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{\sf


\title{ \sf 130 years Boltzmann equation - from classical to nonlocal quantum transport}
\author{\sf K. Morawetz, Max-Planck-Institute for the Physics of Complex Systems\\ \sf
Noethnitzer Str. 38, 01187 Dresden, Germany\\ \sf e-mail: morawetz@mpipks-dresden.mpg.de}
\maketitle

Since the foundation of Ludwig Boltzmann's famous kinetic equation in
1872 there has been a tremendous development to understand
nonequilibrium processes in classical and quantum systems.
A kinetic equation which unifies the achievements of transport in
dense 
gases with the
quantum transport of dense Fermi systems is presented [1,2,7]. The 
quasiparticle drift of Landau's
equation is connected with a dissipation governed by a nonlocal and non-instant
scattering integral in the spirit of Enskog corrections. These corrections
are expressed in terms of shifts in space and time that characterise
non-locality of the scattering process [3]. In this way quantum
transport is possible to recast into a quasiclassical picture.  
The balance equations for the density, momentum and
energy include quasiparticle contributions and the second order quantum
virial corrections beyond the Landau theory. 
The medium effects on binary collisions are shown to
mediate the latent heat, i.e., an energy conversion between correlation
and thermal energy [5,7].
The kinetic equation is now a time delayed
one showing an interplay between continuous differential
equations and discrete mapping such that a chaotic time dependence of the 
distribution function occurs [10].


Compared to the Boltzmann- equation, the presented form of virial
corrections only slightly increases the numerical demands in 
implementations [4].
The charge density distribution with respect to the velocity of matter
produced in peripheral heavy ion reactions around Fermi energy is
calculated [9]. The experimental finding of enhancement of mid-rapidity
matter shows the necessity to include correlations beyond BUU which
was performed in the framework of nonlocal kinetic theory.
Anomalous expansion velocities
appear due to the included correlations which can be
interpreted in terms of Tsallis statistics and which is a
signal of phase transition [8]. 
The numerical solutions of nonlocal and local Boltzmann kinetic
equations
for the simulation of central heavy ion reactions are parameterized in terms
of time dependent thermodynamical variables in the Fermi liquid
sense [6]. This allows one to discuss
dynamical trajectories in phase space. 
We find two different mechanisms of instability, a short time
surface - dominated instability and later a spinodal - dominated volume
instability. The latter one occurs only if the incident energies do
not exceed significantly the Fermi energy and might be attributed to spinodal
decomposition. 
 \\[1ex]

{\small
\noindent[1]
V. \v Spi\v cka, P. Lipavsk\'y, K. Morawetz,
Phys. Rev. B. {\bf 55}, 5084 (1997); 5095 (1997):
{\em  Quasiparticle transport equation with collisional delay:
I. Phenomenological approach, II. Quantum statistical approach}

\noindent[2]
 V. \v{S}pi\v{c}ka, P. Lipavsk\'y, and K. Morawetz, Phys. Lett. {\bf A
 240}, (1998), 160: { \em  Nonlocal corrections to the Boltzmann equation for dense Fermi systems}

\noindent[3]
K. Morawetz, V. Spi\v cka, P. Lipavsk\'y, H.N. Kwong,
Phys. Rev. {\bf C 59},6 (1999) 3052-3059:
{\em Duration and non-locality of a nucleon-nucleon collision},

\noindent[4]
 K. Morawetz, V. \v{S}pi\v{c}ka, P. Lipavsk\'y, G. Kortemeyer,
 Ch. Kuhrts, R. Nebauer, Phys. Rev. Lett. {\bf 82}, (1999), 3767: {\em Virial corrections to simulations of heavy ion reactions},

\noindent[5]
P. Lipavsk\'y, V. Spi\v cka, K. Morawetz,
Phys. Rev. {\bf E 59} (1999) R1291-R1294 rap. comm.: 
{\em Noninstananeous collisions and two concepts of quasiparticles}

\noindent[6]
K. Morawetz,
Phys. Rev. {\bf C 62}, (2000) 44606:
{\em Dynamical constraints on phase transitions}

\noindent[7]
P. Lipavsk{\'y}, K. Morawetz, and V. {\v S}pi{\v c}ka,
(Annales de Physique, Paris,
  2001, No.~{26, 1}):
{\em Kinetic equation
  for strongly interacting dense Fermi systems}

\noindent[8]
K.~Morawetz, M. P{\l o}szajczak, V.D. Toneev,
Phys. Rev. {\bf C 62} (2000), 64602:
{\em Anomalous radial expansion in central heavy-ion reactions}

\noindent[9]
K.~ Morawetz, P.~Lipavsk\'y, J. Normand, D. Cussol, J. Colin, B. Tamain,
Phys. Rev. {\bf C}, nucl-th/0011045:
{\em Mid-rapididty charge distribution in peripheral heavy ion
collisions}


\noindent[10]
K.~ Morawetz, Phys. Rev. Lett., nucl-th/0012028:
{\em Oscillations and the onset of chaos in the kinetic equation for
  interacting Fermi systems}, 
}
}
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