%Paper: 
%From: rau@eu10.mpi-hd.mpg.de (jochen rau)
%Date: Wed, 9 Feb 1994 18:19:40 +0100


% LaTeX-file;
% 2 postscript files are appended. They are marked
% by **FIGURE 1** and **FIGURE 2**
%
%
\documentstyle[12pt]{article}
\parindent=5mm
\begin{document}
%
\def\d{{\rm d}}
\def\e{{\rm e}}
\def\bra#1{\langle #1|}
\def\ket#1{|#1\rangle}
\def\mittel#1{\langle#1\rangle}
\def\r#1{(\ref{#1})}
\def\halb{{\textstyle{1\over 2}}}
%
\baselineskip=15pt
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%
\begin{titlepage}
\begin{flushright}
DUKE-TH-93-55 \\
 \\
February 1994
\end{flushright}
\vspace{1cm}
\begin{center}
{\Huge Pair production in the quantum Boltzmann equation}
\\
\vspace{1cm}
{Jochen Rau\footnote{present address:
Max-Planck-Institut f\"ur Kernphysik,
Postfach 103980, 69029 Heidelberg, Germany}\footnote{e-mail:
rau@eu10.mpi-hd.mpg.de}}
\\
{\sl Physics Department\\
Duke University, Durham, NC 27708--0305}
\end{center}
\vspace{1cm}
\centerline{\bf Abstract}
A source term in the quantum Boltzmann equation, which
accounts for the spontaneous creation of $e^+e^-$-pairs
in external electric fields, is derived from first principles
and evaluated numerically.
Careful analysis of time scales reveals that this source term is
generally non-Markovian.
This implies in particular that there may
be temporary violations of the $H$-theorem.
\\
\vfill\noindent
{\em PACS numbers: 12.38.Mh, 25.75.+r, 52.25.Dg, 05.70.Ln}
%
\end{titlepage}
%
The evolution of the quark-gluon plasma, believed to be formed in the
course of relativistic heavy-ion collisions,
is commonly
described by means of a transport equation
\cite{bialas:czyz:1,bialas:czyz:2,kajantie:matsui,gatoff:etal}.
It is well understood how a transport equation can account for
acceleration in
external fields, scattering,
or \mbox{(hadro-)}chemical reactions of the microscopic constituents.
There is, however, another physical process which becomes increasingly
important at high energies:
regions of very large chromoelectric
field strength may develop and subsequently decay
by emitting quark-antiquark pairs
\cite{ehtamo,biro:etal}.
This gives rise
to the fragmentation of chromoelectric flux tubes (`strings'),
a mechanism
frequently invoked to model hadron production
\cite{casher:etal,glendenning:matsui,banerjee:etal,kornel:string}.
How such spontaneous creation of particles
can be incorporated into a transport equation is still not fully
understood.

Clearly, the transport equation has
to be modified by a source term.
What is this source term? How can it be derived from the
underlying microscopic dynamics?
These issues have recently been approached
in a Wigner function formulation
\cite{eisenberg:kaelbermann,rafelski:etal,kluger:1,kluger:2,kluger:central}.
But, aside from the fact that it lacks an intuitive
probabilistic interpretation, this approach
suffers from several practical limitations.
The source term cannot be determined completely:
it is not known how the
longitudinal momenta of the produced particles are distributed.
It has been suggested that the distribution is a $\delta$-function
\cite{casher:etal};
but while such an ansatz
may be useful for practical purposes
\cite{kluger:1,kluger:2,kluger:central},
it is certainly not exact.
Furthermore, an interplay
of pair creation and collisions
--- possibly leading to a modification of the source term ---
has not yet been considered.
And finally, the Wigner description is not suited for discussing
the apparent irreversibility of the particle creation process or the
associated generation of entropy.
Since pair creation in an
external field is merely a single-particle problem (see below),
the Wigner function retains complete information about the
microscopic state of the system.
Yet irreversibility never manifests itself
on the microscopic level;
it only emerges after a suitable coarse-graining.

I choose a different approach.
In collision experiments one usually
measures the momentum distribution
of the outgoing particles; i.e., one determines
the occupation
$n_\mp(\vec p,t):=\mittel{N_\mp(\vec p)}(t)$
of the various momentum
states,
with the number operators given by
\begin{eqnarray}
N_-(\vec p)&:=&\sum_{m_z} a^\dagger(\vec p,m_z) a(\vec p,m_z)
\quad,
\nonumber \\
N_+(\vec p)&:=&\sum_{m_z} b^\dagger(\vec p,m_z) b(\vec p,m_z)
\quad.
\end{eqnarray}
(Here $a$ and $b$ denote particle and antiparticle field operators,
respectively, $\vec p$ the momentum and $m_z$ the spin component.)
This suggests attempting to describe
the evolution of the occupation numbers $\{n_\mp(\vec p,t)\}$
directly
and to derive a kinetic equation for them
--- including the source term --- from first principles.
I will do so with the help of
a very powerful and broadly applicable tool: the
so-called projection method.
This method, pioneered by Nakajima \cite{nakajima:1},
Zwanzig \cite{zwanzig:1,zwanzig:2,zwanzig:4}
and others
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\cite{mori:1,robertson:pr1,robertson:pr2,robertson:jmp,kawasaki:gunton,grabert:book},
is based on projecting the motion of the quantum system onto
a low-dimensional subspace (the `level of description')
of the space of observables (Liouville space).
It allows for a clear definition of
crucial concepts like the memory time or the
coarse-grained entropy, making it especially suited
for an investigation of the irreversible features of the dynamics.

As in the works cited above, my
investigation is based on a simple model from
quantum electrodynamics.
I consider
the creation of $e^+e^-$-pairs in a
homogeneous, time-independent electric field $\vec E$,
a process often referred to as the Schwinger mechanism
\cite{sauter:1,heisenberg:euler,schwinger:1}.
The starting point is the
Dirac equation
\begin{equation}
i\hbar{\d\over\d t}\ket{\psi(t)}=H\ket{\psi(t)}
\end{equation}
with
\begin{equation}
H=\hat{\vec p}\cdot\vec\alpha+m\beta+qA_0
\end{equation}
and $A_0(\vec r)=-\vec E\cdot\vec r$
($q=-|e|$ for electrons).
We further define
${\vec p(t)}:={\vec p+q\vec E t}$,
the transverse energy
$\epsilon_\perp:=\sqrt{m^2+p_\perp^2}$,
the total kinetic energy
$\epsilon[\vec p(t)]:=
\sqrt{\epsilon_\perp^2+p_{\|}(t)^2}$,
and the dynamical phase
\begin{equation}
\phi_{fi}:=
-{1\over\hbar}\int_{t_i}^{t_f}\d t'\,
\epsilon[\vec p(t')]
\quad.
\end{equation}
`Longitudinal' and `transverse' refer to the direction of the electric field.

