
% Generic article:
\documentclass[12pt]{article} 

% Packages:

\usepackage{latexsym} % Gets \Box etc
\usepackage{amssymb}  % \gtrsim, \geqslant, etc etc: 
                      % see /opt/texmf/tex/ams/doc/amsguide.ps
%\usepackage{amsfonts} % \mathfrak and \mathbb{x} (Blackboard bold)
% \usepackage{amsbsy}   % \pmb and \boldsymbol
%% \usepackage{amsmath} % Screws up \beq and \eeq
%% \usepackage{amstex} 
\usepackage{epsfig}       % For PostScript figures
%\usepackage{rotate}    % rotates PostScript figures
%\usepackage{multirow}  % for multirows in tables
%\usepackage[dvips]{color}

%====== Draft mode
% use this for draft mode
\long\def\draftmode#1{#1}
% use this for final version
% \long\def\draftmode#1{}

\long\def\myput#1{{\draftmode{\bf} #1}}


%============================================================
%  Generic Abbreviations:
%============================================================

\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\ze}{\zeta}
\newcommand{\ka}{\kappa}
\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\ph}{\varphi}
\newcommand{\del}{\nabla}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\renewcommand{\th}{\theta}   % LaTeX: \th already defined
\newcommand{\Up}{\Upsilon}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}

%
\newcommand{\imp}{~\Rightarrow}
%\newcommand{\to}{\rightarrow}
\newcommand{\p}{\partial}
\newcommand{\<}{\langle} 
\renewcommand{\>}{\rangle} % LaTeX: \> already defined
\newcommand{\ul}{\underline}
\newcommand{\txt}{\textstyle}
\newcommand{\dsp}{\displaystyle}
\newcommand{\ns}{\normalsize}
\newcommand{\ad}{\dagger}
\newcommand\eqn[1]{(\ref{#1})}      % parentheses around the LaTex "ref" macro
\newcommand\Eqn[1]{Eq.~(\ref{#1})}  % includes ``Eq.'' in front
\newcommand{\e}{ {\rm e} }
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\ea}{\end{array}}
\newcommand{\eea}{\end{eqnarray}}

\renewcommand{\slash}{\!\!\!\!/\,}
\newcommand{\sslash}{\!\!\!/\,}
\newcommand{\dotprod}{\!\cdot\!}
\newcommand{\vfilll}{\vskip 0pt plus 1filll}
\newcommand{\ol}{\overline}
\newcommand{\nl}{\hfil\break}
\newcommand{\ee}[1]{\times 10^{#1}}
\newcommand\comment[1]{ \hbox{[{\it Comment suppressed here.}\/]} }
\newcommand\hide[1]{}

% Common math/phys abbreviations
\renewcommand{\O}{ {\cal O} }
\newcommand{\tr}{\hbox{tr}}
\newcommand{\Tr}{\hbox{Tr}}
\newcommand{\hc}{ {\rm h.c.} }
\renewcommand{\Re}{ {\rm Re}\, }
\renewcommand{\Im}{ {\rm Im}\, }
\newcommand{\ie}{{i.e.}}
\newcommand{\eg}{{e.g.}}
%\def\sc{\scriptstyle}
%\def\scsc{\scriptscriptstyle}
\newcommand{\seq}{\!=\!}        % for eqs in text

%\newcommand{\bx}{{\bf x}}
%\newcommand{\by}{{\bf y}}
%\newcommand{\bz}{{\bf z}}
%\newcommand{\bp}{{\bf p}}
%\newcommand{\bq}{{\bf q}}
%\newcommand{\bk}{{\bf k}}
%\newcommand{\bv}{{\bf v}}
\newcommand{\bx}{{\vec x}}
\newcommand{\by}{{\vec y}}
\newcommand{\bz}{{\vec z}}
\newcommand{\bp}{{\vec p}}
\newcommand{\bq}{{\vec q}}
\newcommand{\bk}{{\vec k}}
\newcommand{\bv}{{\vec v}}

\newcommand{\bA}{{\bf A}}
\newcommand{\bC}{{\bf C}}

\newcommand{\skipover}[1]{}
\newcommand{\nn}{\nonumber \\}
\newcommand{\order}{{\cal O}}
\newcommand{\C}{{\cal C}}
\newcommand{\half} {{\txt {1\over 2}}}
\newcommand{\third}{{\txt {1\over 3}}}
\newcommand{\twothirds}{{\txt {2\over 3}}}
\newcommand{\sixth}{{\txt {1\over 6}}}
\newcommand{\eighth}{{\txt {1\over 8}}}

% 1/3 neg thin space, for fine-tuning formulae
\def\back{\mskip-.333\thinmuskip}
% Phantom minus sign: very useful for lining things up!
\def\phm{\phantom{-}}

% Otherwise emacs font lock goes crazy:
\newcommand{\percent}{\symbol{'045}}
\newcommand{\dollar}{\symbol{'044}}

%
% "less than or approx" can be done better by 
% \lesssim and \gtrsim from \usepackage{amssymb}
%\newcommand{\lapp}{ {\txt {{\txt <} \atop {\txt \sim}}} }
%\newcommand{\gapp}{ {\txt {{\txt >} \atop {\txt \sim}}} }
%
%
\pretolerance=10000  %No hyphens
\hbadness=2000  %I don't want to hear about underfull hboxes
%\setlength{\arraycolsep}{0.2em} % tighten up arrays for the whole paper

% ============================================================
% Structural customizations:
% ============================================================

\makeatletter %\catcode`\@=11

% 1) Make appendices look normal

\def\appendix{\par                              % Have \appendix say
    \setcounter{section}{0}                     % `Appendix A', not just `A'
    \setcounter{subsection}{0}
    \renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
    \renewcommand{\thesection}{Appendix \Alph{section}}
}

% If you want \ref{app:xxx} to give "A" rather than "Appendix A" then
% use \applabel{app:xxx} to define it instead of \label{app:xxx}
\def\applabel#1{\@bsphack
  \protected@write\@auxout{}%
         {\string\newlabel{#1}{{\Alph{section}}{\thepage}}}%
  \@esphack}
% Use \applabel instead of \label to label appendices.

% 2) Make section, subsection etc headers  smaller.

\def\section{
\setcounter{equation}{0}        % Reset eqn numbers at start of section
\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus 
 -.2ex}{2.3ex plus .2ex}{\large\bf}}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

\def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus 
 -.2ex}{1.5ex plus .2ex}{\normalsize\bf}}

\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-3.25ex plus
 -1ex minus -.2ex}{1.5ex plus .2ex}{\normalsize}}

\makeatother   %\catcode`\@=12

% 3)
%% ===========================================================  M. Alford
%% ============ Showing eqn labels in draft mode: ============  Oct 1995 
%% 
%%
%% Use \beql{chosen label} and \eeql  to begin & end labelled equations.
%% Comment out the indicated line to suppress printing of labels in margin.
%%
\newsavebox{\eqlabel}
%% Define eqn number macro to output the contents of \eqlabel in right margin
%% \eqlabel will have been set by the equationwithlabel environment (see below)

\makeatletter  %\catcode`\@=11
\newlength{\numblen}
\newsavebox{\eqnumb}
%%  base latex:   \def\@eqnnum{\savebox{\eqnumb}{\rm (\theequation)}}
%% If latex, use this:
\def\@eqnnum{\savebox{\eqnumb}{\rm (\theequation)}%
%%  base amstex:  \def\@eqnnum{{\normalfont\normalcolor \tagform@\theequation}}
%% If amstex use this:
%\def\@eqnnum{\savebox{\eqnumb}{\normalfont\normalcolor\tagform@\theequation}%
\settowidth{\numblen}{\usebox{\eqnumb}}%
\makebox[\numblen][l]{\usebox{\eqnumb}~~~\usebox{\eqlabel}}}
\makeatother   %\catcode`\@=12

\newenvironment{equationwithlabel}[1]{ %
%%
%% >>>>>>>>>>>> Comment the next line out to suppress labels <<<<<<<<<<<
  \savebox{\eqlabel}{#1}
  \begin{equation}\label{#1} }{\end{equation}} %\savebox{\eqlabel}{~}}
\newcommand{\beql}[1]{\begin{equationwithlabel}{#1}}
\newcommand{\eeql}{\end{equationwithlabel}}
%%
%%
%% =========== End of showing eqn labels in draft mode ===========  
%% ===============================================================




% ************************************************************************
% **************************** DOCUMENT BEGINS ***************************
% ************************************************************************


\begin{document}

\title{\bf Thermalization of fermionic\\
quantum fields\\[2.ex]}


\author{
J\"urgen Berges\thanks{email: j.berges@thphys.uni-heidelberg.de} 
$\,^a$,\addtocounter{footnote}{1}
Szabolcs Bors\'anyi\thanks{email: mazsx@cleopatra.elte.hu} $\,^{a,b}$ 
and
Julien Serreau\thanks{email: serreau@thphys.uni-heidelberg.de} $\,^a$
\\
[2.ex]
\normalsize{$^a$ Universit\"at Heidelberg, Institut f\"ur 
Theoretische Physik}\\
\normalsize{Philosophenweg 16, 69120 Heidelberg, Germany}\\
[1.ex]
\normalsize{$^b$ E{\"o}tv{\"o}s University, Department of Atomic Physics}\\
\normalsize{H-1117, Budapest, Hungary}\\
}


\newcommand{\preprintno}{
\normalsize HD-THEP-02-47
}
\date{\normalsize{December 30, 2002 $\,\,$}
\\[1ex] 
\preprintno}

\begin{titlepage}
\maketitle
\def\thepage{}          % No page number on title page

\vspace*{-0.5cm}
\begin{abstract}
%       1         2         3         4         5         6 
We solve the nonequilibrium dynamics of a $3+1$ dimensional theory 
with Dirac fermions coupled to scalars via a chirally invariant Yukawa 
interaction. The results are obtained from a systematic coupling expansion 
of the 2PI effective action to lowest non-trivial order, which includes 
scattering as well as memory and off-shell effects. The microscopic dynamics 
is solved numerically without further approximation, for given 
far-from-equilibrium initial conditions. The late time behavior is 
demonstrated to be insensitive to the details of the initial conditions
and to be uniquely determined by the initial energy density.
Moreover, we show that at late time, the system is very well
characterized by a thermal ensemble. In particular, we are able to 
observe the onset of Fermi--Dirac and Bose--Einstein statistics from 
the nonequilibrium dynamics.
\end{abstract}


\end{titlepage}

\renewcommand{\thepage}{\arabic{page}}
%\setcounter{page}{1}

%--------------------------------------- 
                        % Body of paper begins


\section{Introduction}
\label{soverview}

The abundance of experimental data on matter in extreme conditions from
relativistic heavy-ion collision experiments, as well as applications in
astrophysics and cosmology lead to a strong increase of interest in the
dynamics of quantum fields out of equilibrium. Experimental indications 
of thermalization in collision experiments and the justification of current
predictions based on equilibrium thermodynamics, local equilibrium or
\mbox{(non--)linear} response pose major open questions for our theoretical
understanding~\cite{Braun-Munzinger:2001mh,Serreau:2002yr}. One way to resolve 
these questions is to try to understand quantitatively the far-from-equilibrium 
dynamics of quantum fields without relying neither on the assumption of small 
departures from equilibrium, nor on a possible separation of scales, which is at 
the basis of effective kinetic descriptions \cite{Blaizot:2001nr}. In contrast to 
close-to-equilibrium approaches, the quantum-statistical fluctuations 
of the fields are not assumed to be described by a thermal ensemble. 
Moreover, one goes beyond the range of applicability of the usual gradient 
expansion and of the dilute gas approximation.


In contrast to thermal equilibrium, which keeps no
information about the past, nonequilibrium dynamics poses 
an initial value problem: time-translation invariance is explicitly 
broken by the presence of the initial time, where the system has 
been prepared. The question of 
thermalization investigates how the system effectively looses the 
dependence on the details of the initial condition, and becomes 
approximately time-translation invariant at late times.
According to basic principles of equilibrium thermodynamics,
the thermal solution is universal in the sense that it is independent of the
details of the initial condition and is uniquely determined by the values of
the (conserved) energy density and of possible conserved
charges.\footnote{Here we consider closed systems without coupling to a heat
bath or external fields, which could provide sources or sinks of energy.} 
There are two distinct classes of universal behavior, corresponding to 
bosonic and fermionic statistics respectively.

The description of the effective loss of initial conditions and subsequent
approach to thermal equilibrium in quantum field theory requires calculations
beyond so--called ``gaussian'' (leading-order large--$N$, Hartree or mean-field
type) approximations \cite{gaussian1,gaussian2a,gaussian2b}. Similar to the
free-field theory limit, these approximations typically exhibit an infinite
number of additional conserved quantities which are not present in the
underlying interacting theory \cite{Cooper:1996ii,Aarts:2001wi}.  These
spurious constants of motion constrain the time evolution and lead to a
non-universal late-time behavior \cite{Berges:2001fi}. On the contrary, it has
been recently demonstrated in the context of scalar field theories, that the
approach to quantum thermal equilibrium can be described by going beyond these
approximations \cite{Berges:2000ur,Berges:2001fi}. In particular, this has been
achieved by using a systematic coupling expansion up to three loop order
\cite{Berges:2000ur} or a $1/N$--expansion at next-to-leading-order
\cite{Berges:2001fi,Aarts:2002dj} of the two-particle irreducible generating
functional for Green's functions, the so-called 2PI effective action
\cite{Baym,Cornwall:1974vz,Calzetta:1988cq,Ivanov:1998nv}.  In this
context, the corresponding microscopic equations of motion have been shown to
lead to a universal late time behavior, in the sense mentioned above, without
need to have recourse to any kind of coarse-graining.

In this work, we study the far-from-equilibrium time evolution of relativistic
fermionic fields and their subsequent approach to thermal equilibrium. 
Nonequilibrium behavior of fermionic fields has been
previously addressed in several scenarios
\cite{gaussian2a,gaussian2b,Danielewicz:kk,Greene:2000ew,Joyce:2000ed,Lawrie:2000jg},
including the full dynamical problem with fermions coupled to inhomogeneous
{\em classical} bosonic fields \cite{Aarts:1998td}.  Here we go beyond
these approximations and compute the nonequilibrium evolution in a $3+1$
dimensional {\em quantum} field theory of Dirac fermions coupled to scalars in
a chirally invariant way.  As a consequence, we are able to study the approach
to quantum thermal equilibrium. For the considered
nonequilibrium initial conditions we find that the late-time behavior is
universal and characterized by Fermi--Dirac and Bose--Einstein statistics,
respectively.  The results are obtained from a systematic coupling expansion of
the 2PI effective action to lowest nontrivial order, which includes scattering
as well as memory and off-shell effects.  The nonequilibrium dynamics is 
solved numerically without further approximations. We emphasize that, given 
the limitations of a weak-coupling expansion, this is a first-principle 
calculation with no other input than the dynamics dictated by
the considered quantum field theory for given initial conditions.  

