% This paper uses three macros: jnl.tex, reforder.tex, and journals.tex.
% They are provided with the paper. The paper also contains three
% figures. They can be obtained via mail by contacting Dan Stein at
% dls@ccit.arizona.edu.
%
%jnl macro
%% JNL.TEX Doug Eardley
%%
\message{JNL.TEX version 0.95 as of 10/30/86.}
%%
%% This is a set of TeX 82 macros designed to produce scientific
%% papers with a minimum of fuss and using as much of plain.tex as
%% possible. The user need only know what is in the TeXbook, and
%% the macros under ``user definitions'' below. Also, the user
%% definitions are intended to be as simple as possible, so that
%% the user may change them as desired. I have tried to avoid all
%% cleverness, although it may have snuck in here and there.
%%
%% A considerable degree of compatibility with AmSTeX is maintained,
%% although not guaranteed. The intention is that AmSTeX input file
%% should run with only a few changes near the beginning; see
%% discussion below under "AmSTeX compatability".
%%
%% For documentation, see the file JNLHLP.TEX. Optional features are
%% contained in the files PPT.TEX (for two-up preprints), REFORDER.TEX
%% (automatic numbering of references), EQNORDER.TEX (automatic numbering
%% of equations), and TABLEOFC.TEC (automatic generation of table of
%% contents).
% Define a whole menagerie of pseudo-12pt fonts
%%
\font\twelverm=cmr12 \font\twelvei=cmmi12
\font\twelvesy=cmsy10 scaled 1200 \font\twelveex=cmex10 scaled 1200
\font\twelvebf=cmbx12 \font\twelvesl=cmsl12
\font\twelvett=cmtt12 \font\twelveit=cmti12
\font\twelvesc=cmcsc10 scaled 1200
%% removed for Mac \font\twelvesf=amssmc10 scaled 1200
\skewchar\twelvei='177 \skewchar\twelvesy='60
% Define \...point macros to change fonts and spacings consistently
\def\twelvepoint{\normalbaselineskip=12.4pt plus 0.1pt minus 0.1pt
\abovedisplayskip 12.4pt plus 3pt minus 9pt
\belowdisplayskip 12.4pt plus 3pt minus 9pt
\abovedisplayshortskip 0pt plus 3pt
\belowdisplayshortskip 7.2pt plus 3pt minus 4pt
\smallskipamount=3.6pt plus1.2pt minus1.2pt
\medskipamount=7.2pt plus2.4pt minus2.4pt
\bigskipamount=14.4pt plus4.8pt minus4.8pt
\def\rm{\fam0\twelverm} \def\it{\fam\itfam\twelveit}%
\def\sl{\fam\slfam\twelvesl} \def\bf{\fam\bffam\twelvebf}%
\def\mit{\fam 1} \def\cal{\fam 2}%
\def\sc{\twelvesc} \def\tt{\twelvett}
%% removed for Mac \def\sf{\twelvesf}
\textfont0=\twelverm \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm
\textfont1=\twelvei \scriptfont1=\teni \scriptscriptfont1=\seveni
\textfont2=\twelvesy \scriptfont2=\tensy \scriptscriptfont2=\sevensy
\textfont3=\twelveex \scriptfont3=\twelveex \scriptscriptfont3=\twelveex
\textfont\itfam=\twelveit
\textfont\slfam=\twelvesl
\textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf
\scriptscriptfont\bffam=\sevenbf
\normalbaselines\rm}
% tenpoint
\def\tenpoint{\normalbaselineskip=12pt plus 0.1pt minus 0.1pt
\abovedisplayskip 12pt plus 3pt minus 9pt
\belowdisplayskip 12pt plus 3pt minus 9pt
\abovedisplayshortskip 0pt plus 3pt
\belowdisplayshortskip 7pt plus 3pt minus 4pt
\smallskipamount=3pt plus1pt minus1pt
\medskipamount=6pt plus2pt minus2pt
\bigskipamount=12pt plus4pt minus4pt
\def\rm{\fam0\tenrm} \def\it{\fam\itfam\tenit}%
\def\sl{\fam\slfam\tensl} \def\bf{\fam\bffam\tenbf}%
\def\smc{\tensmc} \def\mit{\fam 1}%
\def\cal{\fam 2}%
\textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
\textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
\textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\tenit
\textfont\slfam=\tensl
\textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
\scriptscriptfont\bffam=\fivebf
\normalbaselines\rm}
%%
%% Various internal macros
%%
\def\beginlinemode{\endmode
\begingroup\parskip=0pt \obeylines\def\\{\par}\def\endmode{\par\endgroup}}
\def\beginparmode{\endmode
\begingroup \def\endmode{\par\endgroup}}
\let\endmode=\par
{\obeylines\gdef\
{}}
\def\percentspace{\baselineskip=\normalbaselineskip\divide\baselineskip by 100
\multiply\baselineskip by}
\def\singlespace {\percentspace100}
\def\oneandaquarterspace{\percentspace125}
\def\oneandathirdspace {\percentspace133}
\def\oneandahalfspace {\percentspace150}
\def\doublespace {\percentspace200}
\def\triplespace {\percentspace300}
\newcount\firstpageno\firstpageno=2
\footline={\ifnum\pageno<\firstpageno{\hfil}\else{\hfil\twelverm\folio\hfil}\fi}
\def\toppageno{\global\footline={\hfil}\global\headline
={\ifnum\pageno<\firstpageno{\hfil}\else{\hfil\twelverm\folio\hfil}\fi}}
\let\rawfootnote=\footnote % We must set the footnote style
\def\footnote#1#2{{\rm\singlespace\parindent=0pt\parskip=0pt
\rawfootnote{#1}{#2\hfill\vrule height 0pt depth 6pt width 0pt}}}
\def\raggedleft{\leftskip=0pt plus1fill}
\def\raggedcenter{\leftskip=4em plus 12em \rightskip=\leftskip
\parindent=0pt \parfillskip=0pt \spaceskip=.3333em \xspaceskip=.5em
\pretolerance=9999 \tolerance=9999
\hyphenpenalty=9999 \exhyphenpenalty=9999 }
\def\dateline{\rightline{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\year}}
\def\received{\vskip 3pt plus 0.2fill
\centerline{\sl (Received\space\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\qquad, \number\year)}}
%%
%% Page layout, margins, font and spacing (feel free to change)
%%
\hsize=6.5truein
\hoffset=0.15truein
\vsize=8.9truein
\voffset=0.1truein
\def\landscape{ \hsize=8.9truein\hoffset=0.1truein
\vsize=6.5truein\voffset=0.15truein
\xdef\endit{\immediate\write16
{>>>>To print this, use the command IMPRINT/LAND/DVI \jobname.DVI<<<<.}\endit}}
\parskip=\medskipamount
\def\\{\cr}
\twelvepoint % selects twelvepoint fonts (cf. \tenpoint)
\doublespace % selects double spacing for main part of paper (cf.