The eigenstates $\ket{i,\pm}\equiv \ket{\vec p(t_i),m_z,\pm}$
which correspond to momentum $\vec p(t_i)$,
spin component $m_z$
and positive or negative energy $\pm\epsilon[\vec p(t_i)]$,
evolve according to
\begin{equation}\label{evol:law}
U(t_f,t_i)
\left(
\begin{array}{c}
\ket{i,+} \\
\ket{i,-}
\end{array}
\right)
=
\left(
\begin{array}{cc}
\alpha_{fi} & \beta_{fi} \\
-\beta^*_{fi} & \alpha^*_{fi}
\end{array}
\right)
\left(
\begin{array}{cc}
\e^{+i\phi_{fi}} & 0 \\
0 & \e^{-i\phi_{fi}}
\end{array}
\right)
\left(
\begin{array}{c}
\ket{f,+} \\
\ket{f,-}
\end{array}
\right)
\quad.
\end{equation}
The evolution thus mixes
positive and negative energy eigenstates,
with respective amplitudes $\alpha_{fi}$ and
$\beta_{fi}$;
$|\beta_{fi}|^2$ equals
the probability for having created an $e^+e^-$-pair
with (final) momenta $\pm\vec p(t_f)$ during the time
interval $[t_i,t_f]$.
The amplitudes are determined by
the differential equation
\begin{equation}\label{diff:equation}
\left(
\begin{array}{c}
\dot\alpha_{fi} \\ \dot\beta_{fi}
\end{array}
\right)
=
{qE\over 2}\cdot
{\epsilon_\perp\over \epsilon[\vec p(t_f)]^2}
\left(
\begin{array}{cc}
0 & -\e^{-i2\phi_{fi}} \\
\e^{+i2\phi_{fi}} & 0
\end{array}
\right)
\left(
\begin{array}{c}
\alpha_{fi} \\ \beta_{fi}
\end{array}
\right)
\end{equation}
with initial conditions
$\alpha_{ii}=1$ and $\beta_{ii}=0$,
and the dot indicating differentiation with respect to $t_f$.

In view of applying the projection method,
the above results have to be translated into
the language of field operators.
To do so, I will use the formulation of
quantum statistical mechanics in Liouville space
\cite{fick:processes}.
There the evolution of (Heisenberg picture)
operators is determined by the so-called super-operators
${\cal L}$ (`Liouvillian') and
${\cal U}$;
these super-operators play a role analogous to that
of $H$ and $U$ in Hilbert space.
Employing the shorthand notations
$a_j\equiv a(\vec p(t_j),m_z)$ and $b_{-j}\equiv
b(-\vec p(t_j), -m_z)$ for the particle and antiparticle field operators,
and making use of the general rule
${\cal U}(t_2,t_1)a^\dagger (\psi)=a^\dagger (U(t_2,t_1)\psi)$,
one finds
\begin{equation}\label{bogoliubov:1}
{\cal U}(t_2,t_1)
\left(
\begin{array}{c}
a^\dagger_1 \\
b_{-1}
\end{array}
\right)
=
\left(
\begin{array}{cc}
\alpha_{21} & \beta_{21} \\
-\beta^*_{21} & \alpha^*_{21}
\end{array}
\right)
\left(
\begin{array}{cc}
\e^{+i\phi_{21}} & 0 \\
0 & \e^{-i\phi_{21}}
\end{array}
\right)
\left(
\begin{array}{c}
a^\dagger_2 \\
b_{-2}
\end{array}
\right)
\quad;
\end{equation}
the evolution law for $(a, b^\dagger)$ follows by
Hermitian conjugation.
Thus pair creation can be described by
a time-dependent Bogoliubov transformation
\cite{fetter:book}.

The Liouvillian
\begin{equation}
{\cal L}=i\left.{\partial\over\partial t_2}\right|_{t_2=t_1}
{\cal U}(t_2,t_1)
\end{equation}
may be written as the sum
\begin{equation}
{\cal L}={\cal L}_{\rm diag}+\delta{\cal L}
\end{equation}
of a diagonal part, responsible for acceleration, and an
off-diagonal part which
is responsible for the mixture of particle and antiparticle
states, i.e., for pair creation.
With the definition
$\dot\beta_{11}:=
\left.\dot\beta_{21}\right|_{t_2=t_1}$,
the latter is given by
\begin{equation}
\delta{\cal L}
\left(
\begin{array}{c}
a^\dagger_1 \\
b_{-1}
\end{array}
\right)
=i
\left(
\begin{array}{cc}
0 & \dot\beta_{11} \\
-\dot\beta^*_{11} & 0
\end{array}
\right)
\left(
\begin{array}{c}
a^\dagger_1 \\
b_{-1}
\end{array}
\right)
\quad.
\end{equation}

Starting from the above microscopic equations, we now want to
derive a kinetic equation for the
occupation numbers
$n_\mp(\vec p,t)$.
Provided the initial state is the vacuum,
\begin{equation}
{\rho(t_0)}=\ket 0\bra 0
\quad,
\end{equation}
momentum and charge conservation dictate
$n_+(\vec p,t)=n_-(-\vec p,t)$
for all later times $t$;
it then suffices to consider
the evolution of only, say, the electron occupation
numbers $n_-(\vec p,t)$.
Their evolution equation must have the structure
\begin{equation}
\dot n_-(\vec p,t)
+q\vec E\cdot\nabla_{\vec p}\,n_-(\vec p,t)
=
\dot n_-^{\rm sou}(\vec p,t)
\end{equation}
with some source term
$\dot n_-^{\rm sou}(\vec p,t)$.
Since we know that this source term accounts for transitions
between positive and negative energy eigenstates, it is
tempting to write down a rate equation of the form
\begin{equation}
\dot n^{\rm sou}(\vec p,+\epsilon,t)
=
\halb r(\vec p\,)\cdot\left[
n(\vec p,-\epsilon,t)-n(\vec p,+\epsilon,t)\right]
\quad,
\end{equation}
$r(\vec p\,)$ being the respective transition rate.
If this were correct, the identifications
$n(\vec p,+\epsilon,t)\equiv n_-(\vec p,t)$ and
$n(\vec p,-\epsilon,t)\equiv 2-n_+(-\vec p,t)$
would then lead to
\begin{equation}
\dot n_-^{\rm sou}(\vec p,t)
=
r(\vec p\,)\cdot S(\vec p,t)
\end{equation}
with
\begin{equation}
S(\vec p,t)
:=
1-\halb n_-(\vec p,t)
-\halb n_+(-\vec p,t)
\quad.
\label{factor:s}
\end{equation}
Such a source term, however, can only be correct
in the Markovian limit --- an approximation which is {\em not}
always justified.
Careful investigation
\cite{rau:thesis,rau:muller}
reveals that the above ansatz for the source term has to be
modified:
assuming the quasistationary limit ($t_0\to -\infty$) one finds
\begin{equation}
\dot n_-^{\rm sou}(\vec p,t)
=
\int_0^\infty\d\tau\,R(\vec p,\tau)\cdot S(\vec p-q\vec E\tau,t-\tau)
\quad.
\label{source:ansatz}
\end{equation}
This source term involves an integration over the entire
history of the system,
thus accounting for finite memory effects and rendering
the evolution of the occupation numbers
generally {\em non}-Markovian.