In Sect.~\ref{Sect2PIeffaction}, we review the 2PI effective action for 
fermions, which we use to derive exact time evolution equations for the 
spectral function and the statistical two-point function in 
Sect.~\ref{SectevolrhoF}. We discuss the Lorentz structure of our 
equations in Sect.~\ref{sec:lorentz} and exploit some symmetries in 
Sect.~\ref{sec:symmetries}. 
In Sect.~\ref{sec:model}, we specify to a chiral quark
model for which we solve the nonequilibrium dynamics. 
The initial conditions are discussed in Sect.~\ref{sec:initial}
and some details concerning the numerical implementation are
given in Sect.~\ref{sec:numerics}. The numerical results are 
presented and discussed in Sect.~\ref{Sectfarfromequilibrium} 
and Sect.~\ref{sec:statistics}. We attached two appendices discussing
in details the quasiparticle picture we used to interprete some of
our results.


\section{2PI effective action for fermions}
\label{Sect2PIeffaction}

We consider first a purely fermionic quantum field theory with classical action
\beq
S = \int {\rm d}^4 x \Big(\bar{\psi}_i(x) [ i \partial\,\slash - m_f ] 
\psi_i(x) 
+ V(\bar{\psi},\psi) \Big) \, 
\label{fermact}
\eeq
for $i = 1,\ldots,N_f$ ``flavors'' of Dirac fermions $\psi_i$, a mass parameter
$m_f$ and an interaction term $V(\bar{\psi},\psi)$ to be specified below. Here
$\partial\,\slash \equiv \gamma^\mu \partial_\mu$, with Dirac matrices
$\gamma_\mu$ ($\mu =0,\ldots,3$). Summation over repeated indices and
contraction in Dirac space is implied.  All correlation functions of the
quantum theory can be obtained from the corresponding two-particle-irreducible
(2PI) effective action $\Gamma$. For the relevant case of a vanishing fermionic
``background'' field the 2PI effective action can be written as
\cite{Cornwall:1974vz}
\beq
\Gamma[D] = -i \Tr\ln D^{-1} 
          -i \Tr\, D_0^{-1} D
          + \Gamma_2[D] + {\rm const} \, .
\label{fermioneffact}
\eeq
The exact expression for the functional $\Gamma_2[D]$ contains all 2PI
diagrams with vertices described by $V(\bar{\psi},\psi)$ and propagator 
lines associated to the full fermionic connected two-point function
$D$. In coordinate space the trace $\Tr$ include an integration over a  
closed time path $\C$ along the real axis \cite{Schwinger:1961qe},
as well as integration over spatial coordinates and summation over
flavor and Dirac indices. The free inverse propagator is given by
\beq
i D_{0,ij}^{-1} (x,y) 
= (i \partial\,\slash - m_f )\, \delta^4_{\cal C}(x-y)\, \delta_{ij} \, .
\eeq  
The equation of motion for $D$ in absence of external sources is 
obtained by extremizing the effective action \cite{Cornwall:1974vz}
\beq
\frac{\delta\Gamma[D]}{\delta D_{ij}(x,y)} = 0
\label{fermstat}
\eeq 
According to (\ref{fermioneffact}) one
can write (\ref{fermstat}) as an equation for the propagator
\beq
D_{ij}^{-1}(x,y) = D_{0,ij}^{-1}(x,y) - \Sigma_{ij}(x,y;D) \, ,
\label{fermSD}
\eeq
with the proper self--energy 
\beq 
\Sigma_{ij}(x,y;D) \equiv -i \frac{\delta \Gamma_2[D]}{\delta D_{ji}(y,x)} \, .
\eeq  
Equation (\ref{fermSD}) can be rewritten in a form suitable for an initial 
value problems by convoluting it with $D$. One obtains the following time 
evolution equation for the propagator:
\beq
(i {\partial\,\slash}_{\!x} - m_f ) D_{ij}(x,y) 
- i \int_z \Sigma_{ik}(x,z;D) D_{kj}(z,y) =
i \delta_\C^4(x-y) \delta_{ij}  \,  ,
\label{evolutionC}
\eeq
where we employed the shorthand notation
$\int_z = \int_\C {\rm d}z^0 \int {\rm d}\bz$.
To keep the notation clear, we omit Dirac indices and 
reserve the latin indices $i,j,k,\ldots$ to denote flavor. 


\section{Exact evolution equations for the spectral 
and statistical components of the two-point function}
\label{SectevolrhoF}

To simplify physical interpretation we rewrite (\ref{evolutionC}) 
in terms of equivalent equations for the spectral function, which 
contains the information about the spectrum of the theory,
and the statistical two-point function. The latter will,
in particular, provide an effective description of occupation numbers.
For this we write for the time-ordered two-point function $D_{ij}(x,y)$
and proper self-energy $\Sigma_{ij}(x,y;D)$:\footnote{
If there is a local contribution to the proper self-energy,
we write 
\bea
\Sigma_{ij}(x,y;D) = - i\, 
\Sigma^{\rm (local)}(x;D)\,\delta_{\C}^4(x-y)\,\delta_{ij}
+ \Sigma_{ij}^{\rm (nonlocal)}(x,y;D) \, ,\nonumber
\eea
and the decomposition (\ref{sigbs}) is taken for 
$\Sigma^{\rm (nonlocal)}(x,y;D)$. In this case the local contribution
gives rise to an effective space-time dependent fermion mass term
$\sim m_f + \Sigma^{\rm (local)}(x;D)$.}  
\bea
D_{ij}(x,y)&=& \Theta_{\C}(x^0-y^0) \, D_{ij}^>(x,y) -
\Theta_{\C}(y^0-x^0) \, D_{ij}^<(x,y) \, ,
\label{Dbs}\\[0.2cm]
\Sigma_{ij}(x,y;D)&=&\Theta_{\C}(x^0-y^0) \, \Sigma_{ij}^>(x,y) -
\Theta_{\C}(y^0-x^0) \, \Sigma_{ij}^<(x,y) \label{sigbs} \, .
\eea
Note that for convenience we omit the explicit $D$--dependence in  $\Sigma_{ij}^{>,<}$. 
Inserting the above decompositions into the evolution
equation (\ref{evolutionC}) we
obtain evolution equations for the functions $D_{ij}^>(x,y)$ and
$D_{ij}^<(x,y)$ as well as the identity
\beq
\gamma^0 \left(D_{ij}^>(x,y) + D_{ij}^<(x,y) \right)|_{x^0=y^0} 
= \delta(\bx-\by) \delta_{ij} \, ,
\label{anticomrel}
\eeq
which corresponds to the anticommutation relation for fermionic field
operators. For later use we also note the hermiticity property 
\beq
\left(D_{ji}^>(y,x)\right)^\dagger
=\gamma^0 D_{ij}^>(x,y) \gamma^0 \, , 
\label{hermiticityD}
\eeq
and equivalently for $D_{ij}^<(x,y)$.
Note that here the hermitean conjugation involves complex conjugation and 
taking the transpose in Dirac space only.

Similar to the discussion for scalar fields in
Refs.~\cite{Berges:2001fi,Aarts:2001qa} we introduce the spectral function
$\rho_{ij}(x,y)$ and the statistical propagator $F_{ij}(x,y)$ defined
as\footnote{Equivalently, one can decompose
$$
D_{ij}(x,y) = F_{ij}(x,y) - \frac{i}{2}\, \rho_{ij}(x,y) 
\left[\Theta_\C(x^0-y^0) - \Theta_\C(y^0-x^0) \right] \, . 
$$
}
\bea
\rho_{ij}(x,y) &=&i\, \Big(D_{ij}^>(x,y) + D_{ij}^<(x,y)\Big)\, , \\
F_{ij}(x,y) &=& \frac{1}{2}\, \Big(D_{ij}^>(x,y)-D_{ij}^<(x,y)\Big) \, . 
\eea
The corresponding components of the self-energy are given by\footnote{ Besides
the dynamical field degrees of freedom $D_{ij}$ we introduce quantities which
are functions of these fields. These functions are denoted by either boldface
or Greek letters.}
\bea
\bA_{ij}(x,y) &=&i\, \Big(\Sigma_{ij}^>(x,y) + \Sigma_{ij}^<(x,y)\Big)
\, , \\
\bC_{ij}(x,y) &=& \frac{1}{2}\, 
\Big(\Sigma_{ij}^>(x,y)-\Sigma_{ij}^<(x,y)\Big) \, . 
\eea
With (\ref{hermiticityD}) the two-point functions have the properties 
\bea
\left(\,\rho_{ji}(y,x)\,\right)^\dagger
&=& - \gamma^0 \rho_{ij}(x,y) \gamma^0 \, ,
\label{rhohermiticity}\\
\left(\,F_{ji}(y,x)\,\right)^\dagger
&=& \gamma^0 F_{ij}(x,y) \gamma^0 \, , \label{Fhermiticity} 
\eea
and equivalently for $\bA_{ij}(x,y)$ and $\bC_{ij}(x,y)$.

Using the above notations the evolution equation (\ref{evolutionC})
written for $\rho_{ij}(x,y)$ and $F_{ij}(x,y)$ are given by
\bea 
(i {\partial\,\slash}_{\!x} - m_f ) \rho_{ij} (x,y) &=& 
 \int_{y^0}^{x^0} {\rm d}z\,  \bA_{ik} (x,z)\rho_{kj} (z,y) \, , 
\label{rhoexact}\\
(i {\partial\,\slash}_{\!x} - m_f ) F_{ij}(x,y) &=& 
 \int_0^{x^0} {\rm d}z\, \bA_{ik}(x,z) F_{kj}(z,y)
\nonumber  \\ 
&-& \int_0^{y^0} dz\, \bC_{ik}(x,z)\rho_{kj}(z,y) \, , 
\label{Fexact}
\eea
where we have taken the initial time to be zero. For known self-energies the
equations (\ref{rhoexact}) and (\ref{Fexact}) are exact.  We note that the form
of their RHS is identical to the one for scalar fields \cite{Aarts:2001qa}.
To solve the evolution equations one has to specify initial conditions for the
two-point functions, which is equivalent to specifying a Gaussian initial
density matrix\footnote{We emphasize that a Gaussian initial density matrix
only restricts the initial conditions or the ``experimental setup'' and
represents no approximation for the time evolution.  More general initial
conditions can be discussed using additional source terms in defining the
generating functional for Green's functions 
(see \eg~\cite{Danielewicz:kk,Berges:2000ew}).}.
We note that the fermion anticommutation relation or (\ref{anticomrel})
uniquely specifies the initial condition for the spectral function:
\beq
\gamma^0 \rho(x,y)_{ij}|_{x^0=y^0} = i \delta(\bx-\by)\,\delta_{ij} \, .
\label{rhoinitial}
\eeq
Suitable nonequilibrium initial conditions for the statistical propagator
$F_{ij}(x,y)$ will be discussed below.


\section{Lorentz decomposition}
\label{sec:lorentz}

It is very useful to decompose the fields $\rho_{ij}(x,y)$ and $F_{ij}(x,y)$
into terms that have definite transformation properties under Lorentz
transformation. We will see below that, depending on the symmetry properties of
the initial state and interaction, a number of these terms remain identically
zero under the exact time evolution, which can dramatically simplify the
analysis. Using a standard basis and suppressing flavor indices we write
\beq
\rho = \rho_S + i \gamma_5 \rho_P + 
\gamma_\mu \rho_V^\mu + \gamma_\mu \gamma_5 \rho_A^\mu
+ \frac{1}{2} \sigma_{\mu\nu} \rho_T^{\mu\nu} \, ,
\label{rhodecomp}\\[0.1cm]
\eeq
where $\sigma_{\mu\nu} = \frac{i}{2}[\gamma_\mu,\gamma_\nu]$
and $\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3$.
For given flavor indices the 16 (pseudo-)scalar, (pseudo-)vector and 
tensor components 
\bea
 \rho_S &=& \tilde{\tr}\, \rho \, , \nonumber\\
 \rho_P &=& - i \tilde{\tr}\, \gamma_5 \rho \, , \nonumber\\
 \rho_V^\mu &=& \tilde{\tr}\, \gamma^\mu \rho \, , \label{project}\\
 \rho_A^\mu &=& \tilde{\tr}\, \gamma_5 \gamma^\mu \rho \, , \nonumber\\
 \rho_T^{\mu\nu} &=& \tilde{\tr}\, \sigma^{\mu\nu} \rho  \, , \nonumber
\eea 
are complex two-point functions.  Here we have defined $\tilde{\tr} \equiv \frac{1}{4}
\tr$ where the trace acts in Dirac space. Equivalently, there are  16 complex
components for $F_{ij}$, $\bA_{ij}$ and $\bC_{ij}$ for given flavor indices $i,j$.
Using (\ref{rhohermiticity}) and (\ref{Fhermiticity}), one sees that they obey
\beq
\label{herm}
\rho^{(\Gamma)}_{ij}(x,y)= - \left(\rho^{(\Gamma)}_{ji}(y,x) \right)^*
\quad , \quad F^{(\Gamma)}_{ij}(x,y)= 
\left(F^{(\Gamma)}_{ji}(y,x)\right)^* \, ,
\eeq
where $\Gamma = \{S,P,V,A,T\}$. Inserting the above decomposition 
into the evolution equations (\ref{rhoexact}) and (\ref{Fexact})  
one obtains the respective equations for the various
components Eq.\ (\ref{project}). 
 