% \singlespace, \oneandahalfspace)
\overfullrule=0pt % delete the nasty little black boxes for overfull box
%%
%% The user definitions for major parts of a paper (feel free to change)
%%
\def\draft{\rightline{Draft \timestamp}} % "Draft Timestamp" at right
\def\timestamp{\input timestamp \timestamp} % Timestamp
\def\nsfitp#1{
\rightline{\rm NSF--ITP--#1}} % Preprint number at upper right of title page
\def\title % Title on title page
{\null\vskip 3pt plus 0.2fill
\beginlinemode \doublespace \raggedcenter \bf}
\def\author % Author(s) name(s) on title page
{\vskip 3pt plus 0.2fill \beginlinemode
\singlespace \raggedcenter\sc}
\def\affil % Affiliations (can intermix with \author)
{\vskip 3pt plus 0.1fill \beginlinemode
\oneandahalfspace \raggedcenter \sl}
\def\abstract % Begin abstract
{\vskip 3pt plus 0.3fill \beginparmode
\oneandahalfspace ABSTRACT: }
\def\endtitlepage % End title page, begin body of paper
{\endpage % This subsumes \body
\body}
\let\endtopmatter=\endtitlepage
\def\body % Begin text body; can be used to end
{\beginparmode} % \title, \author, \affil, \abstract,
% \reference, or \figurecaption modes
\def\head#1{ % Head; NOTE enclose the text in {}
\goodbreak\vskip 0.5truein % e.g., \head{I. Introduction}
{\immediate\write16{#1}
\raggedcenter \uppercase{#1}\par}
\nobreak\vskip 0.25truein\nobreak}
\def\subhead#1{ % Subhead; NOTE enclose the text in {}
\vskip 0.25truein % e.g., \subhead{A. History of the Problem}
{\raggedcenter {#1} \par}
\nobreak\vskip 0.25truein\nobreak}
\def\refto#1{$^{#1}$} % For references in text as superscript
\def\references % Begin references -- basic format is Phys Rev
{\head{References} % I.e., volume, page, year (space after commas).
\beginparmode
\frenchspacing \parindent=0pt \leftskip=1truecm
\parskip=8pt plus 3pt \everypar{\hangindent=\parindent}}
\gdef\refis#1{\item{#1.\ }} % Ref list numbers.
% NOTE: \journal and journal abbreviations are now in JRNLS.T
\def\endreferences{\body}
\def\figure#1. (#2) #3\par{ % For inline figures -- leaves #2 of vskip
\goodbreak\midinsert
\hskip-\hoffset\noindent\null\vskip #2\relax
{\tenpoint\singlespace\leftskip=1in\rightskip=0.5in
\noindent Figure #1. #3\par}\endinsert}
\def\figurefile#1. (#2) #3\par{ % For inline figures -- inputs Impress file #2
\goodbreak\midinsert
\hskip-\hoffset\noindent\special{insert #2}\null\vskip4.75in
{\tenpoint\singlespace\leftskip=1in\rightskip=0.5in
\noindent Figure #1. #3\par}\endinsert}
\def\figurecaptions % Begin figure captions
{\endpage
\beginparmode
\head{Figure Captions}
% \parskip=24pt plus 3pt \everypar={\hangindent=4em}
}
\def\endfigurecaptions{\body}
\def\endpage % Eject a page
{\vfill\eject}
\def\endpaper % Ways to say goodbye
{\endmode\vfill\supereject}
\def\endjnl
{\endpaper}
\def\endit
{\endpaper\end}
\def\bye
{\endit}
%%
%% AmSTeX compatability definitions
%%
%% To run a TeX file originally intended for AmSTeX, only small changes
%% should be necessary (I hope). Use the line \input jnl at the start.
%% Remove the lines \input amstex, \documentstyle{itpjnl} at the
%% beginning; also remove all the page layout stuff (\parindent=1cm,
%% \hsize=5.28125in etc.) The page layout is now done automatically.