The kernel $R(\vec p,\tau)$ can be obtained with the help of
the projection method
\cite{rau:thesis,rau:muller}.
One key ingredient in the derivation is the introduction of
a super-operator ${\cal Q}$ which projects
onto the {\em irrelevant} degrees of freedom;
in our case,
\begin{equation}
{\cal Q}N_-={\cal Q}N_+=0
\quad,
\end{equation}
whereas other combinations of field operators are unaffected:
\begin{equation}
{\cal Q}(a^\dagger b^\dagger)=a^\dagger b^\dagger
\quad,\quad
{\cal Q}(ba)=ba
\quad.
\end{equation}
With this definition one finds
\begin{equation}
R(\vec p,\tau)=
-\bra{0}\delta{\cal L}\exp(i{\cal QLQ}\tau)\delta{\cal L}
N_-(\vec p\,)\ket{0}
\quad,
\label{kernel}
\end{equation}
a general result which
holds for {\em arbitrary} field strengths.

The formal expression for the source term can be easily evaluated
in the limit of weak fields.
Provided $E\ll m^2/\hbar q$, then $|\dot\beta|\ll|\dot\phi|$
and hence $\delta{\cal L}$ may be regarded as a small perturbation.
In this case it is legitimate to replace
\begin{equation}
\exp(i{\cal QLQ}\tau)\,\to\,\exp(i{\cal QL}_{\rm diag}{\cal Q}\tau)
\quad,
\end{equation}
leading to
\begin{equation}\label{source}
\dot n_-^{\rm sou}(\vec p,t)=
4\,{\rm Re}\,\int_0^\infty\!\d\tau\,
\dot\beta^*(-\tau,-\tau)\,
\e^{-i2\phi(-\tau,0)}
\,\dot\beta(0,0)\,
S(\vec p-q\vec E\tau,t-\tau)
\end{equation}
(with $\dot\beta(t_2,t_1)\equiv\dot\beta_{21}$
and $\phi(t_2,t_1)\equiv\phi_{21}$).
This source term is consistent with the Schwinger
formula: assuming that
the system is dilute ($S=1$),
using the differential equation \r{diff:equation}
in the limit $\alpha_{fi}\approx 1$,
and employing the Landau-Zener formula
\cite{zener,landau},
one finds that
\begin{equation}
w:=
{1\over h^3}\int\d^3\!p\, \dot n_-^{\rm sou}(\vec p,t)
=
{(qE)^2\over 4\pi^3\hbar^2}\exp\left(-
{\pi m^2\over\hbar qE}\right)
\quad;
\end{equation}
a result which does indeed agree with the leading term
in the Schwinger formula.

The source term \r{source} may be evaluated numerically.
For simplicity I will take the system to be dilute, $S=1$;
the source term then no longer depends on $t$.
It is convenient to introduce
\begin{equation}
a:=\hbar qE/\epsilon_\perp^2
\end{equation}
and to consider, rather than the source term itself, the
dimensionless quantity
\begin{equation}
\eta(a,p_{\|}/\epsilon_\perp):=
{\epsilon_\perp\over qE} \exp\left({\pi\over 2a}\right)\,
\dot n_-^{\rm sou}(\vec p\,)
\quad.
\end{equation}
The pre-factor in its definition has been chosen such that
for $p_{\|}=0$, $\eta$ is of the order one.
As the pre-factor is independent of $p_{\|}$, $\eta$ will
correctly describe the distribution of the longitudinal momenta
of the produced electrons.
One can show that
\begin{equation}
\label{source:explicit}
\eta(a,p_{\|}/\epsilon_\perp)=
\exp\left({\pi\over 2a}\right)
{1\over 2[1+(p_{\|}/\epsilon_\perp)^2]}
\int_{-\infty}^0 \d x
{1\over\cosh^3\varphi(x,p_{\|}/\epsilon_\perp)}
\cos\left({1\over a}x\right)
\end{equation}
with $\varphi$ defined implicitly as the solution of the equation
\begin{equation}
\sinh\varphi\cosh\varphi+\varphi=
x+(p_{\|}/\epsilon_\perp)\sqrt{1+(p_{\|}/\epsilon_\perp)^2}
+\sinh^{-1}(p_{\|}/\epsilon_\perp)
\quad.
\end{equation}

I calculated $\eta$ numerically, using a combination of
Filon's integration formula \cite{handbook} with an efficient
root-finding algorithm.
The results for weak fields ($a<1$; figure 1)
may at first seem surprising.
Clearly the momentum distribution of the produced electrons
is {\em not} narrowly peaked around $p_{\|}=0$;
it is neither a $\delta$-function nor a thermal distribution.
Rather, electrons are being produced predominantly in the direction of
the external field ($p_{\|}>0$).
Electrons moving in the opposite direction ($p_{\|}<0$) are
being annihilated: for them, the production rate is negative.
Of course, such negative production rates are sensible only if
there are electrons available for annihilation.
In the quasistationary limit this is the case:
electrons which have been emitted with positive momentum
are subsequently being decelerated and may then,
as soon as $p_{\|}<0$,
be (partly) annihilated again;
there remains a small surplus which manifests itself as a positive
total production rate.
As another surprising feature, $\eta$ displays (approximately) periodic
oscillations whose period scales with $a$.
This may be
understood qualitatively if one views
pair creation as a tunnelling process
from the negative to the positive energy continuum \cite{casher:etal}.
The barrier between these continua has a spatial width
of the order $\epsilon_\perp/qE$, inducing a `momentum quantisation'
$\Delta p_{\|}\sim\hbar qE/\epsilon_\perp$ and thus
$\Delta(p_{\|}/\epsilon_\perp)\sim a$.
Interference of multiply reflected electron wave functions then leads
to the observed oscillations.
For a strong field ($a>1$; figure 2),
the naive tunnelling picture breaks down;
both the oscillations and the annihilation of particles (negative rates)
become less pronounced.

As we discussed previously, the source term is generally
non-Markovian. It exhibits two characteristic time scales:
(i) the memory time $\tau_{\rm mem}(\vec p)$,
which corresponds to the temporal extent of each
individual creation process
and which indicates
how far back into the past one has to reach
in order to predict future occupation numbers;
and (ii) the production interval $\tau_{\rm prod}(\vec p)$
--- the inverse of the production rate ---,
which corresponds to
the average time that elapses
{\em between} creation processes and thus
constitutes the typical time scale on which the occupation numbers change.
Only if $\tau_{\rm mem}\ll\tau_{\rm prod}$
can memory effects be neglected and the evolution be
considered approximately Markovian.