For a more detailed discussion, we first consider the LHS of the evolution
equations (\ref{rhoexact}) and (\ref{Fexact}). In fact, the approximation of a
vanishing RHS corresponds to the standard mean--field or Hartree--type
approaches frequently discussed in the literature \cite{gaussian1,gaussian2a}.
However, to discuss thermalization we have to go beyond such a ``gaussian''
approximation: it is crucial to include direct scattering which is described
by the nonvanishing contributions from the RHS of the evolution equations.
Starting with the LHS of (\ref{rhoexact}) one finds      
\bea
 \tilde{\tr}\left[(i {\partial\,\slash} - m_f ) \rho \right] &=&
 \left(i \partial_\mu \rho_V^\mu\right) - m_f\, \rho_S \,\, ,
\label{rhoSL}\nonumber\\[0.3cm]
 -i \tilde{\tr}\left[\gamma_5 (i {\partial\,\slash} - m_f ) \rho \right] &=&
 - i \left(i \partial_\mu \rho_A^\mu\right) - m_f\, \rho_P \,\, ,
\label{rhoPL}\nonumber\\[0.3cm]
 \tilde{\tr}\left[\gamma^\mu (i {\partial\,\slash} - m_f ) \rho \right] &=&
 \left(i \partial^\mu \rho_S\right) 
 + i \left(i \partial_\nu \rho_T^{\nu\mu}\right) 
 - m_f\, \rho_V^\mu \,\, ,
\label{rhoVL}\\[0.1cm]
 \tilde{\tr}
\left[\gamma_5 \gamma^\mu (i {\partial\,\slash} - m_f ) \rho \right] &=&
 i \left(i \partial^\mu \rho_P\right) + \frac{1}{2}\, 
 \epsilon^{\mu\nu\gamma\delta} 
\left(i\partial_\nu \rho_{T,\gamma\delta}\right) 
 - m_f\, \rho_A^\mu \,\, ,
\label{rhoAL}\nonumber\\[0.1cm]
 \tilde{\tr}\left[\sigma^{\mu\nu} (i {\partial\,\slash} - m_f ) \rho \right] &=&
 -i \left(i\partial^\mu \rho_V^\nu - i \partial^\nu \rho_V^\mu\right)
 + \epsilon^{\mu\nu\gamma\delta} \left(i\partial_\gamma \rho_{A,\delta}\right)
 - m_f\, \rho_T^{\mu\nu} \, . \qquad
\label{rhoTL}\nonumber
\vspace*{0.1cm}
\eea
The corresponding expressions for the LHS of (\ref{Fexact}) follow from
(\ref{rhoVL}) with the replacement $\rho \to F$. Considering now the various
component (\ref{project}) of the integrand on the RHS of Eq.\ (\ref{rhoexact}),
we find (omitting flavor indices)
\bea
 \tilde{\tr}\left[\bA\, \rho \right] &=& 
 \bA_S\, \rho_S - \bA_P\, \rho_P + \bA_V^\mu\, \rho_{V,\mu}
 - \bA_A^\mu\, \rho_{A,\mu} \nonumber\\
 && + \frac{1}{2} \bA_T^{\mu\nu}\, \rho_{T,\mu\nu} 
 \, ,\quad 
\label{rhoSR}\\[0.1cm]
 -i \tilde{\tr}\left[\gamma_5 \bA\, \rho \right] &=& 
 \bA_S\, \rho_P + \bA_P\, \rho_S - i \bA_V^\mu\, \rho_{A,\mu}
 + i \bA_A^\mu\, \rho_{V,\mu} \nonumber\\
 && + \frac{1}{4}\, \epsilon^{\mu\nu\gamma\delta} \bA_{T,\mu\nu}\, 
 \rho_{T,\gamma\delta} \, ,
\label{rhoPR}\\[0.1cm]
 \tilde{\tr}\left[\gamma^\mu \bA\, \rho \right] &=& 
 \bA_S\, \rho_V^\mu + \bA_V^\mu\, \rho_S - i \bA_P\, \rho_A^\mu 
 + i \bA_A^\mu\, \rho_P + i \bA_{V,\nu}\, \rho_T^{\nu\mu}  \nonumber\\
 && + i \bA_T^{\mu\nu}\, \rho_{V,\nu}
 + \frac{1}{2} \epsilon^{\mu\nu\gamma\delta} \left( \bA_{A,\nu}\, 
 \rho_{T,\gamma\delta} + \bA_{T,\nu\gamma}\, 
 \rho_{A,\delta} \right) \, , \qquad
\label{rhoVR}\\[0.1cm]
 \tilde{\tr}\left[\gamma_5 \gamma^\mu \bA\, \rho \right] &=&
 \bA_S\, \rho_A^\mu + \bA_{A}^\mu\, \rho_S 
 - i \bA_P\, \rho_V^\mu + i \bA_V^\mu\, \rho_P
 + i \bA_{A,\nu}\, \rho_T^{\nu\mu}
  \nonumber\\
 && 
 + i \bA_T^{\mu\nu}\, \rho_{A,\nu} 
 + \frac{1}{2} \epsilon^{\mu\nu\gamma\delta} \left( \bA_{V,\nu}\, 
 \rho_{T,\gamma\delta} + \bA_{T,\nu\gamma}\, 
 \rho_{V,\delta}  \right) \, ,\quad
\label{rhoAR}\\[0.1cm]
 \tilde{\tr}\left[\sigma^{\mu\nu} \bA\, \rho \right] &=& 
 \bA_S\, \rho_T^{\mu\nu} + \bA_T^{\mu\nu}\, \rho_S 
 - \frac{1}{2}\,\epsilon^{\mu\nu\gamma\delta} 
 \left( \bA_P\, \rho_{T,\gamma\delta} + \bA_{T,\gamma\delta}\, 
\rho_P \right)
 \nonumber\\ 
&&
 - i \left(\bA_V^\mu\, \rho_V^\nu - \bA_V^\nu\, \rho_V^\mu \right) 
 + \epsilon^{\mu\nu\gamma\delta} \left( \bA_{V,\gamma}\, \rho_{A,\delta}
 - \bA_{A,\gamma}\, \rho_{V,\delta} \right)
 \nonumber\\[0.15cm] 
&&
 + i \left(\bA_A^\mu\, \rho_A^\nu - \bA_A^\nu\, \rho_A^\mu \right)
 + i \left(\bA_T^{\mu\gamma}\, {\rho_{T,\gamma}}^\nu 
 - \bA_T^{\nu\gamma}\, {\rho_{T,\gamma}}^\mu \right) .\qquad\,\,
\label{rhoTR}
\eea
With the above expressions one obtains the evolution equations for the various
Lorentz components in a straightforward way using (\ref{rhoexact}). We note
that the convolutions appearing on the RHS of the evolution equation
(\ref{Fexact}) for $F$ are of the same form than those computed above for
$\rho$. The respective RHS can be read off Eqs.\ (\ref{rhoSR})--(\ref{rhoTR}) 
by replacing $\rho \to F$ for the first term and $\bA \to \bC$ for the second 
term under the integrals of Eq.\ (\ref{Fexact}).  We have now all the relevant 
building blocks to discuss the most general case of nonequilibrium fermionic 
fields. However, this is often not necessary in practice due to the 
presence of symmetries, which require certain components to vanish 
identically.  

\section{\label{sec:symmetries}Symmetries}


Every symmetry of the microscopic theory described by the action
(\ref{fermact}), if not explicitely broken by the initial condition, is
exactly preserved at all times. In the following, we will exploit the
symmetries of the action and impose some symmetries to the initial 
conditions in order to simplify the fermionic evolution equations 
derived in the previous section. 

\vspace{.2cm} 
{\em Spatial translation invariance and isotropy:}
We will consider spatially homogeneous and isotropic initial conditions. In
this case it is convenient to work in Fourier space and we write
\beq
\rho(x,y) \equiv \rho(x^0,y^0;\bx-\by) = \int \frac{{\rm d}^3p}{(2\pi)^3}\,
e^{i \bp \cdot(\bx-\by)} \rho(x^0,y^0;\bp) \, ,
\eeq
and similarly for all the other two-point functions. Moreover, isotropy implies 
a reduction of the number of independent two-point functions: e.g.~the vector
components of the spectral function can be written as 
\bea
 \rho_V^0(x^0,y^0;\bp) &=& \rho_V^0(x^0,y^0;p) \, , \nonumber \\
 \vec{\rho}_V(x^0,y^0;\bp) &=& \bv\, \rho_V(x^0,y^0;p) \, , \nonumber
\eea
where $p\equiv |\bp|$ and $\bv=\bp/p$. 

\vspace{.2cm}
{\em Parity:} The vector components $\rho_V^0(x^0,y^0;p)$ and $\rho_V(x^0,y^0;p)$ 
are unchanged under a parity transformation, whereas the corresponding axial-vector
components get a minus sign.  Therefore, parity together with rotational
invariance imply that
\beq
 \rho_A^0(x^0,y^0;p) = \rho_A(x^0,y^0;p) = 0 \, .
\eeq
The same is true for the axial-vector components of $F$, $\bA$ and $\bC$.
Parity also implies the pseudo-scalar components of the various two-point
functions to vanish.


\vspace{.2cm}
{\em $CP$--invariance:} For instance, under combined charge conjugation 
and parity transformation the vector component of $\rho$ transforms as
\bea
 \rho_V^0(x^0,y^0;p) &\longrightarrow& \rho_V^0(y^0,x^0;p) \, ,\nonumber\\
 \rho_V(x^0,y^0;p) &\longrightarrow& -\rho_V(y^0,x^0;p) \, ,\nonumber
\eea
and similarly for  $\bA_V^0$ and $\bA_V$.  The $F$--components transform as 
\bea
 F_V^0(x^0,y^0;p) &\longrightarrow& -F_V^0(y^0,x^0;p) \, , \nonumber\\
 F_V(x^0,y^0;p) &\longrightarrow& F_V(y^0,x^0;p) \, , \nonumber
\eea
and similarly for $\bC_V^0$ and $\bC_V$.  Combining this with the hermiticity
relations~(\ref{herm}), one obtains for these components that
\beq
\label{CP}
 \begin{array}{lll} 
  \Re \rho_V^0(x^0,y^0;p) &=& \Im \rho_V(x^0,y^0;p) = 0 \, , \\
  \Re F_V^0(x^0,y^0;p) &=& \Im F_V(x^0,y^0;p) = 0  \, ,\\
  \Re \bA_V^0(x^0,y^0;p) &=& \Im \bA_V(x^0,y^0;p) = 0  \, ,\\
  \Re \bC_V^0(x^0,y^0;p) &=& \Im \bC_V(x^0,y^0;p) = 0 \, ,
 \end{array}
\eeq
for all times $x^0$ and $y^0$ and all individual modes.

\vspace{.2cm}
We stress that a nonequilibrium ensemble respecting a particular symmetry does
not imply that the individual ensemble members exhibit the same symmetry. For
instance, a spatially homogeneous ensemble can be build out of inhomogeneous
ensemble members and clearly includes the associated physics. The most
convenient choice of an ensemble is mainly dictated by the physical problem to
be investigated. Below we will study a ``chiral quark model'' with Dirac
fermions coupled to scalars in a chirally invariant way. Since in this paper we
will restrict the discussion of this model to the phase without spontaneous
breaking of chiral symmetry, it is useful to exploit this symmetry as well. 

\vspace{.2cm}
{\em Chiral symmetry:} 
The only components of the decomposition (\ref{rhodecomp}) allowed by
chiral symmetry are those which anticommute with $\gamma_5$. We therefore 
have
\beq
 \rho_S (x^0,y^0;\bp) = \rho_P (x^0,y^0;\bp) 
 = \rho_T^{\mu \nu} (x^0,y^0;\bp) = 0 \, ,
\eeq
and similarly for the corresponding components of $F$, $\bA$ and $\bC$. 
In particular, chiral symmetry forbids a mass term for fermions and we 
have $m_f\equiv0$.


\subsection*{Equations of motion}

In conclusion, for the above symmetry properties we are left with only four
independent propagators: the two spectral functions $\rho_V^0$ and $\rho_V$ and
the two corresponding statistical functions $F_V^0$ and $F_V$. They are either
purely real or imaginary and have definite symmetry properties under the
exchange of their time arguments $x^0 \leftrightarrow y^0$. These properties as
well as the corresponding ones for the various components of the self energy
are summarized below:

\begin{center}
\begin{tabular}{ll}
 $\rho_V^0$, $\bA_V^0$: & imaginary, symmetric; \\
 $\rho_V$, $\bA_V$: & real, antisymmetric;\\
 $F_V^0$, $\bC_V^0$: & imaginary, antisymmetric;\\
 $F_V$, $\bC_V$: & real, symmetric.
\end{tabular}
\end{center}

The exact evolution equations for the spectral functions
read:\footnote{Actually, the following equations do not rely on the restrictions
(\ref{CP}) imposed by $CP$--invariance: they have the very same form even also
in the case where all the two-point functions are complex.}
\bea
\label{rhoV0eom}
\lefteqn{
i \frac{\partial}{\partial x^0}\, \rho_V^0(x^0,y^0;p)
= p\, {\rho}_V(x^0,y^0;p) } \nonumber\\  
&+& \int_{y^0}^{x^0} {\rm d}z^0 \Big[
\bA_V^0(x^0,z^0;p)\, \rho_V^0(z^0,y^0;p) 
- {\bA}_V(x^0,z^0;p)\, {\rho}_V(z^0,y^0;p) \Big] \, ,
\nonumber\\
\\[0.2cm]
\label{rhoVeom}
\lefteqn{
i \frac{\partial}{\partial x^0}\, {\rho}_V(x^0,y^0;p)
= p\, \rho_V^0(x^0,y^0;p) } \nonumber\\  
&+& \int_{y^0}^{x^0} {\rm d}z^0 \Big[
\bA_V^0(x^0,z^0;p)\, {\rho}_V(z^0,y^0;p) 
- {\bA}_V(x^0,z^0;p)\, \rho_V^0(z^0,y^0;p) \Big] \, .
\nonumber\\
\eea
Similarly, for the statistical two-point functions we obtain
\bea
\label{FV0eom}
\lefteqn{
i \frac{\partial}{\partial x^0}\, F_V^0(x^0,y^0;p)
= p\, {F}_V(x^0,y^0;p) } \nonumber\\  
&+&  \int_0^{x^0} {\rm d}z^0 \Big[
\bA_V^0(x^0,z^0;p)\, F_V^0(z^0,y^0;p) 
- {\bA}_V(x^0,z^0;p)\, {F}_V(z^0,y^0;p) \Big] 
\nonumber\\
&-& \int_0^{y^0} {\rm d}z^0 \Big[
\bC_V^0(x^0,z^0;p)\, \rho_V^0(z^0,y^0;p) 
- {\bC}_V(x^0,z^0;p)\, {\rho}_V(z^0,y^0;p) \Big] \, ,
\nonumber\\
\\[0.2cm]
\label{FVeom}
\lefteqn{
i \frac{\partial}{\partial x^0}\, {F}_V(x^0,y^0;p)
= p\, F_V^0(x^0,y^0;p) } \nonumber\\  
&+& \int_0^{x^0} {\rm d}z^0 \Big[
\bA_V^0(x^0,z^0;p)\, {F}_V(z^0,y^0;p) 
- {\bA}_V(x^0,z^0;p)\, F_V^0(z^0,y^0;p) \Big] 
\nonumber \\
&-& \int_0^{y^0} {\rm d}z^0 \Big[
\bC_V^0(x^0,z^0;p)\, {\rho}_V(z^0,y^0;p) 
- {\bC}_V(x^0,z^0;p)\, \rho_V^0(z^0,y^0;p) \Big] \, .
\nonumber\\
\eea