%% Also OMIT the qualifier \magnification=1200 when you IMPRINT the
%% .dvi file. (\TagsOnRight is harmless, you can take it out or leave
%% it in.) I believe most AmSTeX will work with no change. One problem
%% is \footnote, which is a little different in that it now needs to
%% have an explicit asterisk * (or whatever) included, like this:
%% \footnote*{Text winds up at bottom of page.}
%% This is discussed on p. 116 of the TeXbook. IGNORE the AmSTeX
%% documentation (if you can call it that); refer to the TeXbook.
%%
%% Note that many commands in AmSTeX have their equivalents in the
%% TeXbook, perhaps with different names and slightly differing
%% usage. E.g., the old \align in AmSTeX is replaced by \eqalign
%% (p. 190) and \aligntag is replaced by \eqalignno (p. 192).
%% \align and \aligntag still work, but I recommend that you use
%% \eqalign and \eqalignno in documents run under jnl.
%%
%% See me if you have any problems -- Doug.
%%
\def\TagsOnRight{}
\def\topmatter{}
\def\endtitle{\body}
\def\endauthor{\body}
\def\endaffil{\body}
\def\heading % Heading
{\vskip 0.5truein plus 0.1truein % e.g., \heading I. NOTES \endheading
\beginparmode \def\\{\par} \parskip=0pt \singlespace \raggedcenter}
\def\endheading
{\par\nobreak\vskip 0.25truein\nobreak\beginparmode}
\def\subheading % Subheading
{\vskip 0.25truein plus 0.1truein % e.g., \subheading{A. The Problem}
\beginlinemode \singlespace \parskip=0pt \def\\{\par}\raggedcenter}
\def\endsubheading
{\par\nobreak\vskip 0.25truein\nobreak\beginparmode}
\def\tag#1$${\eqno(#1)$$}
\def\align#1$${\eqalign{#1}$$}
\def\endalign{\cr}
\def\aligntag#1$${\gdef\tag##1\\{&(##1)\cr}\eqalignno{#1\\}$$
\gdef\tag##1$${\eqno(##1)$$}}
\def\endaligntag{}
\def\binom#1#2{{#1 \choose #2}}
\def\stack#1#2{{#1 \atop #2}}
\def\overset #1\to#2{{\mathop{#2}\limits^{#1}}}
\def\underset#1\to#2{{\let\next=#1\mathpalette\undersetpalette#2}}
\def\undersetpalette#1#2{\vtop{\baselineskip0pt
\ialign{$\mathsurround=0pt #1\hfil##\hfil$\crcr#2\crcr\next\crcr}}}
\def\enddocument{\endit}
%%
%% Various little user definitions
%%
\def\ref#1{Ref.~#1} % for inline references
\def\Ref#1{Ref.~#1} % ditto
\def\[#1]{[\cite{#1}]}
\def\cite#1{{#1}}
\def\fig#1{Fig.~{#1}} % For figure numbers
\def\Equation#1{Equation~\(#1)} % For citation of equation numbers
\def\Equations#1{Equations~\(#1)} % ditto
\def\Eq#1{Eq.~\(#1)} % ditto
\def\Eqs#1{Eqs.~\(#1)} % ditto
\let\eq=\Eq\let\eqs=\Eqs % ditto
\def\(#1){(\call{#1})}
\def\call#1{{#1}}
\def\taghead#1{}
\def\frac#1#2{{#1 \over #2}}
\def\half{{\frac 12}}
\def\third{{\frac 13}}
\def\fourth{{\frac 14}}
\def\1{\frac1}\def\2{\frac2}\def\3{\frac3}\def\4{\frac4}\def\5{\frac5}
\def\6{\frac6}\def\7{\frac7}\def\8{\frac8}\def\9{\frac9}
\def\eg{{\it e.g.,\ }}
\def\Eg{{\it E.g.,\ }}
\def\ie{{\it i.e.,\ }}
\def\Ie{{\it I.e.,\ }}
\def\etal{{\it et al.\ }}
\def\etc{{\it etc.\ }}
\def\via{{\it via\ }}
\def\cf{{\sl cf.\ }}
\def\sla{\raise.15ex\hbox{$/$}\kern-.57em}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\twiddle{\lower.9ex\rlap{$\kern-.1em\scriptstyle\sim$}}
\def\bigtwiddle{\lower1.ex\rlap{$\sim$}}
\def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\square{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt
\vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt}
\def\tdot#1{\mathord{\mathop{#1}\limits^{\kern2pt\ldots}}}
\def\super#1{$^{#1}$}
\def\beneathrel#1\under#2{\mathrel{\mathop{#2}\limits_{#1}}}
\def\pmb#1{\setbox0=\hbox{#1}%
\kern-.025em\copy0\kern-\wd0
\kern .05em\copy0\kern-\wd0
\kern-.025em\raise.0433em\box0 }
\def\qed{\vrule height 1.2ex width 0.5em}
\def\const{{\rm const}}
\def\beginitems{\par\begingroup\parskip=0pt\advance\leftskip by 24pt
\def\i##1 {\item{##1}}\def\ii##1 {\itemitem{##1}}\medskip}
\def\enditems{\par\endgroup}
\def\itp{Institute for Theoretical Physics}
\def\itpucsb{\itp\\University of California\\Santa Barbara, California 93106}
\def\itpgrant{This research was supported in part by the National
Science Foundation under Grant No.~PHY82-17853,
supplemented by funds from the National Aeronautics and Space
Administration, at the University of California at Santa Barbara.}
%
%reforder macro
%% REFORDER.TEX 6/7/85 Doug E.
