{solvay}),
when  he was speaking of  a mechanics  consistent with  his dynamical 
fluctuation formula. Indeed, if such a formula could be proven in some mechanics, 
then
 everything would go ``as if'' energy were discretized, while
 a ``real'' discretization
would not be necessary (``the statistical properties of the phenomenon  are the 
same as if energy were transfered through integer numbers of quanta 
$\hbar \omega$'').
 And in fact  there is even one
passage  where he explains
how  the energy level corresponding to the ground state should be understood
in terms of the energy fluctuations equivalent to Planck's law.
 Indeed, having pointed out that the 
formula for the relative energy fluctuations $\sigma^2_E/U^2$ equivalent to 
Planck's formula has the form
$$
\fraz{\sigma^2_E}{U^2}=\fraz{\epsilon}U+\fraz 1N\ ,
$$
which  for a large nunber $N$ of oscillators and small energies becomes 
$$
\fraz{\sigma^2_E}{U^2}\simeq\fraz{\epsilon}U\, ,
$$
he  adds:
``If $U$ becomes of the order of $\hbar\omega$ (namely of $\epsilon$), the relative fluctuation becomes
of the order of unity; in other terms, the fluctuation of energy is of the order
of magnitude of energy itself, i.e. the total energy is alternatively present or 
absent, and consequently behaves as if it were not indefinitely divisible. 
 It is not necessary to make the hypothesis that distinct energy elements of 
a definite magnitude  exist''.



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