\section{Chiral quark--meson model}
\label{sec:model}

As an application we consider a quantum field theory involving  two fermion
flavors (``quarks'') coupled in a chirally invariant way to a scalar
$\sigma$--field and a triplet of pseudoscalar ``pions'' $\pi^a$,
$a=1,\ldots,3$. The classical action reads
\bea
S &=& \int {\rm d}^4 x \Big\{\bar{\psi} i \partial\,\slash \psi 
+\frac{1}{2}\left[\partial_\mu \sigma \partial^\mu \sigma
+ \partial_\mu \pi^a  \partial^\mu \pi^a \right] \nonumber\\
&& \qquad\quad\!\!\!  
+\, g \bar{\psi} \left[\sigma + i\gamma_5 \tau^a \pi^a \right] \psi
- V(\sigma^2 + \pi^2) \Big\} \, ,
\label{chiralfermact}
\eea
where $\pi^2\equiv \pi^a \pi^a$ and where $\tau^a$ denote the standard Pauli matrices. 
The above action is invariant under chiral $SU_L(2)\times SU_R(2)$ transformations.  
For a quartic scalar self-interaction $\sim \left(\sigma^2+ \pi^a\pi^a\right)^2$,
this model corresponds to the well known linear $\sigma$--model \cite{sigma}, which 
has been extensively studied in thermal equilibrium in the literature using various 
approximations \cite{sigma2}. For simplicity we consider here a purely quadratic 
scalar potential:  
\beq
 V  = \frac{1}{2} m^2_0 (\sigma^2 + \pi^2) \, ,
\label{potential}
\eeq
which is sufficient to study thermalization in this model.  We note that
this theory has the same universal properties than the corresponding linear
$\sigma$-model. Extending our study to take into account quartic
self-interaction is straightforward. It gives additional contributions to the
scalar self-energies, Eqs.~(\ref{selfscalarrho}) and (\ref{selfscalarF}) below.
These contributions can be found in Refs.~\cite{Berges:2001fi,Aarts:2002dj} and
we will point out the respective changes below.

\subsection{Equations of motion for the scalar field}

The 2PI effective action for scalar field has been extensively studied in
recent literature \cite{Berges:2001fi,Berges:2000ur,Aarts:2002dj}. 
Here, we briefly recall the main features of the scalar sector and 
stress those aspects which are relevant for the present paper 
(for details see Ref.~\cite{Aarts:2002dj}). For the model considered here, 
Eq.\ (\ref{chiralfermact}), the complete 2PI effective action is a functional 
of the fermionic propagators as well as of scalar propagators.\footnote{As 
emphasized in the previous Section, we do not consider the possibility of a 
broken symmetry in this paper. Therefore we restrict to vanishing scalar 
field average value.} In general, there are various two-point functions 
in the scalar sector and it is useful to group them in a $4 \times 4$ 
matrix ($a,b=1,\ldots,3$)
\beq\label{scalpropmat}
 \mathcal{G} \equiv \left(
 \begin{array}{cc} 
 G_{\sigma \sigma} & G_{\sigma \pi}^b \\
 G_{\pi \sigma}^a & G_{\pi \pi}^{ab} 
 \end{array} \right),
\eeq
where omitted the space-time dependence: $\mathcal{G} \equiv \mathcal{G} (x,y)$.
The 2PI effective action (\ref{fermioneffact}), augmented by the scalar sector,
reads
\beq
 \Gamma[\mathcal{G},D] = \frac{i}{2} \Tr\ln \mathcal{G}^{-1} 
 + \frac{i}{2} \Tr \mathcal{G}_0^{-1}\mathcal{G}
 -i \Tr\ln D^{-1} -i \Tr\, D_0^{-1} D
 + \Gamma_2[\mathcal{G},D] + {\rm const} \, ,
\label{totaleffact}
\eeq
where the free scalar inverse propagator is given by:
\beq
\label{classscalar}
 \mathcal{G}_{0,AB}^{-1} (x,y) = -(\partial^2 + m^2_0 )\, 
 \delta^{(4)}_{\cal C}(x-y) \, \delta_{AB} \, .
\eeq
here, we have introduced the indices $A,B=0,\ldots,3$, which label the matrix
elements in the representation (\ref{scalpropmat}) ($A=0$ corresponds to the
$\sigma$--direction).  The equations of motion for the scalar correlators are
obtained by minimizing the 2PI effective action with respect to $\mathcal{G}$.
They are similar in form to Eq.~(\ref{evolutionC}), with the appropriate
differential operator given by Eq.~(\ref{classscalar}) (see Eqs.\ (\ref{rhoscalar}) 
and (\ref{Fscalar}) below), and where the corresponding self-energies are defined 
by~\cite{Cornwall:1974vz,Berges:2001fi,Aarts:2002dj}
\beq
\label{scalarself}
 \Sigma_{AB} (x,y) = 2i\,
 \frac{\delta\Gamma_2[\mathcal{G},D]}{\delta \mathcal{G}_{BA} (y,x)} \,.
\eeq
Under chiral transformations, the matrix $\mathcal{G}\rightarrow 
\mathcal{R} \, \mathcal{G} \,\mathcal{R}^\dagger$, where $\mathcal{R}$ is 
an $O(4)$ rotation. Therefore, chiral symmetry requires that $\mathcal{G}$ be 
proportional to the unit matrix in $O(4)$--space: 
\beq
\label{scalarsym}
 \mathcal{G}_{AB} (x,y)=G_\phi (x,y) \, \delta_{AB}\,.
\eeq 
The same holds for the corresponding self energies (\ref{scalarself}) as well. 
Similarly, in the fermionic sector, chiral symmetry imply that all fermionic
two-point functions be proportional to unity in flavor space, \eg 
\beq
\label{fermionsym}
 D_{ij} (x,y)=D(x,y) \,\delta_{ij}\,.
\eeq 

The scalar spectral and statistical propagators are defined through
\beq
 G_\phi (x,y) = F_\phi (x,y) -\frac{i}{2} \rho_\phi (x,y)
 \left[\Theta_\C(x^0-y^0) - \Theta_\C(y^0-x^0) \right] \, ,
\eeq
and similarly for the spectral and statistical self-energies
$\Sigma_\phi^{\rho}$ and $\Sigma_\phi^{F}$. These are all real functions and
$F$--like components are symmetric under the exchange of $x$ and $y$, whereas
the $\rho$--like components are antisymmetric.  The equal-time commutation
relation of two scalar field operators imply
\beq
\label{initialrhoscalar}
 \rho_\phi (x,y)|_{x^0=y^0} = 0, \;\;\;\;
 \partial_{x^0}\rho_\phi(x,y)|_{x^0=y^0} = \delta(\bx-\by)\, ,
\eeq 
which uniquely specify the initial conditions for the scalar spectral
propagator. Finally, the equations of motion for the scalar propagators read
(in momentum space):
\bea 
\left[ \partial_{x^0}^2  + p^2 + m^2_0 \right] \rho_\phi (x^0,y^0;p) = 
  -\int_{y^0}^{x^0} {\rm d}z^0\,  
 \Sigma_\phi^{\rho} (x^0,z^0;p)\, \rho_\phi (z^0,y^0;p)\,. &&
 \nonumber\\
\label{rhoscalar}
\\[0.3cm]
 \left[ \partial_{x^0}^2 + p^2 + m^2_0 \right] F_\phi(x^0,y^0;p) = 
 - \int_0^{x^0} {\rm d}z^0\, \Sigma_\phi^{\rho}(x^0,z^0;p)\, 
 F_\phi(z^0,y^0;p)
\nonumber && \\ 
 + \int_0^{y^0} {\rm d}z^0\, \Sigma_\phi^{F}(x^0,z^0;p)\, 
 \rho_\phi(z^0,y^0;p),&&\nonumber\\
\label{Fscalar}
\eea

Note that these are the exact equations of the theory described by the
classical action (\ref{fermact}) together with (\ref{potential}). In the
presence of scalar self-interactions, the self energy receives an additional 
local contribution $\sim \Sigma_\phi^{\rm (local)}(x)\,\delta_\C^{(4)} (x-y)$, 
which simply amounts to a local shift of the bare mass squared appearing in 
the above equations. In that case, the exact equations of motion have the 
same form as above, with the replacement 
$m_0^2 \rightarrow M^2(x) = m_0^2 + \Sigma_\phi^{\rm (local)} (x)$ 
\cite{Berges:2001fi,Aarts:2002dj}.


\subsection{2PI coupling expansion}

The simplest possible truncation of the 2PI effective action
(\ref{totaleffact}) which includes the necessary ingredient for thermalization
corresponds to the first contribution in the Feynman graph expansion of
$\Gamma_2[\mathcal{G},D]$, that is the two--loop contribution depicted in Fig.\
\ref{twoloopfig}. This corresponds to the first non-trivial order in a coupling
expansion.\footnote{We note that
this approximation can also be related to a nonperturbative
$1/N_f$ expansion at next-to-leading order.} 
Making use of Eqs.\ (\ref{scalarsym}) and (\ref{fermionsym}), one
can express the two loop contribution to $\Gamma_2$ directly in terms of
$G_\phi$ and $D$. We obtain\footnote{The relevant self energies can be obtained
directly from this expression by writing:
$$
 2i\,\frac{\delta \Gamma_2}{\delta G_\phi} 
 = 2i\,\frac{\delta \Gamma_2}{\delta \mathcal{G}_{AB}}\,
 \frac{\delta \mathcal{G}_{AB}}{\delta G_\phi} 
 = \delta_{AB} \, \Sigma_{BA} = N_s \, \Sigma_\phi
$$
and similarly for the fermionic self energies $\Sigma_{ij}=\Sigma \,
\delta_{ij}$:
$$
 -i \frac{\delta \Gamma_2}{\delta D}= N_f \Sigma \, .
$$
}
\beq
 \Gamma_2^{\rm (2-loop)} [\mathcal{G},D]
 =-ig^2\frac{N_fN_s}{2}\int_\C {\rm d}^4x \,{\rm d}^4y \,
 \,\tr [D(x,y)\,D(y,x)]\,G_\phi(x,y) \, ,
\eeq
where $N_f=2$ is the number of fermion flavors and $N_s=4$ is the number of
scalar species. From there, it is straightforward to compute the spectral and
statistical components of the two loop self-energies.  We obtain
%\footnote{In writing the following expression, we exploited the 
%$P$ and $CP$--invariance, but not rotational invariance yet. Care must
%be taken concerning the signs in the spatial momentum dependence.}

\begin{figure}[t]
\begin{center}
\epsfig{file=2loop.eps,width=3.cm}
\end{center}
\caption{
Two loop contribution to $\Gamma_2[\mathcal{G},D]$.  The solid and dashed lines
represent respectively the full fermionic ($D$) and bosonic ($\mathcal{G}$)
propagators.}
\label{twoloopfig}
\end{figure}

\bea
\label{selfscalarrho}
 \Sigma_\phi^\rho (x^0,y^0;\bp) = -8 g^2 N_f\int \frac{{\rm d}^3 q}{(2\pi)^3}\,
 \rho_V^\mu(x^0,y^0;\bq)\,F_{V,\mu}(x^0,y^0;\bp-\bq) \, ,&&\\
\label{selfscalarF}
 \Sigma_\phi^F (x^0,y^0;\bp) = -4 g^2 N_f\int \frac{{\rm d}^3 q}{(2\pi)^3}\,
 \Big[F_V^\mu(x^0,y^0;\bq)\,F_{V,\mu}(x^0,y^0;\bp-\bq) &&\nonumber\\
 - \frac{1}{4} \rho_V^\mu(x^0,y^0;\bq)\,\rho_{V,\mu}(x^0,y^0;\bp-\bq) 
 \Big] \, ,&&
\eea

\bea
\bA_V^\mu(x^0,y^0;\bp) = - g^2 N_s\int \frac{{\rm d}^3 q}{(2\pi)^3}\, \Big[
F_V^\mu(x^0,y^0;\bq) \,\rho_\phi(x^0,y^0;\bp-\bq) &&\nonumber\\
+ \rho_V^\mu(x^0,y^0;\bq)\, F_\phi(x^0,y^0;\bp-\bq)\Big] \, ,&&
\\[0.2cm] 
\bC_V^\mu(x^0,y^0;\bp) = - g^2 N_s\int \frac{{\rm d}^3q}{(2\pi)^3}\, \Big[
F_V^\mu(x^0,y^0;\bq) \,F_\phi(x^0,y^0;\bp-\bq) &&\nonumber\\
-\frac{1}{4} \rho_V^\mu(x^0,y^0;\bq)\, \rho_\phi(x^0,y^0;\bp-\bq)\Big]\, .&&
\label{selffermionF}
\eea

Finally, let us mention that, in the case of non-vanishing scalar
self-interactions, the only additional two loop contribution gives 
rise to a local mass shift as described above. In particular, at this 
order, there is no contribution to the RHS of Eqs.\ (\ref{rhoscalar}) and 
(\ref{Fscalar}), which is the relevant part to describe thermalization.