%% (mods: 3/25/87 R.G.Palmer)
%%
%% This macro package is intended for use with JNL.
%% It will automatically order and sort the references in a paper
%% by order of first citation.(!!) To use, say \input reforder
%% after \input jnl (and after all definitions of \refto etc.,
%% in particular after any use of the \refstyleXX macros),
%% but before any use of \refto etc. Use \refto{} (or \ref{} and
%% \Ref{}) to cite references in the text. Use \refis{} to supply
%% the references, SKIP A LINE after each reference. Open the
%% reference listing with \references and close it with \endreferences.
%%
%% REFORDER depends on the
%% JNL macros \refto{}, \ref{}, \Ref{} to identify citation of references.
%% REFORDER also contains a macro \cite{} which can be used to cite
%% references; e.g., ``Reference \cite{19} blah...'' will produce
%% output ``Reference 19 blah''. Multiple citations can be separated
%% by commas. E.g., \refto{24,26,27} and \cite{3,7}
%% are legal. Also legal is \refto{3-7}, which expands to mean the same
%% as \refto{3,4,5,6,7}. Reference ``numbers'' can in general be any
%% alphanumeric string; e.g. BjorkenAndDrell is perfectly OK used in
%% the form\ref{BjorkenAndDrell}; such strings should contain no blanks.
%%
%% If you have your own pet macros to cite references such as, e.g.,
%% \def\referpet#1{$^(#1)$)}, you can bring it to the attention of
%% REFORDER so all \referpet's will be properly cited simply by
%% declaring ``\citeall\referpet'' once near the beginning, after
%% \referpet is defined and
%% after \input reforder. This has the effect of redefining the macro
%% as e.g., \def\referpet#1{$^(\cite{#1})$}. (Such \citeall'ed macros
%% must have exactly one argument #1, as in \referpet.) See e.g.,
%% the end of this file where \refto, \ref and \Ref are \citeall'ed.
%%
%% REFORDER depends on the macro \refis{} to supply each reference.
%% \refis{} can be used to supply a reference anywhere in the paper
%% after its first citation. The macro \endreferences actually triggers
%% sorting and listing of references. Skip a line after a reference
%% listing (or, alternatively, end each listing in \par).
%%
%% Use \ignoreuncited after \input reforder if you wish to ignore
%% references that are supplied but not cited. This is particularly
%% useful if you maintain a master file of references (each supplied
%% with \refis{}) but only use a subset of these in a given paper.
%% Include your reference file (with \input) between \references
%% and \endreferences.
%%
%% Use \referencefile after \input reforder if you want an ordered
%% source listing of the references in file .ref
%%
%% The \reftorange macro can be used to produce a superscript
%% reference range, like $^{10-15}$. (The \refto macro always
%% lists the references one by one, even for e.g. \refto{10-15}).
%% Use e.g. \reftorange{10}{11-14}{15} -- the references in the
%% middle group are cited but only 10-15 appears in the text.
%% Note that \reftorange does NOT check for increasing order.
\catcode`@=11
\newcount\r@fcount \r@fcount=0
\newcount\r@fcurr
\immediate\newwrite\reffile
\newif\ifr@ffile\r@ffilefalse
\def\w@rnwrite#1{\ifr@ffile\immediate\write\reffile{#1}\fi\message{#1}}
\def\writer@f#1>>{}
\def\referencefile{% Stuff to write .REF file
\r@ffiletrue\immediate\openout\reffile=\jobname.ref%
\def\writer@f##1>>{\ifr@ffile\immediate\write\reffile%
{\noexpand\refis{##1} = \csname r@fnum##1\endcsname = %
\expandafter\expandafter\expandafter\strip@t\expandafter%
\meaning\csname r@ftext\csname r@fnum##1\endcsname\endcsname}\fi}%
\def\strip@t##1>>{}}
\let\referencelist=\referencefile
\def\citeall#1{\xdef#1##1{#1{\noexpand\cite{##1}}}}
\def\cite#1{\each@rg\citer@nge{#1}} % Variable No. of args, separated by ","
\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg} % each@ag is a general purpose
\def\apply@rg#1,{\ifx\end#1\let\next=\relax% variable no. of arg. macro.