In the weak field limit
both time scales can be extracted from the source term \r{source}.
First the {\em memory time:}
The factor
\begin{equation}
\dot\beta^*(-\tau,-\tau)\propto
{(\epsilon_\perp/qE)\over (\tau-p_{\|}/qE)^2+(\epsilon_\perp/qE)^2}
\end{equation}
constitutes a Lorentz distribution
in $\tau$, centered around $p_{\|}/qE$ with width $\epsilon_\perp/qE$.
Significant contributions to the source term thus come from
times $\tau$ which are smaller than $(p_{\|}+\epsilon_\perp)/qE$.
As the typical momentum scale is set by $\Delta p_{\|}\sim\hbar qE/
\epsilon_\perp$, we may conclude
\begin{equation}
\tau_{\rm mem}\sim
{\hbar\over\epsilon_\perp}+{\epsilon_\perp\over qE}
\quad.
\end{equation}
The memory time combines two time scales of different origin.
(i) The time $\hbar/\epsilon_\perp$ is proportional to $\hbar$ and
therefore of quantum mechanical origin.
It corresponds (via the time-energy uncertainty relation)
to the time needed to create a {\em virtual} particle-antiparticle
pair,
and may thus be regarded as the `time between two
production attempts.'
(ii) The time $\epsilon_\perp/qE$, on the other hand,
is independent of $\hbar$ and therefore classical.
It can be interpreted in various
ways, depending on the picture employed to visualize
the pair creation process.
If pair creation is viewed as a
tunnelling process,
the classical memory time coincides with the
time needed for the wave function to
traverse the barrier with the speed of light \cite{olkhovsky}.
Alternatively, pair creation may be viewed as
a non-adiabatic transition between the two time-dependent
energy levels $\pm\epsilon[\vec p(t)]$.
In that case the classical memory time corresponds to
the width of the transition region, i.e., the region of
closest approach of the two levels.
Finding the {\em production interval} is less straightforward.
Assuming $p_{\|}=0$ for simplicity,
again invoking the weak field limit, and exploiting the
fact that the source term must be consistent with the
Schwinger formula, one can show that
\begin{equation}
\label{tauprod}
\tau_{\rm prod}(0,\vec p_\perp)\sim
{\epsilon_\perp\over qE}
\exp\left({\pi\epsilon_\perp^2\over 2\hbar qE}\right)
\quad.
\end{equation}

As long as $E\ll m^2/\hbar q\le
\epsilon_\perp^2/\hbar q$,
the particle creation process is Markovian:
$\tau_{\rm mem}\ll\tau_{\rm prod}$.
In the weak field limit, therefore, the entropy
associated with the coarse-grained level of description
spanned by $\{N_\pm(\vec p)\}$,
\begin{eqnarray}
S_{\rm c.g.}(t)
&:=&
-2k{\Omega\over h^3}\int\d^3\!p\,\left[
{n_-(\vec p,t)\over 2}\ln {n_-(\vec p,t)\over 2}+
\left(1-{n_-(\vec p,t)\over 2}\right)
\ln \left(1-{n_-(\vec p,t)\over 2}\right)\right]
+
\nonumber \\
&&
\,+\,(n_-\leftrightarrow n_+)
\quad,
\end{eqnarray}
obeys an $H$-theorem.
($\Omega$ denotes the volume.)
The monotonous increase of the coarse-grained entropy
explains why spontaneous pair creation is perceived as irreversible.
This apparent irreversibility is, of course, a consequence
of the coarse-graining:
information is being transferred from accessible (slow) to
inaccessible (fast) degrees of freedom.
The slow degrees of freedom
are the occupation
numbers of the various momentum states.
{}From these, information gradually `leaks' into unobserved
degrees of freedom: correlations and rapidly oscillating phases
which entangle the respective wave functions of the members
of a particle-antiparticle pair.

As soon as $E\ge m^2/\hbar q$, the situation changes.
The source term \r{source}, which was derived in the
weak field limit, is then only a rough estimate.
Already this weak-field estimate becomes non-Markovian:
at $E=m^2/\hbar q$
the production interval and the memory time
are of the same order $\tau\sim\hbar/m$.
This is a clear indication that at this point conventional Markovian
transport theories must break down.
There may be temporary violations of the $H$-theorem:
the coarse-grained entropy, while still increasing on average,
may now oscillate (on the same scale $\tau\sim\hbar/m$).
Such oscillations have indeed been observed in numerical
simulations \cite{kluger:central}.
A systematic study of these memory effects
should proceed from the general equations \r{source:ansatz}
and \r{kernel}.
Although such an enterprise is beyond the
scope of this letter, we can already say that
(i) memory effects become significant at large field strengths;
and (ii) the projection method can account for these memory effects
and thus appears to be a suitable tool for their investigation.

The above analysis can be extended to include binary collisions
of the produced particles.
This is done by replacing
${\delta{\cal L}}\to{\delta{\cal L}}+{\cal V}$,
where ${\cal V}$ contains the two-body interaction.
To lowest order perturbation theory,
pair creation and collisions do not interfere \cite{rau:thesis};
the additional interaction
gives rise to a separate collision term.
Like the source term, this collision term
is generally non-Markovian and
must be subjected to
a time scale analysis, leading again to a criterion for the
validity of the Markovian approximation.
One finds that there are two contributions to the
memory time: the average time needed for a particle to pass
through an interaction range, and the typical
`off-shell' time given by the time-energy uncertainty relation.
For the Markovian approximation to be valid, these have to be
smaller than the average time that elapses between two successive
collisions \cite{rau:thesis,daniel}.