\section{Initial conditions}
\label{sec:initial}

The time evolution for the fermions is described by first-order 
(integro-)differential equations for $F$ and $\rho$: Eqs.\ 
(\ref{rhoV0eom})--(\ref{FVeom}). As pointed out above, the 
initial spectral function $\rho(t,t;p)|_{t=0}$ is completely 
determined by the equal-time anticommutation relation of fermionic 
field operators (cf.~Eq.~(\ref{rhoinitial})). 
To uniquely specify the time evolution for $F$ we have to set the initial
conditions. The most general (gaussian) initial conditions for $F$ respecting
spatial homogeneity, isotropy, parity, charge conjugation and chiral symmetry 
can be written as 
\bea
 F_V(t,t',p)|_{t=t'=0} &=& \frac{1}{2}- n_0^{f}(p) \, ,
\label{initialFV}\\
 F_V^0(t,t',p)|_{t=t'=0} &=& 0
\label{initialFV0} \, .
\eea
Here $n_0^{f}(p)$ denotes the initial particle number distribution and ranges 
between 0 and 1 (the definition of  effective particle number
distribution in terms of equal-time two-point functions is detailed in
\ref{sec:quasiparticle}). At late times, when thermal equilibrium 
is approached, this leads to a canonical description with a vanishing chemical 
potential.\footnote{For nonzero chemical potential or net charge density the BCS
mechanism can lead to the condensation of Cooper pairs of fermions, which will
be discussed elsewhere.}    

The evolution equations (\ref{Fscalar}) and (\ref{rhoscalar}) for the scalar
correlators are second-order in time and one needs to specify initial conditions
for the propagators and their time derivatives. As for the fermions, the initial
conditions for the scalar spectral function is completely specified by the
field commutation relations which yield $\rho_\phi(t,t';p)|_{t=t'}=0$ and
$\partial_t \rho_\phi(t,t';p)|_{t=t'}=1$.
For the scalar two-point function $F_\phi$ we consider 
(cf.~also Refs.~\cite{Berges:2001fi,Aarts:2001qa,Berges:2002cz})
\bea
 F_\phi(t,t',p)|_{t=t'=0} & = & \frac{1}{\epsilon_0(p)}
 \Big[n_0(p)+\frac{1}{2}\Big] \, , \nonumber\\
 \partial_{t}F_\phi(t,t',p)|_{t=t'=0} & = & 0 \, ,
\label{initialFphi}\\
 \partial_{t}  \partial_{t'} F_\phi(t,t',p)|_{t=t'=0} & = & 
 \epsilon_0(p)\,\Big[n_0(p)+\frac{1}{2}\Big] \, , \nonumber
\eea
with an initial particle number distribution $n_0(p)$ and initial mode energy
$\epsilon_0(p)$ (see \ref{sec:quasiparticle}).

It is instructive to consider for a moment the solution of the {\em free field} equations,
which can be obtained from Eqs.\ (\ref{rhoV0eom})--(\ref{rhoVeom}) and
(\ref{FV0eom})--(\ref{FVeom}) by neglecting the memory integrals on their RHS.
The solution of the fermionic free field equations with the above initial
conditions reads
\bea
 F_V(t,t',p)&=&\left(\frac{1}{2}-n_0^{f}(p)\right)\,\cos[p(t-t')] \, ,\nonumber\\
 F_V^0(t,t',p)&=& - i\,\left(\frac{1}{2}- n_0^{f}(p)\right)\,\sin[p(t-t')]\, \, \quad\,\,  \nonumber
\eea
and for the spectral functions
\beq
 \rho_V(t,t',p)= \sin[p(t-t')] \quad , \qquad
 \rho_V^0(t,t',p)= i \cos[p(t-t')]\, . \nonumber
\eeq
One observes that each mode of the equal-time correlator $F_V(t,t,p)$ is strictly 
conserved in the absence of the memory integrals. This correlator being directly 
related to particle number (see \ref{sec:quasiparticle}), this means that the 
latter is {\em conserved mode by mode} in this approximation. Although this 
is expected in the free field limit, this is of course not the case in the 
fully interacting theory. However, we emphasize that such additional 
conservation laws do not only appear in the free field limit, but are a 
generic property of mean-field-type approximations \cite{gaussian2a}, 
which may include local corrections to the bare mass, but which neglect the 
scattering contributions described by the memory integrals on the RHS of 
Eqs.~(\ref{rhoV0eom})-(\ref{FVeom}). In these approximations, the existence 
of this infinite number of spurious conserved quantities ({\it i.e.} which 
are not present in the fully interacting theory) prevents the system to 
approach the correct thermal equilibrium limit at late times. The same 
is true for scalar fields as well \cite{Berges:2001fi}. It is therefore 
crucial to go beyond such ``gaussian'' approximations in order to correctly 
describe the time evolution of the system in the interacting theory.


\section{Numerical implementation}
\label{sec:numerics}

We numerically solve the evolution equations
(\ref{rhoV0eom})--(\ref{FVeom}) and (\ref{rhoscalar})--(\ref{Fscalar}), 
together with the self-energies Eqs.\ (\ref{selfscalarrho})--(\ref{selffermionF}).
The structure of the fermionic equations is reminiscent of the form of classical
canonical equations. In this analogy, $F_V(t,t')$ plays the role of the canonical 
coordinate and $F_V^0(t,t')$ is analogous to the canonical momentum. This suggests 
to discretize $F_V(t,t')$ and $\rho_V(t,t')$ at $t-t'=2n a_t$ (even) and 
$F_V^0(t,t')$ and $\rho_V^0(t,t')$ at $t-t'=(2n+1)a_t$ (odd) time-like lattice 
sites with spacing $a_t$. This is a generalization of the ``leap-frog'' 
prescription for temporally inhomogeneous two-point functions. This implies
in particular that the discretization in the time direction is coarser
for the fermionic two-point functions than for the bosonic ones. This 
``leap-frog'' prescription may be easily extended to the memory integrals 
on the RHS of Eqs.\ (\ref{rhoV0eom})--(\ref{FVeom}) as well.


We emphasize that the discretization does not suffer from the
problem of so--called fermion doublers \cite{montvay}. The spatial 
doublers do not appear since (\ref{rhoV0eom})--(\ref{FVeom})
are effectively second order in $\vec{x}$-space. Writing the equations
for $\vec F_V(t,t',\vec x)$ and $\vec \rho_V(t,t',\vec x)$
starting from (\ref{rhoV0eom})--(\ref{FVeom}) one realizes that instead of
first order spatial derivatives there is a Laplacian appearing.
Hence we have the same Brillouin zone for the fermions and scalars.
Moreover, time-like doublers are easily avoided by using a sufficiently 
small stepsize in time $a_t/a_s$.

The fact that Eqs. (\ref{rhoV0eom})--(\ref{FVeom}) and
(\ref{rhoscalar})--(\ref{Fscalar}) contain memory integrals makes numerical
implementations expensive. Within a given numerical precision it is typically
not necessary to keep all the past of the two-point functions in the memory.
A single PIII desktop workstation with
2GB memory let us use a memory array with 470 timesteps (with 2 temporal
dimensions: $t$ and $t'$). We have checked for the presented runs that a 30\%
change in the memory interval length did not alter the results.  For a typical
run 1-2 CPU-days were necessary.

The shown plots are calculated on a $470\times470\times32^3$ lattice.
(The dimensions refer to the $t$ and $t'$ memory arrays and the momentum-space
discretization, respectively.) By exploiting the spatial symmetries detailed in
Section~\ref{sec:symmetries} the memory need could be reduced by a factor
of~30.
We have checked that the infrared cutoff is well below any other mass scales
and that the UV cutoff is greater than the mass scales at least by a factor of
three.

To extract physical quantities we follow the time evolution
of the system for a given lattice cutoff up to late times and 
measure the renormalized scalar mass $m$ which is then used to 
set the scale. In the evolution equations we analytically subtract 
only the respective quadratically divergent terms obtained from a 
standard perturbative analysis. 
We emphasize that for the results presented below we use the late-time 
thermal mass to set the scale, and not the vacuum mass for convenience. 

We made runs for a range of couplings $g^2=0.49$ -- $1$ which show very similar
qualitative behavior. Below, we present plots corresponding to $g=1$, for which
the time needed to closely approach thermal equilibrium is the shortest. 
This allows us to obtain an accurate thermalization with the lowest numerical
cost.



\section{Far-from-equilibrium dynamics}
\label{Sectfarfromequilibrium}

\begin{figure}[t]
\begin{center}
\epsfig{file=join_fermion.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{The time evolution of the fermionic two-point 
function $F_V(t,t;p)$ for three values of the
momentum $p$, in units of the renormalized scalar thermal mass $m$. 
The evolution is shown for two very different initial conditions 
with the {\em same} initial energy density. One observes that
the dynamics becomes rather quickly insensitive to the initial 
distributions displayed in the insets -- much before the modes 
settle to their final values. The long-time behavior is shown on a 
logarithmic scale for $t \ge 30 m^{-1}$.}
\label{lostinitialfig}
\end{figure}
In Fig.~\ref{lostinitialfig} we present the time evolution 
of the fermion equal-time two-point function $F_V(t,t;p)$ for three 
momenta $p$. Results are given for two very different
initial particle number distributions, which are displayed 
in the insets (see also Eqs. (\ref{initialFV})--(\ref{initialFphi})). 
The (conserved) energy density is taken to be the same for both runs. 
In this case, since thermal equilibrium is uniquely specified by the 
value of the energy density, the correlator modes should approach 
universal values at late times if thermalization occurs. 

\begin{figure}[t]
\begin{center}
\epsfig{file=join_scalar.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{The same as in Fig.~\ref{lostinitialfig} but for the bosonic 
two-point function $F_\phi(t,t;p)$ for three different momenta.
The initial particle number distributions for the two runs denote
by ``A'' and ``B'' are displayed in the insets. For the employed
parameters one observes that, in the scalar sector, the
time needed to become effectively insensitive to the initial 
distributions is comparable to the time scale describing the approach 
to the universal late-time value.}
\label{scalarlostinitialfig}
\end{figure}

It is striking to observe from Fig.~\ref{lostinitialfig} 
that after a comparably short time, much before the correlation 
modes reach their late-time values, the dynamics becomes rather 
insensitive to the details of the initial conditions: for a given momentum,
the curves corresponding to the two different runs come
close to each other rather quickly. During the slow 
subsequent evolution the system is still far away from 
equilibrium before the approach to the late-time values sets in.
From both runs one observes that the characteristic time needed to
effectively loose the information about the details of the initial conditions
is much shorter than the time needed to approach the late-time
result. Moreover, the latter is found to be universal in the
sense that the different runs agree with each other to very good 
precision.     

Fig.~\ref{scalarlostinitialfig} shows the corresponding behavior of the
scalar correlator modes $F_\phi(t,t;p)$. The respective initial
particle number distributions in the scalar sector for the two runs
are given in the inset. For the two different runs the modes having the 
same momenta approach each other rather slowly as compared to the fermionic 
sector. However, they reach their final values on a time scale which is 
comparable to that observed for the fermions in Fig~\ref{lostinitialfig}.


\begin{figure}[t]
\begin{center}
\epsfig{file=spectral.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{The fermionic two-point function $F_V(t,t';p)$ as a function
of $t-t'$ at late time $t=600\, m^{-1}$. One observes a very good
agreement with an exponential behavior. The rate is well described by the
width of the corresponding spectral function in frequency space
as shown in the inset (cf.\ the text for details).}
\label{spectralfig}
\end{figure}
To characterize these time scales in more detail we
consider in Fig.~\ref{spectralfig} the unequal-time two-point 
function $F_V(t,t';p)$. As expected if the system is to become insensitive 
to the details of the initial conditions, we observe that the correlation 
between some time $t$ and another time $t'$ should be suppressed for 
sufficiently large $t-t'$. We note that the oscillation envelope of 
$F_V(t,t',p)$ can be well described in terms of an exponential for 
sufficiently late times. The corresponding damping rate approaches 
a constant value. To estimate the asymptotic rate we show
in Fig.~\ref{spectralfig} the unequal-time two-point function
$F_V(t,t';p)$ as a function of $t-t'$ for the late time
$mt = 600$. The fit to an exponential 
yields the damping rate $\gamma^{\rm (damp)}_f(p=0.78m) = 0.03(1)\, m$.  We
find a moderate momentum dependence of this rate with $\gamma^{\rm (damp)}_f(0)
= 0.067(1)m$ and $\gamma^{\rm (damp)}_f(0) \gtrsim \gamma^{\rm (damp)}_f(p >
0)$.

We emphasize that the rate $\gamma^{\rm (damp)}_f(p)$ can be 
related to the width of the Fourier transform of the spectral function
with respect to the time difference $t-t'$. In principle, the latter involves 
an integration over an infinite time interval. However, since we 
consider an initial-value problem for finite times we know 
$\rho_V(t,t',p)$ only on a finite interval. To overcome
this problem, we fit the data for $\rho_V(t,t',p)$ by a $7$--parameter
formula that is capable to account for the observed oscillations and damping, 
but which is more general than the usual 2-parameter Breit-Wigner 
formula. We perform the Fourier transformation on the extrapolated 
data. The resulting function $\rho_V(\omega,p)$ is displayed as a function
of frequency $\omega$ in the inset of Fig.~\ref{spectralfig}. One clearly 
observes a nonzero width of the spectral function, the value of which may 
be obtained from a fit to a Breit-Wigner formula. By doing so, we obtain 
a very good agreement of the damping rate inferred from the width of the 
spectral function on the one hand and from the linear fit on the log-plot 
for $F_V(t,t';p)$ on the other hand.

We can use the fermion damping rate to quantify the time
scale characterizing the effective loss of the details of the initial
conditions: Comparing with Fig.~\ref{lostinitialfig}, we observe 
that the inverse fermion damping rate at $p\approx0$
$\left(1/\gamma^{\rm (damp)}_f = 15(1) m^{-1}\right)$ 
characterizes well the time for which the dynamics becomes rather 
insensitive to the initial distributions. In contrast, we find 
that this time scale does not characterize the late-time
behavior. For the latter, we observe, to very good approximation, an 
exponential relaxation of each mode $F_V(t,t;p)$ to their universal 
late-time values. Carrying out the measurement for different modes we
observe that the corresponding rate is almost independent of momentum 
and is given by $1/\gamma^{\rm (therm)}_f=95(5) m^{-1}$. The corresponding
time scale $1/\gamma^{\rm (therm)}_f$ is therefore much larger
than the characteristic damping time $1/\gamma^{\rm (damp)}_f$.

A similar analysis can be performed for the scalar sector.
Here we find at $p\approx0$ the corresponding values
$1/\gamma^{\rm (damp)}_\phi = 50(5)\,m^{-1}$ 
and $1/\gamma^{\rm (therm)}_\phi = 90(5) m^{-1}$. We note that for the scalars 
both rates agree rather well.\footnote{Similar studies in $1+1$ dimensional
quantum \cite{Berges:2001fi} and classical \cite{Aarts:2000wi} scalar theories 
typically find a substantial difference between damping and thermalization rates.
However, this is a consequence of stringent phase-space restrictions and 
is therefore particular to $1+1$ dimensional systems.}
In particular, one sees that, although the observed damping rates for fermions 
and bosons are very different, the respective thermalization rates 
are rather similar. This suggests that in $3+1$ dimensions, the 
time needed to approach the universal asymptotic values is determined
by the longest damping rate present in the system, which is
provided by the scalars. 

We finally note that approximate rates describing the early time
behavior may be determined by an exponential fit to the functions
$F_\phi(t,0)$, $F_V^0(t,0)$ and $F_V(t,0)$ in a finite time interval. 
The inferred rates depend on time and approach the late-time values 
given above.  At very early times we observe an approximate exponential 
damping with a rate about twice as big as the late-time value for the 
fermions, and about half the late-time rate for the scalars.