\else,\thecsname{#1}\let\next=\apply@rg\fi\next}% args separated by commas
\def\citer@nge#1{\citedor@nge#1-\end-} % Check for M-N range (M and N numbers)
\def\citer@ngeat#1\end-{#1}
\def\citedor@nge#1-#2-{\ifx\end#2\r@featspace#1 % Single argument
\else\citel@@p{#1}{#2}\citer@ngeat\fi} % M-N range of arguments
\def\citel@@p#1#2{\ifnum#1>#2{\errmessage{Reference range #1-#2\space is bad.}%
\errhelp{If you cite a series of references by the notation M-N, then M and
N must be integers, and N must be greater than or equal to M.}}\else%
{\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\r@fcite\the\count0,%
\loop\advance\count0 by1\relax% Loop from M to N
\ifnum\count0<\count1,\expandafter\r@fcite\the\count0,%
\repeat}\fi}
\def\r@featspace#1#2 {\r@fcite#1#2,} % Eat spaces at beginning or end of arg
\def\r@fcite#1,{\ifuncit@d{#1}% Cite individual reference
\newr@f{#1}%
\expandafter\gdef\csname r@ftext\number\r@fcount\endcsname%
{\message{Reference #1 to be supplied.}%
\writer@f#1>>#1 to be supplied.\par}%
\fi%
\csname r@fnum#1\endcsname}
\def\ifuncit@d#1{\expandafter\ifx\csname r@fnum#1\endcsname\relax}%
\def\newr@f#1{\global\advance\r@fcount by1%
\expandafter\xdef\csname r@fnum#1\endcsname{\number\r@fcount}}
\let\r@fis=\refis % Save old \refis, redefine
\def\refis#1#2#3\par{\ifuncit@d{#1}% Use two params #2 #3 to strip blank
\newr@f{#1}%
\w@rnwrite{Reference #1=\number\r@fcount\space is not cited up to now.}\fi%
\expandafter\gdef\csname r@ftext\csname r@fnum#1\endcsname\endcsname%
{\writer@f#1>>#2#3\par}}
\def\ignoreuncited{% redefine \refis if ignoring uncited references
\def\refis##1##2##3\par{\ifuncit@d{##1}%
\else\expandafter\gdef\csname r@ftext\csname r@fnum##1\endcsname\endcsname%
{\writer@f##1>>##2##3\par}\fi}}
\def\r@ferr{\endreferences\errmessage{I was expecting to see
\noexpand\endreferences before now; I have inserted it here.}}
\let\r@ferences=\references
\def\references{\r@ferences\def\endmode{\r@ferr\par\endgroup}}
\let\endr@ferences=\endreferences
\def\endreferences{\r@fcurr=0% Save old \endreferences, redefine
{\loop\ifnum\r@fcurr<\r@fcount% Loop over refnum and produce text
\advance\r@fcurr by 1\relax\expandafter\r@fis\expandafter{\number\r@fcurr}%
\csname r@ftext\number\r@fcurr\endcsname%
\repeat}\gdef\r@ferr{}\endr@ferences}
% Save old \endpaper, redefine it to write parting message.
\let\r@fend=\endpaper\gdef\endpaper{\ifr@ffile
\immediate\write16{Cross References written on []\jobname.REF.}\fi\r@fend}
\catcode`@=12
\def\reftorange#1#2#3{$^{\cite{#1}-\setbox0=\hbox{\cite{#2}}\cite{#3}}$}
\citeall\refto % These macros will generate citations
\citeall\ref %
\citeall\Ref %
%
%journals macro
\gdef\journal#1, #2, #3, 1#4#5#6{{\sl #1 }{\bf #2}, #3 (1#4#5#6)}
\def\same#1, #2, 1#3#4#5{{\bf #1}, #2 (1#3#4#5)}
\def\refstyleprnp{ % Input like pr, output like np!!
\gdef\refto##1{ [##1]} % Reference in text []
\gdef\refis##1{\item{##1)\ }} % Reference list numbers)
\gdef\journal##1, ##2, ##3, 1##4##5##6{ % Journal reference
{\sl ##1~}{\bf ##2~}(1##4##5##6) ##3}}
\def\pr{\journal Phys. Rev., }
\def\pra{\journal Phys. Rev. A, }
\def\prb{\journal Phys. Rev. B, }
\def\prc{\journal Phys. Rev. C, }
\def\prd{\journal Phys. Rev. D, }
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\title
A New Look at Broken Ergodicity
\vskip .25in
\author D.L. Stein
\affil
Department of Physics
University of Arizona
Tucson, AZ 85721
\vskip .25in
\author C.M. Newman
\affil
Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
\vskip .25in
\abstract
We study the nature and mechanisms of broken ergodicity (BE) in specific random walk models
corresponding to diffusion on random potential surfaces, in both one and high dimension. Using
both rigorous results and nonrigorous methods, we confirm several aspects of the standard BE
picture and show that others apply in one dimension, but need to be modified in higher
dimensions. These latter aspects include the notions that at fixed temperature confining
barriers increase logarithmically with time, that ``components'' are necessarily bounded regions
of state space which depend on the observational timescale, and that the system continually
revisits previously traversed regions of state space. We examine our results in the context of
several experiments, and discuss some implications of our results for the dynamics of disordered
and/or complex systems.
\endtitlepage
\body
\noindent 1. $\underline {\rm Introduction}$.
\medskip
When a system has many metastable states, it may become trapped for long times in some subset
of its total state space, making it difficult to compare experimental results with
calculations based on the usual Gibbs formalism. A viewpoint commonly called ``broken
ergodicity'' (BE) has evolved to serve as a qualitative guide for the understanding of some of
the dynamical and thermal properties of these systems. This has been extremely useful in
several respects, but we are still hampered by the lack of a real theory.
Much of the problem is the difficulty of characterizing the nature of metastability in real
systems. As a result, the standard picture of BE which has emerged (to be described below) is
based largely on intuition and simple pictures of what these state spaces may look like. All
basically involve diffusion of a particle (the system) on a rugged landscape, which may or
may not possess correlations. These pictures can all be described as diffusion in a strongly
inhomogeneous environment.
While many of the results obtained in this way are compelling (and, as we will discuss
below, almost certainly correct in a wide variety of situations), progress has been slow, at
least partially because of the lack of specific models to test these ideas on. In this paper
we will attempt to do just that; we will examine some simple, well-known models and see how
broken ergodicity arises in them.
These models are representative of uncorrelated random potentials. Reasoning based on
random walks on such potentials has guided much of the thinking about how BE operates in
disordered systems.\refto{models} We will not address in this paper the question of the
accuracy of such assumptions; \ie whether random walks on rugged landscapes are useful for
modelling dynamics of some disordered systems. Our only goal here is to introduce clear,
well-defined models and to study their long-time behavior in the strongly inhomogeneous limit.