Let me summarize the main conclusions.
(i) The source term in the quantum Boltzmann
equation can be derived in an unambiguous
fashion by employing the projection method.
(ii) To lowest order, the
source term is not altered by the presence of collisions.
(iii) In the weak field limit, $E\ll m^2/\hbar q$, the source term
is given by \r{source} or \r{source:explicit}, respectively.
It is then Markovian, and the coarse-grained entropy increases
monotonically.
As information is continuously being transferred to inaccessible
degrees of freedom, spontaneous pair creation appears irreversible.
(iv) But as soon as $E\ge m^2/\hbar q$, there may be sizeable
memory effects, leading to temporary violations of the $H$-theorem.
Their description is beyond the scope of
conventional Markovian transport theories.
A more suitable starting point appears to be
the projection method, in particular equations \r{source:ansatz}
and \r{kernel}.
\\\\
{\sl Acknowledgements.}
I thank B. M\"uller for very valuable help and advice,
and C. Waigl for assistance with the numerical work.
Financial support by the Studienstiftung des deutschen Volkes
and by the U.S. Department of Energy (grant no. DE-FG05-90ER40592)
is gratefully acknowledged.
%
\begin{thebibliography}{33}
\bibitem{bialas:czyz:1}
A. Bialas and W. Czyz,
{Phys. Rev. D} {\bf 31}, 198 (1985).
\bibitem{bialas:czyz:2}
A. Bialas and W. Czyz,
{Nucl. Phys.} {\bf B 267}, 242 (1986).
\bibitem{kajantie:matsui}
K. Kajantie and T. Matsui,
{Phys. Lett.} {\bf 164B}, 373 (1985).
\bibitem{gatoff:etal}
G. Gatoff, A. K. Kerman, and T. Matsui,
{Phys. Rev. D} {\bf 36}, 114 (1987).
\bibitem{ehtamo}
H. Ehtamo, J. Lindfors, and L. McLerran,
{Z. Phys. C} {\bf 18}, 341 (1981).
\bibitem{biro:etal}
T. S. Bir\'o, H. B. Nielsen, and J. Knoll,
{Nucl. Phys.} {\bf B245}, 449 (1984).
\bibitem{casher:etal}
A. Casher, H. Neuberger, and S. Nussinov,
{Phys. Rev. D} {\bf 20}, 179 (1979).
\bibitem{glendenning:matsui}
N. K. Glendenning and T. Matsui,
{Phys. Rev. D} {\bf 28}, 2890 (1983).
\bibitem{banerjee:etal}
B. Banerjee, N. K. Glendenning, and T. Matsui,
{Phys. Lett.} {\bf 127B}, 453 (1983).
\bibitem{kornel:string}
K. Sailer, T. Sch\"onfeld, Z. Schram, A. Sch\"afer, and W. Greiner,
{Int. J. Mod. Phys. A} {\bf 6}, 4395 (1991).
\bibitem{eisenberg:kaelbermann}
J. M. Eisenberg and G. K\"albermann,
{Phys. Rev. D} {\bf 37}, 1197 (1988).
\bibitem{rafelski:etal}
I. Bialynicki-Birula, P. Gornicki, and J. Rafelski,
{Phys. Rev. D} {\bf 44}, 1825 (1991).
\bibitem{kluger:1}
Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,
{Phys. Rev. Lett.} {\bf 67}, 2427 (1991).
\bibitem{kluger:2}
Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,
{Phys. Rev. D} {\bf 45}, 4659 (1992).
\bibitem{kluger:central}
F. Cooper, J. M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky,
{Phys. Rev. D} {\bf 48}, 190 (1993).
\bibitem{nakajima:1}
S. Nakajima,
{Prog. Theor. Phys.} {\bf 20}, 948 (1958).
\bibitem{zwanzig:1}
R. Zwanzig,
{J. Chem. Phys.} {\bf 33}, 1338 (1960).
\bibitem{zwanzig:2}
R. Zwanzig,
{Phys. Rev.} {\bf 124}, 983 (1961).
\bibitem{zwanzig:4}
R. Zwanzig,
{Physica (Utrecht)} {\bf 30}, 1109 (1964).
\bibitem{mori:1}
H. Mori,
{Prog. Theor. Phys.} {\bf 33}, 423 (1965).
\bibitem{robertson:pr1}
B. Robertson,
{Phys. Rev.} {\bf 144}, 151 (1966).
\bibitem{robertson:pr2}
B. Robertson,
{Phys. Rev.} {\bf 160}, 175 (1967); Erratum: {\bf 166}, 206 (1968).
\bibitem{robertson:jmp}
B. Robertson,
{J. Math. Phys.} {\bf 11}, 2482 (1970).
\bibitem{kawasaki:gunton}
K. Kawasaki and J. D. Gunton,
{Phys. Rev. A} {\bf 8}, 2048 (1973).
\bibitem{grabert:book}
H. Grabert,
{\em Projection Operator Techniques in Nonequilibrium Statistical Mechanics}
(Springer, Berlin, Heidelberg, 1982).
\bibitem{sauter:1}
F. Sauter,
{Z. Phys.} {\bf 69}, 742 (1931).
\bibitem{heisenberg:euler}
W. Heisenberg and H. Euler,
{Z. Phys.} {\bf 98}, 714 (1936).
\bibitem{schwinger:1}
J. Schwinger,
{Phys. Rev.} {\bf 82}, 664 (1951).
\bibitem{fick:processes}
E. Fick and G. Sauermann,
{\em The Quantum Statistics of Dynamic Processes}
(Springer, Berlin, Heidelberg, 1990).
\bibitem{fetter:book}
A. L. Fetter and J. D. Walecka,
{\em Quantum Theory of Many-Particle Systems}
(McGraw-Hill, New York, 1971).
\bibitem{rau:thesis}
J. Rau,
Ph.D. thesis, Duke University, 1993 (unpublished).
\bibitem{rau:muller}
J. Rau and B. M\"uller,
to be published.
\bibitem{zener}
C. Zener,
{Proc. Roy. Soc. A} {\bf 137}, 696 (1932).
\bibitem{landau}
L. D. Landau and E. M. Lifshitz,
{\em Quantum Mechanics}
(Pergamon, Oxford, 1965).
\bibitem{handbook}
M. Abramowitz and I. A. Stegun (eds.),
{\em Handbook of Mathematical Functions}
(National Bureau of Standards, 1964).
\bibitem{olkhovsky}
The proper definition of a `tunnelling time' is
in fact controversial; for a review, see
V. S. Olkhovsky and E. Ricami,
{Phys. Rep.} {\bf 214}, 339 (1992).
\bibitem{daniel}
P. Danielewicz,
{Ann. Phys.} {\bf 152}, 239 (1984).
%
\end{thebibliography}
\newpage
%
\noindent
{\large\bf Figure Captions}
\\\\
Figure 1: The re-scaled production rate $\eta$ as a function of
$p_{\|}/\epsilon_\perp$ for weak fields ($a=0.2, 0.3, 0.7$).
\\\\
Figure 2: The re-scaled production rate $\eta$ as a function of
$p_{\|}/\epsilon_\perp$ for a strong field ($a=2.9$).
%
%
\end{document}