\section{Onset of Bose-Einstein and Fermi-Dirac statistics}
\label{sec:statistics}

In the previous section, we have seen that the out-of-equilibrium 
evolution of the system leads to a universal late-time behavior,
uniquely characterized by the initial energy density. We now
analyze in details this universal behavior and study the onset of 
thermalization. The latter can be studied directly on the level of 
correlation functions by following the nonequilibrium evolution 
to sufficiently late-times and by comparing their late-time values 
with those of the corresponding correlators directly calculated in 
thermal equilibrium \cite{Aarts:2000wi,LOinh,Berges:2000ur}. 
An equivalent way to demonstrate thermalization\footnote{We emphasize 
that, because the microscopic dynamics we study here is time-reversal
invariant (in the sense that the time evolution is unitary), neither thermalization 
nor late time independence on the details of initial conditions can be strictly 
established. However, our results demonstrate that, for the initial conditions
considered here, thermal equilibrium can be approached very closely at sufficiently 
late time. Indeed, for practical purposes, the late time result is indistinguishable
from strict thermal equilibrium. At finite times the latter cannot be reached on 
a fundamental level without some kind of coarse graining, which is a matter of 
principle and not a question of approximation. For a more detailed discussion see
Ref.~\cite{Berges:2001fi}.}
is to show the onset of Bose-Einstein and Fermi-Dirac statistics. 
The information on the statistics can be unambiguously extracted as
follows. In thermal equilibrium, the spectral function
and the statistical two-point function are not independent from each
other, but are related by the fluctuation-dissipation relation. The latter
is an exact relation, which can be stated in $4$--dimensional Fourier space 
as:
\beq
F^{(\rm eq)}_{S/F}(\omega,p) = \left(n_{S/F}(\omega) \pm \frac{1}{2}\right)
\varrho_{S/F}(\omega,p) \, ,
\label{eq:fluctdiss}
\eeq 
where the frequency $\omega$ is the Fourier conjugate of $t-t'$ (recall that,
in equilibrium, time-translation invariance implies that two-point functions
only depend on $t-t'$). The subscript $S/F$ refers to bosonic and fermionic 
two-point functions respectively. The corresponding distribution functions 
are given by the Bose--Einstein distribution 
$n_S= n_{BE} = 1/[\exp (\omega/T) -1]$
for scalars, and by the Fermi--Dirac distribution
$n_F= n_{FD} = 1/[\exp (\omega/T) + 1]$ for fermions. The value of the
temperature $T$ is determined by the energy density of the system. 
Out of equilibrium, the spectral and statistical two-point functions are 
completely independent in general. However, if thermal equilibrium
is to be approached, they should become related by the fluctuation-dissipation
relation at late times. We stress that this means first, that they should 
become related to each other and second, that this relation should be 
well characterized by either the Bose--Einstein or the Fermi--Dirac 
statistics.

\begin{figure}[t]
\begin{center}
\epsfig{file=fluctdiss.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{The late-time ratio of the statistical two-point function  
and the spectral function in frequency space, both for
fermions ($F_V/\rho_V$) and for scalars ($F_\phi/\rho_\phi$).  
In thermal equilibrium the quotient corresponds to the Bose-Einstein
(BE) distribution function for scalars and to the Fermi-Dirac (FD) 
distribution for fermions -- independently of any assumption
on a quasi-particle picture (see Eq.\ (\ref{eq:fluctdiss})). 
The BE/FD distributions are displayed by the continuous curves 
parametrized by the same temperature. The value of the latter, $T=0.94m$,
is actually not fitted but has been taken from the inverse slope 
of Fig.~\ref{compslopefig}. This shows the correspondence with the 
quasi-particle picture described in the text.}
\label{fig:fluctdiss}
\end{figure}

A unique answer about the late-time statistics, which does not rely 
on any assumption, can be obtained from $F$ and $\rho$ in
Wigner coordinates. For this we express $\rho_V(t,t';p)$ in terms
of the center coordinate $X^0=(t+t')/2$ and the relative
coordinate $s^0=t-t'$ and write \cite{Aarts:2001qa}
\beq
i\varrho_V(X^0;\omega,p) =  \int_{-2 X^0}^{2 X^0}
{\rm d}s^0\, e^{i \omega s^0}\, 
\rho_V(X^0+s^0/2,X^0-s^0/2;p) \, .
\eeq  
The factor $i$ is introduced such that $\varrho_V(X^0;\omega,p)$
is real. The equivalent transformation is done for $F_V$, however,
without the factor $i$. Since we consider an initial-value problem,
the time integral over $s^0$ is bounded by $\pm 2 X^0$ (cf.~also the 
discussion in Ref.~\cite{Aarts:2001yn}).
If thermal equilibrium is approached for sufficiently large $X^0$
then the correlators do no longer depend on $X^0$ and a Fourier
transform with respect to $t-t'$ can be performed to very good 
approximation (cf.~the discussion in Sect.~\ref{Sectfarfromequilibrium}). 
The distribution functions may then be extracted from
the quotient of the Wigner transformed two-point functions. 

In Fig.~\ref{fig:fluctdiss} we show the $n_F(\omega)$ and $n_S(\omega)$
functions measured using Eq.~\ref{eq:fluctdiss} at late time.
Both functions are in good agreement with the equilibrium distributions.
We emphasize that the displayed continuous curves are no fits, they 
are the Bose-Einstein and Fermi-Dirac distribution functions
parametrized by the same temperature, the latter being extracted as 
explained below.

We stress that the above procedure to extract the distribution 
functions is independent of any assumption about a quasi-particle 
picture. However, for many practical purposes it is very convenient 
to have an effective description of particle number and mode
energy directly in real time -- without the need of a Fourier transform. 
An efficient description is elaborated in \ref{sec:quasiparticle}. The 
value we used for the temperature in the thermal equilibrium distributions 
of Fig.~\ref{fig:fluctdiss} has been actually measured with good accuracy 
based on this quasi-particle picture. In \ref{sec:quasiparticle} we define 
the effective particle number and energy to be used, both for the fermionic 
and bosonic fields. In the latter case, these have already been used 
previously to investigate thermalization in bosonic scalar field 
theories~\cite{Berges:2001fi,Aarts:2001yn,Aarts:2001qa}.


\begin{figure}[t]
\begin{center}
\epsfig{file=evoldist_FD.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{The time-dependent fermionic quasi-particle distribution 
$n_f(t,p)$ as a function of mode energy $p$ for various times $t$. 
We have plotted the inverse slope function $\log(1/n_f -1)$, which 
reduces to a straight line intersecting the origin when $n_f(t,p)$ 
approaches a Fermi-Dirac distribution. This plot shows the data for 
run ``A'' of Fig.~\ref{lostinitialfig}.}
\label{finvslopefig}
\end{figure}

\begin{figure}[t]
\begin{center}
\epsfig{file=evoldist_BE.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{The time-dependent bosonic quasi-particle distribution 
$n(t,p)$ as a function of mode energy $\epsilon(t,p)$ (see text) 
for various times. In this case the inverse slope function is 
$\log(1/n + 1)$, which reduces to a straight line in case of a 
Bose-Einstein distribution. This plot shows the data for run ``A''
of Fig.~\ref{scalarlostinitialfig}.}
\label{sinvslopefig}
\end{figure}

In Figs.~\ref{finvslopefig}~and~\ref{sinvslopefig} we show the 
effective quasi-particle number distributions defined as 
\beq
 \frac{1}{2} - n_f(t,p) = F_V(t,t;p)
\label{fermionnumber}
\eeq
for fermions and 
\bea
 \frac{1}{2} + n(t,p) &=& \epsilon(t,p) \, F_\phi(t,t;p) \\
\label{bosonnumber}
 \epsilon(t,p) &=& 
 \left(\frac{\partial_t\partial_{t'}F_\phi(t,t';p)}{F_\phi(t,t';p)}
 \right)_{t=t'}^{1/2}
\label{bosonenergy}
\eea
where $\ep(t,p)$ is the quasi-particle mode energy for bosons as discussed in
\ref{sec:quasiparticle} (because of chiral symmetry, the fermionic quasi-particle
mode energy is simply $p$).  The curves correspond to the initial conditions of
run ``A'' shown in Figs.~\ref{lostinitialfig}~and~\ref{scalarlostinitialfig}.
One observes how the effective fermionic and bosonic particle numbers change with 
time, approaching Fermi--Dirac and Bose--Einstein distributions respectively.
To emphasize this point, we plot the corresponding ``inverse slope functions'' 
$\log(1/n_f -1)$ and $\log(1/n + 1)$, which reduce to straight lines for 
Fermi--Dirac and Bose--Einstein distributions respectively. The associated 
inverse slopes correspond to the temperature. We see in 
Figs.~\ref{finvslopefig}~and~\ref{sinvslopefig} that both inverse slope 
functions approach straight lines at late times. The associated temperatures 
for fermions and bosons are independent of time and agree very well with 
each other, as shown in Fig.~\ref{compslopefig}. Moreover, for comparison, 
we display in Fig.~\ref{compslopefig} the fermionic inverse slope function
evaluated with the bosonic effective particle number, and vice versa. This
illustrate the degree of sensitivity of the inverse slope functions to the
different statistics and in turn the degree of precision with which we are
able to probe thermalization.


\begin{figure}[t]
\begin{center}
\epsfig{file=finaldist.eps,width=12.cm}
\end{center}
\vspace*{-0.5cm}
\caption{Bosonic and fermionic quasi-particle distributions at late times
as a function of mode energy ($n_p\equiv n(t,p)$ and 
$\epsilon_p\equiv \epsilon(t,p)$ for bosons
and $n_p\equiv n_f(t,p)$ and $\epsilon_p\equiv p$ for fermions). 
Both inverse slope formulas introduced in 
Figs.~\ref{finvslopefig}~and~\ref{sinvslopefig} have been used in order to
demonstrate that the fermions clearly do not follow the Bose-Einstein
statistics, nor the scalars follow the Fermi-Dirac statistics. For
the respective correct statistics the curves  lie on top of each other, 
being described by the same temperature. The value of the latter is used
in Fig. \ref{fig:fluctdiss} to draw the Bose--Einstein and Fermi--Dirac 
curves.}
\label{compslopefig}
\end{figure}






\section{\label{sec:conclusions} Conclusions}


In this paper we have discussed the far-from-equilibrium 
dynamics and subsequent thermalization of a system of coupled 
fermionic and bosonic quantum fields. We solved the nonequilibrium
dynamics beyond mean-field type approximations by calculating the 
complete lowest non-trivial order in a coupling expansion of the 2PI 
effective action, which includes direct scattering as well as memory and 
off-shell effects. To our knowledge this is the first time that such a 
calculation is performed without further approximations. As a result, 
we show that, for various far-from-equilibrium initial conditions,
the late time behavior is universal and uniquely determined by the value 
of the initial energy density. Moreover, we are able to probe thermalization 
with great accuracy by observing the onset of Fermi--Dirac and Bose--Einstein 
statistics. We emphasize that in the present calculation, there is no other 
input than the dynamics dictated by the considered quantum field theory for 
given nonequilibrium initial conditions.

This work can be extended in many directions. The equations we 
derive are also valid in the phase with spontaneous symmetry breaking.
Combined with earlier work on scalar theories \cite{Aarts:2002dj}
(see also \cite{Berges:2002cz}), this provides a description of the 
nonequilibrium dynamics of the linear $\sigma$-model for QCD with 
two quark flavors. The model has served for many years as a valuable 
testing ground for ideas on the equilibrium phase structure of low 
energy QCD at nonzero temperature and density \cite{sigma2}. With the present 
techniques a quantitative understanding of the out-of-equilibrium 
physics of this model is within reach. 


 



\subsection*{Acknowledgment}
We thank C.~Wetterich for many discussions and collaboration
on related work. We are also grateful to W.~Wetzel for his continuous
support with computers. Sz.~B.\ acknowledges the hospitality of the Institute 
f\"ur Theoretische Physik, Heidelberg. His work was supported by 
the short term scholarship program of the DAAD.


\appendix
\section{\label{sec:quasiparticle}Effective particle number and energy}


There are many equivalent ways one can introduce the notion of particle 
number density in the free theory. For example, one can define it as the 
average energy per mode divided by the energy of the corresponding mode. 
In the fully interacting theory, this procedure becomes ambiguous because 
the expression of the total energy receives contributions from interactions 
and the average energy per mode is not uniquely defined. This is a manifestation 
of the fact that the notion of particle number is, strictly speaking, only 
well-defined in the free theory.
However, in many cases, physical systems can be well described by weakly
interacting quasi-particles and the notion of effective (quasi-)particle
number becomes useful again. There are several particle number definitions 
in the literature. Most recently it has been proposed to use some 
derivatives of $F(t,t;p)$ (in our notation) \cite{Garbrecht:2002pd}. Most
of these constructions however suffer from the problem described above.

Here we propose an elegant way to circumvent this difficulty. We consider the 
case of charged fields and construct the effective particle number starting 
from the conserved $U(1)$ currents instead of energy density.\footnote{As 
we shall see below for the scalar fields, the expression for the effective 
particle number we obtain in this way can be directly transposed to the case of
a neutral field.} The former have the nice property that their exact 
expression in terms of the fields do not contain any explicit dependence 
on the interaction part. Moreover, they have a very simple physical 
interpretation in terms of effective particle number in a quasi-particle 
picture. Therefore, by identifying their exact expression with that of 
the quasi-particle picture, one obtains a very simple, general expression 
for the effective particle number density, independently of the interactions. 
Of course, this expression is only meaningful in the case the interacting 
theory is well-described by a quasi-particle picture. The important point 
is that this procedure allows one to construct a general expression for 
the effective particle number density without knowing {\it a priori} which 
part of the interaction is to be considered as the ``dressing'' of the 
the quasi-particles and which part describe their ``residual'' interactions.
Directly applying the present construction to the case of neutral scalar
fields, we recover the particle number definition used in previous studies 
to exhibit thermalization \cite{Aarts:2001qa,Berges:2001fi} (see also 
\cite{Berges:2002cz}).
Another nice feature of the present construction is that it can be explicitely 
shown that the effective particle numbers we define are always positive 
and, for the fermionic case, smaller than one (see \ref{sec:bounded}).