The analysis will be based on some rigorous results obtained in an earlier paper,\refto{NS1}
hereafter referred to as I, and some nonrigorous results obtained in another,\refto{NS2}
hereafter referred to as II. We will find that while many of the central ideas of standard BE
apply to these models, there are some surprising deviations from important elements of the
conventional picture. We will find that this is at least partially due to the fact that
while all workers in the field recognize that the relevant state spaces for physical systems are
high-dimensional, much of the intuition about BE is nevertheless based on what are ultimately
one-dimensional pictures. We will see explicitly how the presence of many dimensions
considerably changes the standard analysis.
The paper is organized as follows: In Section 2, we review some of the basic features of BE
pertinent to the analysis contained below. In Section 3, we introduce two simple models
of a random walk in a random environment (RWRE), and review our earlier results within the
context of these models. In Section 4, we analyze the behavior of broken ergodicity in these
models, in both one and high dimensions. In Section 5, we discuss and summarize these
results, and make a few brief remarks about experiment.
\medskip
\noindent 2. $\underline {\rm Broken\ Ergodicity}$.
\medskip
Because the phenomenon of broken ergodicity has been discussed at great
length in the literature, we here review only those aspects of it which are relevant for the
cases under consideration. The importance of non-ergodicity in disordered systems,
particularly spin glasses, was emphasized early on by Anderson\refto {PWA1,PWA2}. Early analyses
and applications were given by J\"ackle,\refto{Jackle} Palmer\refto{Palmer1, Palmer2}, and van
Enter and van Hemmen\refto{vEvH}. The presentation by Palmer is especially comprehensive and
accessible; most of the discussion in this section follows his treatment. We are concerned here
only with some of the central ideas of BE; for a complete overview, we refer the reader to
the above papers.
We are primarily interested in cases where ergodicity is broken because the observational
timescale ($\tobs$) falls within a continuum of relaxational or equilibrational timescales
intrinsic to the system, as is commonly believed to occur for glasses and spin glasses.
(See \Ref{Palmer1} and \Ref{vEvH} for other examples, including the more familiar situation
of broken symmetry.) This may occur either in the presence or absence of a phase
transition. The former is typically indicated by the state space breaking up into two or
more disjoint components, separated by free energy barriers which diverge in the
thermodynamic limit. Broken ergodicity can and does occur, however, when the
system possesses {\it metastable\/} states surrounded by {\it finite\/} free energy barriers.
Because the typical timescale for escape from a metastable state grows exponentially with the
barrier, these need not be large for ergodicity to be broken on laboratory timescales.
The central idea is that state space can be decomposed into {\it components\/} which
are not necessarily intrinsic to the system, but rather depend on the timescale. Components
are defined by the probability of confinement on some timescale $\tau$: if the system is in a given
component at time $0$, then the probability that it has not escaped from the
component by time $\tau$ is greater than some specified (fixed)
probability $p_0$.\refto{Palmer1} Clearly, the definition of component depends both on the
specified timescale $\tau$ and the probability $p_0$. It is also assumed that on the same
timescale, the system is ergodic {\it within\/} the component; \ie the system visits a
representative sampling of states within the component so that the state space average
equals the time average, so long as one confines the averaging to states within the
component.\refto{Palmer1}
What is the confinement mechanism? We are interested here in {\it structural\/} confinement
mechanisms rather than dynamical (\ie the existence of possible constants of the motion.)
In the former case, the system is confined to a component because the smallest free energy
barrier that must be surmounted in order to escape corresponds to an escape time large
compared to the observational time. The standard picture envisions a very mountainous
terrain, with a series of isolated lakes and puddles in various valleys. The
``water level'' corresponds to the largest free energy scale which
the system can sample on a given temperature and time scale. If temperature is
held fixed, and time is allowed to increase, the water level steadily rises, with lakes
merging into bigger lakes and bigger lakes into oceans, leading to a hierarchical merging of
components\refto{Krey, Palmer1, Palmer2, PS, Sibani}; see Fig.~1. (One can arrive at the same
picture by fixing time $t$ and letting temperature $T$ increase; the height of the water
level scales as $T\log t$.)
Until the system surmounts the highest barriers, ergodicity remains broken; as soon as the
system surmounts some free energy barrier, it finds itself in a larger component which is
confined by higher free energy barriers.\refto{Palmer1, Palmer2, PS} So the confining free
energy barriers which the system must surmount, at fixed temperature, increase
logarithmically with the time. Also, because the system is now ergodic within the larger
component, it will continually revisit the previous smaller one, which is now a subset of
the portion of state space which it currently explores.
We will now examine these {\it ans\"atze\/} in two specific
models, both of which possess a continuum of free energy barriers --- and therefore a
continuum of relaxational timescales. We will find that the above
picture needs to be modified in several respects: while it
precisely describes a {\it one-\/}dimensional version of our models,
there are important differences in higher dimensions.
These high-dimensional models are indeed the relevant ones, since within the context of BE
one is usually referring to the evolution of a system in some high-dimensional state
space. As in I, one often models this state space as some graph ${\cal G}$.
The vertices of the graph correspond to the states themselves, and the
edges correspond to transitional paths between pairs of states. The
dimensionality of the graph scales with $N$, the number of degrees of
freedom in the system. We will follow this general procedure here.