   **FIGURE 1**
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Courier
%%BoundingBox: 50 50 554 770
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/gnulinewidth 5.000 def
/vshift -46 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 def
/vpt2 vpt 2 mul def
/hpt2 hpt 2 mul def
/Lshow { currentpoint stroke moveto
  0 vshift rmoveto show } def
/Rshow { currentpoint stroke moveto
  dup stringwidth pop neg vshift rmoveto show } def
/Cshow { currentpoint stroke moveto
  dup stringwidth pop -2 div vshift rmoveto show } def
/DL { Color {setrgbcolor [] 0 setdash pop}
 {pop pop pop 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def
/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def
/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def
/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def
/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/M {moveto} def
/L {lineto} def
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub moveto
  0 currentlinewidth rlineto  stroke } def
/D { stroke [] 0 setdash  2 copy  vpt add moveto
  hpt neg vpt neg rlineto  hpt vpt neg rlineto
  hpt vpt rlineto  hpt neg vpt rlineto  closepath  stroke
  P  } def
/A { stroke [] 0 setdash  vpt sub moveto  0 vpt2 rlineto
  currentpoint stroke moveto
  hpt neg vpt neg rmoveto  hpt2 0 rlineto stroke
  } def
/B { stroke [] 0 setdash  2 copy  exch hpt sub exch vpt add moveto
  0 vpt2 neg rlineto  hpt2 0 rlineto  0 vpt2 rlineto
  hpt2 neg 0 rlineto  closepath  stroke
  P  } def
/C { stroke [] 0 setdash  exch hpt sub exch vpt add moveto
  hpt2 vpt2 neg rlineto  currentpoint  stroke  moveto
  hpt2 neg 0 rmoveto  hpt2 vpt2 rlineto stroke  } def
/T { stroke [] 0 setdash  2 copy  vpt 1.12 mul add moveto
  hpt neg vpt -1.62 mul rlineto
  hpt 2 mul 0 rlineto
  hpt neg vpt 1.62 mul rlineto  closepath  stroke
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
90 rotate
0 -5040 translate
0 setgray
/Courier findfont 140 scalefont setfont
newpath
LTa
1008 2823 M
6969 2823 L
3989 491 M
3989 4689 L
LTb
1008 491 M
1071 491 L
6969 491 M
6906 491 L
924 491 M
(-25) Rshow
1008 957 M
1071 957 L
6969 957 M
6906 957 L
924 957 M
(-20) Rshow
1008 1424 M
1071 1424 L
6969 1424 M
6906 1424 L
924 1424 M
(-15) Rshow
1008 1890 M
1071 1890 L
6969 1890 M
6906 1890 L
924 1890 M
(-10) Rshow
1008 2357 M
1071 2357 L
6969 2357 M
6906 2357 L
924 2357 M
(-5) Rshow
1008 2823 M
1071 2823 L
6969 2823 M
6906 2823 L
924 2823 M
(0) Rshow
1008 3290 M
1071 3290 L
6969 3290 M
6906 3290 L
924 3290 M
(5) Rshow
1008 3756 M
1071 3756 L
6969 3756 M
6906 3756 L
924 3756 M
(10) Rshow
1008 4223 M
1071 4223 L
6969 4223 M
6906 4223 L
924 4223 M
(15) Rshow
1008 4689 M
1071 4689 L
6969 4689 M
6906 4689 L
924 4689 M
(20) Rshow
1604 491 M
1604 554 L
1604 4689 M
1604 4626 L
1604 351 M
(-2) Cshow
2796 491 M
2796 554 L
2796 4689 M
2796 4626 L
2796 351 M
(-1) Cshow
3989 491 M
3989 554 L
3989 4689 M
3989 4626 L
3989 351 M
(0) Cshow
5181 491 M
5181 554 L
5181 4689 M
5181 4626 L
5181 351 M
(1) Cshow
6373 491 M
6373 554 L
6373 4689 M
6373 4626 L
6373 351 M
(2) Cshow
LTb
1008 491 M
6969 491 L
6969 4689 L
1008 4689 L
1008 491 L
140 2590 M
currentpoint gsave translate 90 rotate 0 0 moveto
(eta) Cshow
grestore
3988 211 M
(p_long/epsilon_trans) Cshow
LT0
LT0
6486 4486 M
(a=0.2) Rshow
6570 4486 M
6822 4486 L
6969 2828 M
6898 2829 L
6821 2830 L
6745 2832 L
6670 2833 L
6598 2835 L
6527 2838 L
6458 2840 L
6390 2843 L
6324 2847 L
6259 2851 L
6195 2855 L
6133 2860 L
6073 2867 L
6013 2873 L
5955 2881 L
5898 2890 L
5843 2901 L
5788 2913 L
5735 2926 L
5682 2941 L
5631 2958 L
5581 2976 L
5532 2998 L
5483 3021 L
5436 3047 L
5390 3076 L
5344 3109 L
5299 3144 L
5255 3183 L
5212 3225 L
5170 3271 L
5128 3321 L
5087 3374 L
5047 3431 L
5008 3492 L
4969 3557 L
4931 3625 L
4893 3696 L
4856 3770 L
4819 3846 L
4783 3923 L
4748 4002 L
4712 4080 L
4678 4156 L
4644 4231 L
4610 4302 L
4576 4368 L
4543 4427 L
4511 4479 L
4478 4521 L
4446 4551 L
4414 4568 L
4383 4571 L
4352 4557 L
4321 4526 L
4290 4477 L
4259 4407 L
4229 4317 L
4198 4206 L
4168 4074 L
4138 3922 L
4108 3751 L
4078 3563 L
4048 3358 L
4018 3141 L
3989 2914 L
3959 2680 L
3929 2444 L