\vspace{.3cm}
\noindent
{\bf Fermions}
\vspace{.3cm}

\noindent
The $4$--current associated with the $U(1)$ symmetry for each given flavor is 
$\sim \bar\psi \gamma^\mu \psi$. In momentum space, the expectation value of 
the latter can be written as $J_f^\mu (t,p) = \tr [\gamma^\mu D^<(t,t,p)]$, where 
the subscript stands for fermions. In terms of the equal time statistical two 
point function, its temporal and spatial components read:\footnote{The constant
factor in the temporal component comes from the fact that we define the current
without the standard normal ordering.}
\bea
 J_f^0 (t,p) & = & 2 \, [1 - 2\,F_V^0(t,t;p) ] \, ,\nonumber \\
 \vec J_f (t,p) & = & -4\bv \, F_V(t,t;p)\, .\nonumber
\eea
We want to identify these expressions with the corresponding ones in a 
quasi-particle picture, that is with the free field expressions. These
are given by:\footnote{Here, we have made explicit use of our 
assumptions of parity and rotational invariance, which imply that the 
different spin states contribute the same, therefore the factor $2$.}
\bea
 J_{f,{\rm QP}}^0 (t,p) & = & 2 \, [1+Q_f(t,p)] \, ,\nonumber \\
 \vec J_{f,{\rm QP}} (t,p) & = & -2 \bv \,[1- 2N_f(t,p)] \, ,\nonumber
\eea
where $Q_f(t,p)=n_f-\bar{n}_f$ is the difference between particle and anti-particle
effective number densities and $N_f(t,p)=(n_f+\bar{n}_f)/2$ is their half-sum. The 
physical content of these expressions is simple: the temporal component $J^0$ 
directly represents the net-charge density per mode $Q_f(t,p)$, whereas the spatial 
part $\vec J$ is the net current density per mode and is therefore sensitive to
the sum of particle and anti-particle number densities.
Identifying the above expressions, we define
\bea
 \frac{1}{2}\,Q_f(t,p) &=& -F_V^0(t,t;p) \, ,\label{genericcharge} \\
 \frac{1}{2}-N_f(t,p) &=& F_V(t,t;p) \, .\label{genericparticle}
\eea
Note that the equal time two-point functions on the RHS of these equations
are real by definition (see Eq.~(\ref{CP})). Moreover, using the anticommutation 
relation for the fermion fields, one can show that these general definitions 
always satisfy $0 \le N_f(t,p) \le 1$ and $-1 \le Q_f(t,p) \le 1$ (see 
\ref{sec:bounded} below).
These properties are important for the above definitions to be physically 
meaningful. 

Introducing a non-vanishing net charge density per mode was useful for the
above general construction. However, in the present paper, we consider only
$CP$--invariant systems, which imply that the latter should vanish. Indeed,
we see from Eqs.~(\ref{CP}) that the requirement of $CP$--invariance imply
that our above definition of net charge density per mode vanishes
identically for all times and for all modes. Therefore, the effective 
particle and anti-particle numbers are equal we have
\bea
 Q_f(t,p) &=&0 \, ,\nonumber\\
 \frac{1}{2}-n_f(t,p) &=& F_V(t,t;p) \, .\nonumber
\eea
where $n_f(t,p)$ is the effective particle number.

\vspace{.3cm}
\noindent
{\bf Bosons}
\vspace{.3cm}

\noindent
Following the same lines as above, let us consider for a moment the case
of a single charged scalar field $\phi$.
The $4$-current associated the corresponding $U(1)$ symmetry is 
$\sim i[\phi^\dagger (\partial^\mu \phi) - (\partial^\mu \phi^\dagger) \phi]$ 
and, as in the case of fermions, its expectation value has a simple expression 
in terms of the equal-time statistical two-point function of the charged field. 
In momentum space, it reads:
\bea
 J_b^0 (t,p) & = &  i (\partial_t-\partial_{t'}) \,F_\phi(t,t';p) |_{t'=t} -1
 \nonumber \\
 \vec J_b (t,p) & = & 2 \,\bp\, F_\phi(t,t;p)\, .\nonumber
\eea
where, as before, the constant contribution comes from the fact that we define
the current without the usual normal ordering. Here, the statistical two-point
function for the charged scalar field is defined as the expectation value of
the anticommutator of two fields operators: 
$F_\phi(t,t';p)=\mbox{$\frac{1}{2}
\langle[\phi(t,\bp);\phi^\dagger(t',\bp)]_+\rangle$}$. It has the symmetry
property $F_\phi(t,t';p)=F_\phi^*(t',t;p)$, so that the above expressions are real.
The corresponding quasi-particle expressions read:
\bea
 J_{b,{\rm QP}}^0 (t,p) & = & Q_b(t,p)-1 \, ,\nonumber \\
 \vec J_{b,{\rm QP}} (t,p) & = & \frac{\bp}{\epsilon(t,p)} \, [1+2N_b(t,p)] 
 \, ,\nonumber
\eea
where $Q_b(t,p)$ and $N_b(t,p)$ have the same meaning as before in terms
of effective particle and anti-particle number densities and where
$\epsilon(t,p)$ is the quasi-particle energy. We therefore define
\bea 
 Q_b(t,p) &=& i (\partial_t-\partial_{t'}) \,F_\phi(t,t';p) |_{t'=t} \, ,
\label{app_scalarcharge} \\
 1+ 2N_b(t,p) &=& 2 \, \epsilon(t,p) \, F_\phi(t,t;p)\, .
\label{app_scalarnumber}
\eea
Note that the RHS of both expressions are real quantities, as they should if
the LHS are to be interpreted as charge and quasi-particle number densities
respectively.

We now need to define an effective quasi-particle energy.\footnote{Note 
that this was not necessary in the fermionic case, because of our assumption
of chiral symmetry. In fact, in this case, one can repeat the following 
argument to get an expression for the fermionic effective quasi-particle 
energy. One obtains $\epsilon_f (t,p) = p$, as expected.} For this purpose, 
we use the free field inspired expression for the average energy per mode, 
which, in the case of a single charged scalar field, can be expressed as:
$$
 \partial_t \partial_{t'} F_\phi(t,t';p) |_{t'=t} +
 \epsilon^2(t,p) F_\phi(t,t;p) \equiv \epsilon(t,p) \, \Big[2n(t,p)+1\Big] \, ,
$$
in terms of the statistical two-point function.
Therefore, we obtain, for the effective quasi-particle energy:
\beq
\label{app_scalarenergy}
 \epsilon^2(t,p) = \left(\frac{\partial_t\partial_{t'}F_\phi(t,t';p)}{F_\phi(t,t';p)}
 \right)_{t=t'} \, ,
\eeq
It is easy to show that the combination of equal time correlators
appearing on the RHS of the above equation is indeed positive. Moreover, 
using commutation relations for the scalar field, one can show that our
effective particle number $N_b(t,p)$, as given by
Eqs.~(\ref{app_scalarnumber}) and (\ref{app_scalarenergy}), is a positive
quantity (see~\ref{sec:bounded} below).  Let us also mention that in the case of
$CP$--invariant systems, our effective charge density per mode
(\ref{app_scalarcharge}) vanishes, as it should.\footnote{This immediately
follows from the behavior of the statistical propagator under
$CP$--transformation: $F_\phi (t,t';p) \longrightarrow F_\phi (t',t;p)$.} 

When dealing with neutral scalar fields, as it is the case in the present
paper, we simply use the same formula we derived above. Notice that this is
consistent since in that case one has $F_\phi(t,t';p)=F_\phi(t',t;p)$, which
directly implies that $Q_b(t,p) = 0$. Therefore, in the present paper, we use
the definitions
\beq
\label{app_neutral}
 \frac{1}{2}+ n(t,p) = \epsilon(t,p) \, F_\phi(t,t;p) \\
\eeq
together with (\ref{app_scalarenergy}) for the bosonic effective particle number
$n(t,p)$ and energy $\epsilon(t,p)$ for each individual scalar 
species.\footnote{There is actually a
rationale for transposing the formula we derived above directly to the neutral
case. Indeed, one can apply the above construction to the charged pion field
$\pi^\pm =\frac{1}{\sqrt 2} (\pi_1 \pm i\pi_2)$. Expressing everything in terms
of the neutral fields $\pi_1$ and $\pi_2$ and exploiting the symmetry of our
model, one obtains that the effective particle numbers corresponding to the
$\pi_1$ and $\pi_2$ fields are degenerate and given by the formula
(\ref{app_neutral}). Exploiting the full $O(4)$ symmetry of the model, we
conclude that the same formula holds for the $\sigma$ and $\pi_3$ fields as well.}


\section{\label{sec:bounded}}


Here we show that the combinations of equal time two-point functions we 
used in this paper to define effective particle number densities are 
always positive and, for the fermionic case, smaller than one. This is 
actually a simple consequence of the (anti-)commutation relations of field 
operators.

\vspace{.3cm}
\noindent
{\bf Fermions}
\vspace{.3cm}

\noindent
As a first exercise, let us show that for any operators $\varphi$ and 
$\varphi^\dagger$ satisfying the anticommutation relations 
$$
 [\varphi,\varphi^\dagger]_{_{_+}}=1 
$$
and 
$$
 [\varphi,\varphi]_{_+}=[\varphi^\dagger,\varphi^\dagger]_{_+}=0 \, ,
$$
one has
\beq
\label{App_ineqfermion}
 0 \,\le\, \<\varphi^\dagger\,\varphi\> \,\le\, 1 \, ,
\eeq
where the brackets denote an average with respect to any density matrix.

The left inequality above is trivially obtained for example by inserting a
complete sum of states between the operators $\varphi^\dagger$ and $\varphi$.
One obtains a sum of positive quantities which is of course positive. To show
the second inequality, we introduce the hermitian operator $N=\varphi^\dagger
\varphi$. Using the anticommutation relations above, it is easy to see that
$N^2=N$, from which it follows that 
$$
 \Delta n^2 \equiv \Big\<\,\Big(N-\<N\>\Big)^2\,\Big\> = \<N^2\> - \<N\>^2 = 
 \<N\> \Big(1-\<N\>\Big) \, .
$$
It is clear that $\Delta n^2 \ge 0$ and we have just showed that $\<N\> \ge 0$.
We therefore conclude from the above equation that $\<N\> \le 1$ as announced.

Let us now come to our effective number densities.  Using
Eqs.~(\ref{genericcharge})
and (\ref{genericparticle}) and recalling that $Q_f\equiv n_f-\bar n_f$ and
$N_f\equiv(n_f+\bar n_f)/2$, we get, for the effective particle and
anti-particle number densities:
\bea
 \frac{1}{2}-n_f(t,p)&=&F_V(t,t,p)+F_V^0(t,t,p) 
 = \frac{1}{4} \tr \Big[(\gamma_0 + \bv\cdot\vec\gamma)F(t,t,p) \Big]
 \nonumber\\
 \frac{1}{2}-\bar{n}_f(t,p)&=&F_V(t,t,p)-F_V^0(t,t,p)
 =-\frac{1}{4} \tr \Big[(\gamma_0 - \bv\cdot\vec\gamma)F(t,t,p) \Big]\, .  
 \nonumber
\eea
In terms of the fermionic field operators $\psi(t,\bp)$ and $\bar\psi(t,\bp)$,
which satisfy the anticommutation relations
$$
 [\psi(t,\bp),\bar\psi(t,\bp)]_{_+}=\gamma_0
$$
and 
$$
 [\psi(t,\bp),\psi(t,\bp)]_{_+}=[\bar\psi(t,\bp),\bar\psi(t,\bp)]_{_+}=0 \, , 
$$
one has 
$$
 F(t,t,p)=\frac{1}{2} \Big\< [\psi(t,\bp),\bar\psi(t,\bp)]_{_-} \Big\> \, .
$$
Therefore one can rewrite (from now on, we drop the explicit $t$ and $\bp$
dependence):
\bea
 n_f&=&\frac{1}{4} \,\<\bar\psi (\gamma_0 + \bv\cdot\vec\gamma) \psi \> 
 \nonumber\\
 1-\bar n_f&=&\frac{1}{4} \,\<\bar\psi (\gamma_0 - \bv\cdot\vec\gamma) \psi \> 
 \nonumber
\eea
Now we introduce the operator
$$
 \varphi=\frac{\gamma_0 + \bv\cdot\vec\gamma}{\sqrt2}\,\psi \, ,
$$
in term of which,
\beq
\label{App_fpart}
 n_f=\frac{1}{4} \,\sum_{\alpha=1}^4 
 \<\varphi^\dagger_\alpha \varphi_\alpha\> \, ,
\eeq
where we explicitely wrote the sum over Dirac indices.  For a given Dirac
indice, it is easy to check that
\beq
\label{App_anticom}
 [\varphi_\alpha,\varphi^\dagger_\alpha]_{_{_+}}=
 \Big(1-\gamma_0\, \bv\cdot\vec\gamma\Big)_{\alpha \alpha} = 1
\eeq
and 
$$
 [\varphi_\alpha,\varphi_\alpha]_{_+}=
 [\varphi^\dagger_\alpha,\varphi^\dagger_\alpha]_{_+}=0 \, ,
$$
where we specialized to the Dirac basis to write the last equality of
Eq.~(\ref{App_anticom}).\footnote{This is more convenient, but not necessary.
It is simple to adapt the argument to any basis.} Using (\ref{App_ineqfermion})
for each individual $\alpha$, we conclude from (\ref{App_fpart}) that
$$
 0 \le n_f (t,p) \le 1
$$
for any time $t$ and any momentum $p$. It is straightforward to repeat the above
arguments to show that 
$$ 
0 \le \bar n_f (t,p)  \le 1\, .
$$

\vspace{.3cm}
\noindent
{\bf Bosons}
\vspace{.3cm}

\noindent
In the case of bosons, we first use Eqs.~(\ref{app_scalarenergy}) and
(\ref{app_scalarenergy}), to rewrite
$$
 \frac{1}{2} + n(t,p) = \sqrt{F_\phi(t,t;p) \, 
 [\partial_t \partial_{t'} F_\phi(t,t';p)]_{t'=t}} \, .
$$
Recalling the definition of the statistical propagator in terms of field
operators:
\bea
 F_\phi(t,t;p) &=& \<\phi^\dagger (t,\bp) \, \phi(t,\bp) \> \, , \nonumber\\
 \partial_t \partial_{t'} F_\phi(t,t';p)]_{t'=t} &=&
 \<\Pi^\dagger (t,\bp) \, \Pi(t,\bp) \> \, , \nonumber 
\eea
where $\Pi(t,\bp)$ and $\Pi^\dagger(t,\bp)$ are the canonical momenta conjugate
to $\phi^\dagger(t,\bp)$ and $\phi(t,\bp)$ respectively (e.g.~$\Pi(t,\bp)=\dot\phi^\dagger(t,\bp)$), 
we see that 
$$
 n(t,p) \ge 0 \,\,<=>\,\,
 \<\phi^\dagger \, \phi \> \,
 \<\Pi^\dagger  \, \Pi \> \ge \frac{1}{2} \, ,
$$
where we dropped the explicit time and momentum dependence of field operators. 
The second inequality is nothing but Heisenberg's uncertainty principle, a direct
consequence of the equal time canonical commutation relations of field 
operators \cite{BohmQM}.