We begin with a description of our dynamical models, which are just
specific examples of a much studied process -- the random walk in a random
environment.\reftorange{RWRE}{Sinai, PV}{DL}
\medskip
\noindent 3. $\underline {\rm Inhomogeneous\ Random\ Walk\ as\ Invasion\ Percolation}$.
\medskip
In this section we review the results of earlier work. We consider two different dynamical
models within the overall context of the RWRE:
1) Model A --- ``edge'' model. Here we consider a graph ${\cal G}$ in which the sites correspond
to states and the (nondirected) edges to the dynamical pathways which connect them. For
specificity, we take ${\cal G}$ to be the lattice $Z^d$, although this is unnecessary for our
results.\refto{hypercube} We assign nonnegative, independent, identically distributed random
variables to each of the edges; these represent the energy barriers which must be surmounted in order
to travel between pairs of sites connected via the edges. (We assume that the distribution of
these variables is continuous so that all barriers have distinct energies.)
If $W_{xy}=W_{yx}$ is the value assigned the edge
connecting sites $x$ and $y$, then the rate to travel {\it in either direction\/} between
$x$ and $y$ is taken to scale with inverse temperature $\b$ as
$$
r_{xy}(\beta)\sim\exp[-\b W_{xy}]\quad . \eqno(3.1)
$$
2) Model B --- ``site'' model. Here we assign random variables to {\it both\/} sites
and edges of ${\cal G}=Z^d$. Because we wish to view each site as corresponding to a locally stable
state (\ie as the minimum energy configuration within a ``valley''), and each edge as
again corresponding to an energy barrier, the values assigned to sites and edges cannot be
identically distributed --- both sites touching an edge must have lower assigned energy
values than that of the edge itself. (As in model A, we simplify matters by choosing each distinct edge
to connect a single pair of sites.) A simple way to implement this
is to choose the site variables independently from a single negative distribution, and the
edge variables independently from a single positive distribution. However, if
the model is to correspond to any sort of physical (random) potential, the
distribution for the site variables, whatever its form, {\it must be bounded from below.\/}
This is relevant to the analysis given below. The distribution for the edge
variables, of course, need not be bounded from above.
If $W_x$ is the variable assigned to site $x$, then the equilibrium
probability density over sites $\pi_x(\b)$ scales with $\b$ as $$ \pi_x(\b)\sim\exp[-\b
W_x]\quad . \eqno(3.2) $$ Detailed balance then requires that
$$
\pi_x(\beta)r_{xy}(\beta)=\pi_y(\beta)r_{yx}(\beta)\quad ,
\eqno(3.3)
$$
where $r_{xy}$ is understood as the rate to go from $x$ to $y$. The rates,
satisfying \eq{3.3}, are chosen so that
$$
r_{xy}(\beta)\sim\exp\left[-\beta(W_{xy}-W_x)\right] \eqno(3.4)
$$
and
$$
r_{yx}(\beta)\sim\exp\left[-\beta(W_{xy}-W_y)\right]\quad .
\eqno(3.5)
$$
We note that for detailed balance to hold in model A, the probability density over sites
must be site-independent; that is, model A corresponds to model B
with all energy minima degenerate. Despite the extreme simplicity of model A, it has a rich
and suggestive dynamical behavior, and we therefore include it in our analysis.
Our main result in I is a rigorous statement about the {\it order in which sites are visited
for the first time\/}, given an arbitrary starting site. We will see later that it is easily
extended to a result about which sites are visited (or not visited) on a given timescale, and
the nature of that process, which is of central interest in a BE treatment.
The assignment of random variables in both models defines an ordering on the
(undirected) edges of ${\cal G}$ in
which $\{x,y\}<\{x',y'\}$ if $W_{xy}8$, there are {\it
infinitely\/} many disjoint invasion regions. That is, given two starting
sites far from each other, their invasion regions will totally miss each
other with high probability. The asymptotic dimension of these invasion
regions in high $d$ is four.
There is a picturesque way to view this: Let $p_c$ denote the critical value
for independent bond percolation on $Z^d$ and let $w_c$ denote the energy level
such that Prob$\ (W_{xy}\le w_c)=p_c$. Then from any starting site the invasion
process, as time increases, focuses on the so-called incipient infinite
cluster at $p_c$ in the corresponding independent bond
percolation problem. Once the process finds an infinite cluster of edges
with energy levels $\le w_0$, where $w_0>w_c$, it never
again crosses an edge with $W_{xy}>w_0$. One can then consider the invasion
process from any point as following a ``path'' which eventually leads to
``the sea'' at infinity. In less than eight dimensions, all invasion regions from different points eventually
follow the same path to the sea. Along the way, all individual paths merge, some sooner,
some later.
However, in greater than eight dimensions, there are an infinite number of
disjoint paths to infinity. Indeed one should think of infinitely many
distinct seas, each of which has many tributaries,
\ie invasion regions which flow into it, or equivalently, an (infinite) set of sites whose
invasion regions connect to it. The process flows into one of these seas,
and {\it it will never visit any sites which connect, via the invasion process, to any of the
other seas.\/} Because models such as A and B are often used to (abstractly) describe state
spaces in very high dimensions, this picture is a crucial component in what follows.
We close this section with an important remark about the theorem from I described above. It is
well known that the RWRE asymptotically approaches
ordinary diffusion at long times.\refto{PV} (This is also the case above two
dimensions for RWRE's which, unlike those treated here, do not
satisfy detailed balance\refto{DL}. It need not be the case in such models
for one dimension or in detailed balance models with sufficiently correlated
environments\refto{Sinai}.) But our picture seems to contradict the diffusion picture. In fact,
both are consistent, because each corresponds to a different method of taking the limits
time$\to\infty$ and $\beta\to\infty$, and the behavior of each model is sensitive to this.