3899 2209 L
3869 1981 L
3839 1763 L
3809 1561 L
3779 1379 L
3748 1221 L
3718 1090 L
3687 988 L
3656 919 L
3625 881 L
3594 875 L
3563 899 L
3531 950 L
3499 1023 L
3466 1115 L
3434 1217 L
3401 1326 L
3367 1433 L
3333 1534 L
3299 1623 L
3265 1698 L
3229 1757 L
3194 1801 L
3158 1833 L
3121 1859 L
3084 1884 L
3046 1916 L
3008 1961 L
2969 2023 L
2930 2103 L
2890 2197 L
2849 2300 L
2807 2400 L
2765 2486 L
2722 2547 L
2678 2579 L
2633 2582 L
2587 2566 L
2541 2547 L
2494 2543 L
2445 2566 L
2396 2618 L
2346 2686 L
2295 2747 L
2242 2778 L
2189 2770 L
2134 2734 L
2079 2703 L
2022 2706 L
1964 2750 L
1904 2805 L
1844 2828 L
1782 2802 L
1718 2760 L
1653 2756 L
1587 2801 L
1519 2840 L
1450 2820 L
1379 2778 L
1307 2788 L
1232 2835 L
1156 2831 L
1079 2788 L
1008 2804 L
LT1
6486 4346 M
(a=0.3) Rshow
6570 4346 M
6822 4346 L
6969 2824 M
6898 2825 L
6821 2825 L
6745 2825 L
6670 2825 L
6598 2826 L
6527 2826 L
6458 2826 L
6390 2827 L
6324 2828 L
6259 2828 L
6195 2829 L
6133 2830 L
6073 2831 L
6013 2832 L
5955 2833 L
5898 2835 L
5843 2836 L
5788 2838 L
5735 2841 L
5682 2843 L
5631 2846 L
5581 2849 L
5532 2852 L
5483 2856 L
5436 2860 L
5390 2865 L
5344 2870 L
5299 2876 L
5255 2882 L
5212 2889 L
5170 2896 L
5128 2904 L
5087 2913 L
5047 2922 L
5008 2932 L
4969 2942 L
4931 2954 L
4893 2965 L
4856 2978 L
4819 2990 L
4783 3003 L
4748 3017 L
4712 3031 L
4678 3044 L
4644 3058 L
4610 3071 L
4576 3084 L
4543 3096 L
4511 3108 L
4478 3118 L
4446 3127 L
4414 3134 L
4383 3139 L
4352 3142 L
4321 3142 L
4290 3140 L
4259 3135 L
4229 3126 L
4198 3114 L
4168 3098 L
4138 3079 L
4108 3056 L
4078 3029 L
4048 2999 L
4018 2965 L
3989 2927 L
3959 2887 L
3929 2844 L
3899 2799 L
3869 2753 L
3839 2705 L
3809 2658 L
3779 2611 L
3748 2565 L
3718 2522 L
3687 2483 L
3656 2447 L
3625 2417 L
3594 2393 L
3563 2375 L
3531 2365 L
3499 2363 L
3466 2368 L
3434 2382 L
3401 2404 L
3367 2433 L
3333 2468 L
3299 2509 L
3265 2554 L
3229 2602 L
3194 2650 L
3158 2696 L
3121 2739 L
3084 2776 L
3046 2805 L
3008 2825 L
2969 2835 L
2930 2834 L
2890 2824 L
2849 2805 L
2807 2781 L
2765 2754 L
2722 2728 L
2678 2707 L
2633 2696 L
2587 2696 L
2541 2709 L
2494 2734 L
2445 2768 L
2396 2806 L
2346 2840 L
2295 2864 L
2242 2872 L
2189 2863 L
2134 2838 L
2079 2806 L
2022 2777 L
1964 2763 L
1904 2770 L
1844 2796 L
1782 2831 L
1718 2857 L
1653 2861 L
1587 2839 L
1519 2806 L
1450 2785 L
1379 2793 L
1307 2824 L
1232 2851 L
1156 2847 L
1079 2817 L
1008 2797 L
LT2
6486 4206 M
(a=0.7) Rshow
6570 4206 M
6822 4206 L
6969 2823 M
6898 2823 L
6821 2824 L
6745 2824 L
6670 2824 L
6598 2824 L
6527 2824 L
6458 2824 L
6390 2824 L
6324 2824 L
6259 2824 L
6195 2825 L
6133 2825 L
6073 2825 L
6013 2825 L
5955 2826 L
5898 2826 L
5843 2826 L
5788 2827 L
5735 2827 L
5682 2828 L
5631 2828 L
5581 2829 L
5532 2830 L
5483 2831 L
5436 2832 L
5390 2833 L
5344 2834 L
5299 2835 L
5255 2836 L
5212 2838 L
5170 2839 L
5128 2841 L
5087 2843 L
5047 2845 L
5008 2847 L
4969 2850 L
4931 2852 L
4893 2855 L
4856 2858 L
4819 2861 L
4783 2864 L
4748 2867 L
4712 2871 L
4678 2874 L
4644 2878 L
4610 2882 L
4576 2885 L
4543 2889 L
4511 2893 L
4478 2897 L
4446 2901 L
4414 2905 L
4383 2909 L
4352 2912 L
4321 2916 L
4290 2919 L
4259 2922 L
4229 2925 L
4198 2927 L
4168 2929 L
4138 2930 L
4108 2931 L
4078 2932 L
4048 2932 L
4018 2931 L
3989 2930 L
3959 2928 L
3929 2925 L
3899 2921 L
3869 2917 L
3839 2912 L
3809 2906 L
3779 2900 L
3748 2893 L
3718 2885 L
3687 2876 L
3656 2867 L
3625 2857 L
3594 2847 L
3563 2836 L
3531 2825 L
3499 2814 L
3466 2802 L
3434 2791 L
3401 2779 L
3367 2767 L
3333 2756 L
3299 2745 L
3265 2735 L
3229 2725 L
3194 2716 L
3158 2708 L
3121 2701 L
3084 2695 L
3046 2691 L
3008 2688 L
2969 2687 L
2930 2687 L
2890 2689 L
2849 2694 L
2807 2700 L
2765 2708 L
2722 2717 L
2678 2729 L
2633 2742 L
2587 2756 L
2541 2772 L
2494 2788 L
2445 2804 L
2396 2820 L
2346 2836 L
2295 2850 L
2242 2862 L
2189 2871 L
2134 2877 L
2079 2879 L
2022 2878 L
1964 2873 L
1904 2864 L
1844 2851 L
1782 2837 L
1718 2822 L
1653 2808 L
1587 2795 L
1519 2787 L
1450 2784 L
1379 2787 L
1307 2796 L
1232 2809 L
1156 2824 L
1079 2838 L
1008 2847 L
stroke
grestore
end
showpage
%%Trailer
%%Pages: 1