\vspace*{-0.3cm}
\begin{thebibliography}{10}

\bibitem{Braun-Munzinger:2001mh} 
P.~Braun-Munzinger and J.~Stachel,
%``Particle ratios, equilibration, and the QCD phase boundary,''
J.\ Phys.\ G {\bf 28} (2002) 1971;
%,
%CITATION = ;%
%\cite{Heinz:2002un}
%\bibitem{Heinz:2002un}
U.~W.~Heinz and P.~F.~Kolb,
\textit{``Two RHIC puzzles: Early thermalization and the HBT problem''},
arXiv:
Proceedings of the 18th Winter Workshop on Nuclear Dynamics. 
Edited by R. Bellwied, J. Harris, and W. Bauer. EP Systema, Debrecen, Hungary, 2002. pp. 205-216.
%CITATION = ;%

\bibitem{Serreau:2002yr}
J.~Serreau,
\textit{``Kinetic equilibration in heavy ion collisions: The role of elastic  processes''},
arXiv:.
%%CITATION = %
Proceedings of Quark Matter 2002, to appear in Nucl. Phys. {\bf A};
%\bibitem{Serreau:2001xq} 
J.~Serreau and D.~Schiff, 
%``Kinetic equilibration in heavy ion collisions: The role of elastic processes,'' 
JHEP {\bf 0111} (2001) 039 
; 
%%CITATION = ;%% 
%\bibitem{Baier:2002bt} 
R.~Baier, A.~H.~Mueller, D.~Schiff and D.~T.~Son, 
%``Does parton saturation at high density explain hadron multiplicities at RHIC?,'' 
Phys.\ Lett.\ B {\bf 539} (2002) 46 
. 
%%CITATION = ;%%



\bibitem{Blaizot:2001nr} 
For a review see J.~P.~Blaizot and E.~Iancu,
%``The quark-gluon plasma: Collective dynamics and hard thermal loops,''
Phys.\ Rept.\  {\bf 359} (2002) 355,
%
and references therein;
%CITATION = ;%
See also: 
%\bibitem{Arnold:2002zm} 
P.~Arnold, G.~D.~Moore and L.~G.~Yaffe, 
\textit{``Effective kinetic theory for high temperature gauge theories''},
arXiv:. 
%%CITATION = ;%% 


\bibitem{gaussian1}
F.\ Cooper, S.\ Habib, Y.\ Kluger, E.\ Mottola, J.P.\ Paz,
P.R.\ Anderson, Phys.\ Rev.\ {\bf D50} (1994) 2848;
D.~Boyanovsky, H.~J.~de Vega, R.~Holman, J.~Salgado,
Phys.\ Rev.\ {\bf D59} (1999) 125009;

\bibitem{gaussian2a}  
%\cite{Cooper:2002yv}
%\bibitem{Cooper:2002yv}
F.~Cooper and V.~M.~Savage,
%``Dynamics of the chiral phase transition in the 2+1 dimensional  Gross-Neveu model,''
Phys.\ Lett.\ B {\bf 545} (2002) 307
%;
%CITATION = ;%
%\cite{Chodos:2000cc}
%\bibitem{Chodos:2000cc}
A.~Chodos, F.~Cooper, W.~Mao and A.~Singh,
%``Equilibrium and non-equilibrium properties associated with the chiral  
%phase transition at finite density in the Gross-Neveu model,''
Phys.\ Rev.\ D {\bf 63} (2001) 096010
%.
%CITATION = ;%
%\cite{Boyanovsky:2001va}
%\bibitem{Boyanovsky:2001va}
D.~Boyanovsky, H.~J.~De Vega, R.~Holman and M.~R.~Martin,
%``Non-equilibrium large N Yukawa dynamics: Marching through the Landau  pole,''
Phys.\ Rev.\ D {\bf 65} (2002) 045007
%.
%CITATION = ;%


\bibitem{gaussian2b} 
%\cite{Baacke:1998di}
%\bibitem{Baacke:1998di}
J.~Baacke, K.~Heitmann and C.~Patzold,
%``Nonequilibrium dynamics of fermions in a spatially homogeneous scalar  background field,''
Phys.\ Rev.\ D {\bf 58} (1998) 125013
%;
%CITATION = ;%
%\cite{Baacke:1999nq}
%\bibitem{Baacke:1999nq}
J.~Baacke and C.~Patzold,
%``Renormalization of the nonequilibrium dynamics of fermions in a flat  FRW universe,''
Phys.\ Rev.\ D {\bf 62} (2000) 084008
%.
%CITATION = ;%


\bibitem{Cooper:1996ii} 
F.~Cooper, S.~Habib, Y.~Kluger and E.~Mottola,
%``Nonequilibrium dynamics of symmetry breaking in lambda Phi**4 field  
%theory,''
Phys.\ Rev.\ D {\bf 55} (1997) 6471
%%;
%CITATION = ;%
.

\bibitem{Aarts:2001wi} 
G.~Aarts, G.~F.~Bonini and C.~Wetterich,
%``Exact and truncated dynamics in nonequilibrium field theory,''
Phys.\ Rev.\ D {\bf 63} (2001) 025012
%.         

\bibitem{Berges:2001fi} 
J.~Berges,
%``Controlled nonperturbative dynamics of quantum fields out of  equilibrium,''
Nucl.\ Phys.\ A {\bf 699} (2002) 847
%.
%CITATION = ;%

\bibitem{Berges:2000ur} 
J.~Berges and J.~Cox,
%``Thermalization of quantum fields from time-reversal invariant evolution  equations,''
Phys.\ Lett.\ B {\bf 517} (2001) 369
%.
%CITATION = ;%

\bibitem{Aarts:2002dj} 
G.~Aarts, D.~Ahrensmeier, R.~Baier, J.~Berges and J.~Serreau,
%``Far-from-equilibrium dynamics with broken symmetries from the 2PI-1/N  expansion,''
Phys.\ Rev.\ D {\bf 66} (2002) 045008
%.
%CITATION = ;%

\bibitem{Baym} 
J.~M.~Luttinger and J.~C.~Ward, Phys.\ Rev.\ {\bf 118} (1960) 1417;
G.~Baym, Phys.\ Rev.\ {\bf 127} (1962) 1391.
%CITATION = PHRVA,D10,2428;%          

\bibitem{Cornwall:1974vz} 
J.~M.~Cornwall, R.~Jackiw and E.~Tomboulis,
%``Effective Action For Composite Operators,''
Phys.\ Rev.\ D {\bf 10} (1974) 2428%;
%CITATION = PHRVA,D10,2428;%

\bibitem{Calzetta:1988cq} 
E.~Calzetta and B.~L.~Hu,
%``Nonequilibrium Quantum Fields: Closed Time Path Effective Action, 
%Wigner Function And Boltzmann Equation,''
Phys.\ Rev.\ D {\bf 37} (1988) 2878.
%CITATION = PHRVA,D37,2878;%

\bibitem{Ivanov:1998nv} 
Y.~B.~Ivanov, J.~Knoll and D.~N.~Voskresensky,
%``Self-consistent approximations to non-equilibrium many-body theory,''
Nucl.\ Phys.\ A {\bf 657} (1999) 413
%.
%CITATION = ;%

\bibitem{Danielewicz:kk} 
For reviews, see
P.~Danielewicz,
%``Quantum Theory Of Nonequilibrium Processes. I,''
Annals Phys.\  {\bf 152} (1984) 239;
%CITATION = APNYA,152,239;%
%\cite{Brown:1998zx}
%\bibitem{Brown:1998zx}
for relativistic fermions, see also
%\cite{Elze:1989un}
%\bibitem{Elze:1989un}
H.~T.~Elze and U.~W.~Heinz,
%``Quark - Gluon Transport Theory,''
Phys.\ Rept.\  {\bf 183} (1989) 81;
%CITATION = PRPLC,183,81;%
D.~A.~Brown and P.~Danielewicz,
%``Partons in phase space,''
Phys.\ Rev.\ D {\bf 58} (1998) 094003
%.
%CITATION = ;%

\bibitem{Greene:2000ew} 
P.~B.~Greene and L.~Kofman,
%``On the theory of fermionic preheating,''
Phys.\ Rev.\ D {\bf 62} (2000) 123516
%.
%CITATION = ;%

\bibitem{Joyce:2000ed} 
M.~Joyce, K.~Kainulainen and T.~Prokopec,
%``Fermion propagator in a nontrivial background field,''
JHEP {\bf 0010} (2000) 029
% .
%CITATION = ;%
%\cite{Kainulainen:2002th}
%\bibitem{Kainulainen:2002th}
K.~Kainulainen, T.~Prokopec, M.~G.~Schmidt and S.~Weinstock,
%``Semiclassical force for electroweak baryogenesis: Three-dimensional  derivation,''
Phys.\ Rev.\ D {\bf 66} (2002) 043502
%.
%CITATION = ;%

\bibitem{Lawrie:2000jg} 
%[damping and relaxation times in a Yukawa model]
I.~D.~Lawrie and D.~B.~McKernan,
%``Nonequilibrium perturbation theory for spin-1/2 fields,''
Phys.\ Rev.\ D {\bf 62} (2000) 105032
%.
%CITATION = ;%


\bibitem{Aarts:1998td} 
G.~Aarts and J.~Smit,
%``Real-time dynamics with fermions on a lattice,''
Nucl.\ Phys.\ B {\bf 555} (1999) 355
%.
%CITATION = ;%


\bibitem{Schwinger:1961qe} 
J.~Schwinger,
%``Brownian Motion Of A Quantum Oscillator,''
J.\ Math.\ Phys.\ {\bf 2} (1961) 407;
%CITATION = JMAPA,2,407;%
%\cite{Keldysh:1964ud}
%\bibitem{Keldysh:1964ud}
L.~V.~Keldysh,
%``Diagram Technique For Nonequilibrium Processes,''
Zh.\ Eksp.\ Teor.\ Fiz.\ {\bf 47} (1964) 1515 
[Sov.\ Phys.\ JETP {\bf 20} (1965) 1018].
%CITATION = ZETFA,47,1515;%
%\bibitem{Chou:1985es} 
K.~C.~Chou, Z.~B.~Su, B.~L.~Hao and L.~Yu,
%``Equilibrium And Nonequilibrium Formalisms Made Unified,''
Phys.\ Rept.\  {\bf 118} (1985) 1.
%CITATION = PRPLC,118,1;%


\bibitem{Aarts:2001qa} 
G.~Aarts and J.~Berges,
%``Nonequilibrium time evolution of the spectral function in quantum field  theory,''
Phys.\ Rev.\ D {\bf 64} (2001) 105010
%.
%CITATION = ;%

\bibitem{Berges:2000ew} 
J.~Berges, N.~Tetradis and C.~Wetterich,
%``Non-perturbative renormalization flow in quantum field theory and  statistical physics,''
Phys.\ Rept.\  {\bf 363} (2002) 223
%.

\bibitem{montvay} 
I.~Montvay, G.~M\"unster, \textit{``Quantum Fields on a Lattice''},
Cambridge University Press, 1994


\bibitem{Berges:2002cz} 
J.~Berges and J.~Serreau,
\textit{``Parametric resonance in quantum field theory,''}
arXiv:.
%CITATION = ;%

\bibitem{Aarts:2000wi} 
G.~Aarts, G.~F.~Bonini and C.~Wetterich, 
%``Exact and truncated dynamics in nonequilibrium field theory,'' 
Phys.\ Rev.\ D {\bf 63} (2001) 025012 
. 
%%CITATION = ;%% 


 
\bibitem{LOinh}  
G.~Aarts, J.~Smit,
%``Particle production and effective thermalization in inhomogeneous mean field theory,''
Phys.\ Rev.\ {\bf D61} (2000) 025002;
%\bibitem{Salle:2000jb}
M.~Salle, J.~Smit and J.~C.~Vink,
%``Staying thermal with Hartree ensemble approximations,''
Nucl.\ Phys.\ B {\bf 625} (2002) 495;
%.
%\bibitem{Bettencourt:2001xg}
L.~M.~Bettencourt, K.~Pao and J.~G.~Sanderson,
%``Dynamical behavior of spatially inhomogeneous relativistic lambda  
%phi**4 quantum field theory in the Hartree approximation,''
Phys.\ Rev.\ D {\bf 65} (2002) 025015
%.

\bibitem{Aarts:2001yn} 
G.~Aarts and J.~Berges,
%``Classical aspects of quantum fields far from equilibrium,''
Phys.\ Rev.\ Lett.\  {\bf 88} (2002) 041603
%.
%CITATION = ;%

\bibitem{Garbrecht:2002pd} 
B.~Garbrecht, T.~Prokopec and M.~G.~Schmidt,
\textit{``Particle number in kinetic theory''},
arXiv:.
%CITATION = ;%

\bibitem{sigma}
M.~Gell-Mann and M.~Levy, 
%``The Axial Vector Current In Beta Decay,'' 
Nuovo Cim.\ {\bf 16} (1960) 705; 
%%CITATION = NUCIA,16,705;%% 
C.~Itzykson and J.-B. Zuber, \textit{``Quantum Field Theory''},
McGraw-Hill International Editions.

\bibitem{sigma2} 
R.~D.~Pisarski and F.~Wilczek, 
%``Remarks On The Chiral Phase Transition In Chromodynamics,'' 
Phys.\ Rev.\ D {\bf 29} (1984) 338; 
%%CITATION = PHRVA,D29,338;%% 
%\bibitem{Berges:1998sd} 
J.~Berges, D.~U.~Jungnickel and C.~Wetterich, 
%``The chiral phase transition at high baryon density from nonperturbative flow equations,'' 
Eur.\ Phys.\ J.\ C {\bf 13} (2000) 323 
;
%%CITATION = ;%% 
%\bibitem{Scavenius:2000qd} 
O.~Scavenius, A.~Mocsy, I.~N.~Mishustin and D.~H.~Rischke, 
%``Chiral phase transition within effective models with constituent quarks,'' 
Phys.\ Rev.\ C {\bf 64} (2001) 045202 
. 
%%CITATION = ;%% 

\bibitem{BohmQM} 
Arno Bohm, 
\textit{``Quantum Mechanics: Foundations and Applications''},
Springer-Verlag, Third Edition.


\end{thebibliography}


\end{document}