Previous treatments\refto{PV, DL} studied the case where temperature is fixed and time
goes to infinity; in that case, the RWRE will exhibit normal diffusive behavior.
In our picture, we first focus on a particular site $y_0$. Suppose that $y_0$
is the $157^{\rm th}$ site invaded by the invasion process described earlier.
Our theorem states that, as $\b\to\infty$, the probability that $y_0$ is also the
$157^{\rm th}$ site visited by the RWRE converges to one.
This implies that there exists a temperature-dependent
timescale --- an ergodic time, so to speak --- beyond which our picture breaks down and
normal diffusion takes over (or equivalently, ergodicity is restored).\refto{erg} The ergodic time
diverges as temperature goes to zero. A timescale of this type is
a common feature in most systems which break ergodicity. We will discuss this further in the following
sections, but meanwhile note the rigorous illustration, in a specific model, of an important
feature\refto{Palmer1} of broken ergodicity --- the way in which limits are taken is crucial!
\medskip
\noindent 4. $\underline {\rm Broken\ Ergodicity\ in\ the\ RWRE}$.
\medskip
\noindent A. {\it One-Dimensional Picture}
We first consider the RWRE in one dimension. It will be sufficient to
consider only Model A in this case, because here there is no significant
qualitative difference between the two models. This is not quite true
in higher dimensions.
For specificity, let the edge random variables, which correspond to
barriers, be chosen independently from the positive half of a Gaussian
distribution with mean zero and variance one. Consider
the behavior of the diffusing particle for large time and low temperature. It is easy to see
that in this case, all of the assertions made in Section 2 are correct
after some initial transient time (the larger $\b$ is, the shorter this
transient time becomes). On some observational timescale $\tobs$, the
particle is trapped with high probability between two barriers, neither of
which are surmountable (with some prespecified probability) on a timescale
of the order of $\tobs$. If one is willing to wait considerably longer (on
a logarithmic timescale), then the length of the line segment the particle
explores is correspondingly larger, surrounded at each end by suitably
large barriers. It is not hard to show that these grow in the manner
specifed in \Ref{Palmer1} and \Ref{PS}: $\Delta F_{\rm esc}\sim\log\tobs$.
If one were to watch a greatly speeded up movie
of the particle motion, it would look something like the following. After
diffusing to the right (say) some distance, the particle encounters a
barrier significantly larger (compared to $1/\b$) than any it has
previously crossed. The particle is effectively reflected to the left,
where it undergoes a net diffusive motion until it encounters a new
barrier significantly larger than any previous ones, including the
original reflecting barrier. (Prior to this, however, it may have
encountered barriers smaller than the first reflecting barrier but larger
than any others and subsequently have bounced back and forth a number of
times.) The particle ``reflects'' off this barrier and begins a net
diffusive motion to the right. Eventually, well to the right of the first
reflecting barrier, it encounters a new barrier of yet greater magnitude
than any previous ones, which reflects it back to the left, and so on.
Informally, the process resembles a game of diffusive ping-pong with
asymmetrically receding paddles.
\medskip
In two and higher dimensions, the picture changes dramatically. We will
see that in both models A and B, several of the standard BE assumptions break down.
Among the most important of these is that as time increases, while
components grow larger, {\it they do not contain previously visited
portions of state space}. Perhaps more surprisingly, in Model A the
confining barriers (\ie outlets --- see below) do not increase with time; they instead {\it
decrease}, asymptotically approaching a constant from above. In Model B, a constant
barrier value is also approached (although not necessarily monotonically).\refto{Note} In neither case do the barriers grow
logarithmically with time. In order to see where these surprising features come from, we
return to a more extensive discussion of invasion percolation before examining the models.
\medskip
\noindent B. {\it Ponds and Outlets}
\medskip
We briefly digress from our discussion of broken ergodicity in the RWRE
to examine the process whereby invasion percolation ``finds a path'' to
infinity. In accordance with our theorem proved in I, this will be
equivalent to the behavior of the diffusing particle in the RWRE under
an appropriate range of temperature and timescale. The picture
presented in this section holds irrespective of whether there is one or
infinitely many disjoint invasion regions.
We first present the standard argument which connects the asymptotic
geometry of the invasion region to that of the incipient infinite
cluster at $p_c$ in the corresponding independent percolation problem.
(See Appendix A.) We consider bond percolation on $Z^d$ in both cases.
Hereafter the term ``invasion region'' should be understood to mean
``invasion region starting from some arbitrarily chosen initial point
$x_0$''. As in Appendix A, we can and do confine ourselves to the case
where the bond, or edge, variables are chosen independently from the uniform
distribution on $[0,1]$.
Given $x_0$, and a configuration of the bond variables, there exists a unique invasion route to infinity. Consider
all bonds whose values (\ie magnitudes of assigned random variables) are
smaller than some $p_1>p_c$. By correspondence with the associated
independent bond percolation problem, these comprise a unique infinite
cluster, in addition to finite clusters of varying
sizes.\refto{Grimmett} Therefore, once the invasion process reaches {\it
any\/} of the bonds within this infinite cluster, {\it it will never
again cross any bond whose value is greater than $p_1$\/}.
Consider next all bonds whose values are less than some $p_2$, where
$p_c