   **FIGURE 2**
%!PS-Adobe-2.0
%%Creator: gnuplot
%%DocumentFonts: Courier
%%BoundingBox: 50 50 554 770
%%Pages: (atend)
%%EndComments
/gnudict 40 dict def
gnudict begin
/Color false def
/gnulinewidth 5.000 def
/vshift -46 def
/dl {10 mul} def
/hpt 31.5 def
/vpt 31.5 def
/vpt2 vpt 2 mul def
/hpt2 hpt 2 mul def
/Lshow { currentpoint stroke moveto
  0 vshift rmoveto show } def
/Rshow { currentpoint stroke moveto
  dup stringwidth pop neg vshift rmoveto show } def
/Cshow { currentpoint stroke moveto
  dup stringwidth pop -2 div vshift rmoveto show } def
/DL { Color {setrgbcolor [] 0 setdash pop}
 {pop pop pop 0 setdash} ifelse } def
/BL { stroke gnulinewidth 2 mul setlinewidth } def
/AL { stroke gnulinewidth 2 div setlinewidth } def
/PL { stroke gnulinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 0 1 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def
/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def
/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def
/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def
/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def
/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/M {moveto} def
/L {lineto} def
/P { stroke [] 0 setdash
  currentlinewidth 2 div sub moveto
  0 currentlinewidth rlineto  stroke } def
/D { stroke [] 0 setdash  2 copy  vpt add moveto
  hpt neg vpt neg rlineto  hpt vpt neg rlineto
  hpt vpt rlineto  hpt neg vpt rlineto  closepath  stroke
  P  } def
/A { stroke [] 0 setdash  vpt sub moveto  0 vpt2 rlineto
  currentpoint stroke moveto
  hpt neg vpt neg rmoveto  hpt2 0 rlineto stroke
  } def
/B { stroke [] 0 setdash  2 copy  exch hpt sub exch vpt add moveto
  0 vpt2 neg rlineto  hpt2 0 rlineto  0 vpt2 rlineto
  hpt2 neg 0 rlineto  closepath  stroke
  P  } def
/C { stroke [] 0 setdash  exch hpt sub exch vpt add moveto
  hpt2 vpt2 neg rlineto  currentpoint  stroke  moveto
  hpt2 neg 0 rmoveto  hpt2 vpt2 rlineto stroke  } def
/T { stroke [] 0 setdash  2 copy  vpt 1.12 mul add moveto
  hpt neg vpt -1.62 mul rlineto
  hpt 2 mul 0 rlineto
  hpt neg vpt 1.62 mul rlineto  closepath  stroke
  P  } def
/S { 2 copy A C} def
end
%%EndProlog
%%Page: 1 1
gnudict begin
gsave
50 50 translate
0.100 0.100 scale
90 rotate
0 -5040 translate
0 setgray
/Courier findfont 140 scalefont setfont
newpath
LTa
1008 1424 M
6969 1424 L
3989 491 M
3989 4689 L
LTb
1008 491 M
1071 491 L
6969 491 M
6906 491 L
924 491 M
(-0.4) Rshow
1008 957 M
1071 957 L
6969 957 M
6906 957 L
924 957 M
(-0.2) Rshow
1008 1424 M
1071 1424 L
6969 1424 M
6906 1424 L
924 1424 M
(0) Rshow
1008 1890 M
1071 1890 L
6969 1890 M
6906 1890 L
924 1890 M
(0.2) Rshow
1008 2357 M
1071 2357 L
6969 2357 M
6906 2357 L
924 2357 M
(0.4) Rshow
1008 2823 M
1071 2823 L
6969 2823 M
6906 2823 L
924 2823 M
(0.6) Rshow
1008 3290 M
1071 3290 L
6969 3290 M
6906 3290 L
924 3290 M
(0.8) Rshow
1008 3756 M
1071 3756 L
6969 3756 M
6906 3756 L
924 3756 M
(1) Rshow
1008 4223 M
1071 4223 L
6969 4223 M
6906 4223 L
924 4223 M
(1.2) Rshow
1008 4689 M
1071 4689 L
6969 4689 M
6906 4689 L
924 4689 M
(1.4) Rshow
1434 491 M
1434 554 L
1434 4689 M
1434 4626 L
1434 351 M
(-3) Cshow
2285 491 M
2285 554 L
2285 4689 M
2285 4626 L
2285 351 M
(-2) Cshow
3137 491 M
3137 554 L
3137 4689 M
3137 4626 L
3137 351 M
(-1) Cshow
3989 491 M
3989 554 L
3989 4689 M
3989 4626 L
3989 351 M
(0) Cshow
4840 491 M
4840 554 L
4840 4689 M
4840 4626 L
4840 351 M
(1) Cshow
5692 491 M
5692 554 L
5692 4689 M
5692 4626 L
5692 351 M
(2) Cshow
6543 491 M
6543 554 L
6543 4689 M
6543 4626 L
6543 351 M
(3) Cshow
LTb
1008 491 M
6969 491 L
6969 4689 L
1008 4689 L
1008 491 L
140 2590 M
currentpoint gsave translate 90 rotate 0 0 moveto
(eta) Cshow
grestore
3988 211 M
(p_long/epsilon_trans) Cshow
LT0
LT0
6486 4486 M
(a=2.9) Rshow
6570 4486 M
6822 4486 L
6969 1426 M
6921 1426 L
6845 1427 L
6772 1427 L
6700 1428 L
6629 1428 L
6561 1429 L
6494 1430 L
6429 1431 L
6365 1432 L
6302 1433 L
6241 1434 L
6182 1436 L
6124 1438 L
6067 1439 L
6011 1442 L
5957 1444 L
5904 1447 L
5852 1450 L
5802 1453 L
5752 1457 L
5704 1462 L
5656 1467 L
5610 1472 L
5565 1478 L
5521 1485 L
5477 1492 L
5435 1500 L
5393 1509 L
5353 1519 L
5313 1530 L
5274 1542 L
5236 1555 L
5198 1570 L
5162 1585 L
5126 1603 L
5091 1621 L
5056 1642 L
5022 1664 L
4989 1687 L
4957 1713 L
4925 1741 L
4893 1771 L
4863 1803 L
4832 1837 L
4803 1874 L
4773 1914 L
4745 1956 L
4717 2000 L
4689 2047 L
4661 2097 L
4634 2150 L
4608 2206 L
4582 2264 L
4556 2326 L
4531 2390 L
4506 2457 L
4481 2526 L
4456 2598 L
4432 2673 L
4408 2750 L
4385 2829 L
4361 2910 L
4338 2993 L
4315 3078 L
4293 3163 L
4270 3250 L
4248 3337 L
4226 3425 L
4204 3513 L
4182 3600 L
4160 3686 L
4138 3771 L
4117 3855 L
4095 3937 L
4074 4016 L
4052 4092 L
4031 4165 L
4010 4234 L
3989 4300 L
3967 4360 L
3946 4416 L
3925 4467 L
3903 4513 L
3882 4552 L
3860 4586 L
3839 4614 L
3817 4635 L
3795 4650 L
3773 4658 L
3751 4660 L
3729 4655 L
3707 4643 L
3684 4625 L
3662 4600 L
3639 4569 L
3616 4531 L
3592 4488 L
3569 4439 L
3545 4384 L
3521 4323 L
3496 4258 L
3471 4187 L
3446 4112 L
3421 4033 L
3395 3949 L
3369 3862 L
3343 3772 L
3316 3678 L
3288 3582 L
3260 3483 L
3232 3381 L
3204 3278 L
3174 3174 L
3145 3068 L
3114 2960 L
3084 2852 L
3052 2744 L
3020 2635 L
2988 2526 L
2955 2417 L
2921 2308 L
2886 2201 L
2851 2094 L
2815 1988 L
2779 1883 L
2741 1780 L
2703 1680 L
2664 1581 L
2624 1485 L
2584 1391 L
2542 1301 L
2500 1214 L
2456 1132 L
2412 1053 L
2367 980 L
2321 912 L
2273 850 L
2225 795 L
2175 746 L
2125 706 L
2073 674 L
2020 651 L
1966 638 L
1910 635 L
1853 644 L
1795 664 L
1736 695 L
1675 738 L
1612 794 L
1548 860 L
1483 938 L
1416 1024 L
1348 1118 L
1277 1218 L
1205 1320 L
1132 1421 L
1056 1518 L
1008 1572 L
stroke
grestore
end
showpage
%%Trailer
%%Pages: 1

