Nonlocal quantum kinetic theory
P. Lipavsky (Charles University Prague)  
The quantum version of the 130 yearsold Boltzmann equation remains still the basis of modern transport theories. Extensions become necessary for transienttime effects like the femtosecond response and for strongly correlated systems. At short time scales higher correlations have no time to develop yet and femtosecond laser excitation of collective modes in semiconductors as well as quenches of cold atoms in optical lattices can be described even analytically by fluctuations of the meanfield. For plasma systems exposed to a sudden switching analytical results are available from the timedependent Fermi’s Golden Rule in
good agreement with the results of twotime Greenfunctions solving the Kadanoff and Baym equation. At later times when correlations develop, a kinetic equation of nonlocal and noninstantaneous character is derived which unifies the achievements of the transport in dense quantum gases with the Landau theory of quasiclassical transport in dense Fermi systems. The numerical solution is not more expensive than solving the Boltzmann equation since large cancellations in the offshell motion appear which are hidden usually in nonMarkovian behaviors. The quasiparticle drift of Landau’s equation is connected with a dissipation governed by a nonlocal and noninstant scattering integral in the spirit of Enskog corrections. These corrections are expressed in terms of shifts in space and time that characterize nonlocality of the scattering process. 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 correlated twoparticle contributions 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. V. Spicka, P. Lipavsky, K. Morawetz, Phys. Rev. B. 55, 5084 (1997); 5095 (1997): Quasiparticle transport equation with collisional delay: I. Phenomenological approach, II. Quantum statistical approach V. Spicka, P. Lipavsky, and K. Morawetz, Phys. Lett. A 240, (1998), 160: Nonlocal corrections to the Boltzmann equation for dense Fermi systems K. Morawetz, V. Spicka, P. Lipavsky, H.N. Kwong, Phys. Rev. C 59,6 (1999) 30523059: Duration and nonlocality of a nucleonnucleon collision, K. Morawetz, V. Spicka, P. Lipavsky, G. Kortemeyer, Ch. Kuhrts, R. Nebauer, Phys. Rev. Lett. 82, (1999), 3767: Virial corrections to simulations of heavy ion reactions, K. Morawetz, Phys. Rev. C 62, (2000) 44606: Dynamical constraints on phase transitions, K. Morawetz, M. P loszajczak, V.D. Toneev, Phys. Rev. C 62 (2000), 64602: Anomalous radial expansion in central heavyion reactions, K. Morawetz, P. Lipavsky, J. Normand, D. Cussol, J. Colin, B. Tamain, Phys. Rev. C 63 (2001) 034619: Midrapididty charge distribution in peripheral heavy ion collisions P. Lipavsky, V. Spiˇcka, K. Morawetz, Phys. Rev. E 59 (1999) R1291R1294: Noninstananeous collisions and two concepts of quasiparticles Book: P. Lipavský, V. Spicka, K. Morawetz; Kinetic equation for strongly interacting dense Fermi systems Annales de Physique, Vol. 26 No. 01 (February 2001), ISBN 2868835414  


Damping of giant resonances
Rainer Walke (Rostock), Michale Vogt (Dresden/Zurich)  
The giant resonances in symmetric and asymmetric nuclear matter [1] are described using a conserving relaxation time approximation to include collision effects and compared with experimental data of hot resonances. The isovector and isoscalar modes are coupled due to asymmetric nuclear meanfield acting on neutrons and protons differently. A further coupling is observed caused by collisional correlations. The latter one leads to the appearance of a third mode [2]. We had suggested that this mode might be observable in neutron rich systems like 11Be while in proton rich isobars this mode should appear with strong damping which was observed later as Pygmy resonances. Using a pseudoparticle simulation technique we simulate large amplitude isoscalar octupole excitations in a finite nuclear system [3]. Dependent on the initialization we can either observe clear octupole modes or overdamped octupole modes which decay immediately into quadrupole ones. Octupole modes should be present in central asymmetric collisions of heavy ions with mass relation 3:7. The contribution of surface scattering is compared with the contribution from the interparticle collisions. A unified response function is derived which includes surface damping via the Lyapunov exponent [4] as well as collisional damping. The former one is calculated for different shape deformations of quadrupole and octupole type [5]. The surface as well as the collisional contribution each reproduce the experimental value, therefore we propose a proper weighting between both contributions related to their relative occurrence due to collision frequency between particles and of particles with the surface. We find that for low and high temperatures the collisional contribution dominates whereas the surface damping is dominant around temperatures of a third of the centroid energy [5]. [1] U. Fuhrmann, K. Morawetz, R. Walke, Phys. Rev. C 58 (1995), 1473 [2] K. Morawetz, U. Fuhrmann, R. Walke, M. DiToro, Phys. Rev. C 57 (1998), R2813 [3] R. Walke, K. Morawetz, Phys. Rev. C 60 (1998) 17301; K. Morawetz, U. Fuhrmann, R. Walke, Nucl. Phys. A 649 (1998) 348c [4] K. Morawetz, Phys. Rev. E 61 (1998) 2555 [5] K. Morawetz, M. Vogt, U. Fuhrmann, P. Lipavsk, V. Spicka, Phys. Rev. C 60 (1999) 54601  


Nonlocal kinetic theory and simulation of heavy ion collisions P. Lipavsky (Charles University Prague)  
With the nonrelativistic Green'sfunction approach to the kinetic
equations for Fermi liquids far from equilibrium a consistent treatment is reached
of the offshell motion between collisions
and on the noninstant and nonlocal picture of binary collisions.
The resulting kinetic equation is of the Boltzmann type, and it
represents
an interpolation between the theory of transport in metals and the
theory of moderately dense gases. The free motion of particles is
renormalised by various mean field and mass corrections in the spirit
of Landau's quasiparticles in metals. The collisions are nonlocal in
the spirit of Enskog's theory of nonideal gases. The collisions are
moreover noninstant, a feature which is absent in the theory of gases,
but which is
shown to be important for dense Fermi systems.
In spite of its formal complexity, the presented theory has a simple
implementation within the MonteCarlo simulation schemes. Applications
in nuclear physics are given for heavyion reactions and the results are
compared with the former theory and recent experimental data.
movie: Ta +Au at 33 MeV movie: central collisions movie: Au+Au (40150 MeV, 012 fm impact) 

Dynamical constraints on phase transitions
J. Normand, D. Cussol, J. Colin, B. Tamain (LPC Caen), P. Lipavsky (Charles University Prague)  
The numerical solutions of nonlocal and local Boltzmann kinetic equations for the simulation of central heavy ion reactions are parametrized in terms of timedependent thermodynamical variables in the Fermi liquid sense. This allows one to discuss dynamical trajectories in phase space. The nonequilibrium state is characterized by nonisobaric, nonisochoric, etc., conditions, shortly called isonothing conditions. Therefore a com bination of thermodynamical observables is constructed which allows one to locate instabilities and points of possible phase transition in a dynamical sense. We find two different mechanisms of instability, a short time surfacedominated instability and later a spinodaldominated 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. In contrast the fast surface explosion occurs far outside the spinodal region and pertains also in the cases where the system develops too fast to suffer a spinodal decomposition and where the system approaches equilibrium outside the spinodal region.  


Midrapidity charge distribution
J. Normand, D. Cussol, J. Colin, B. Tamain (LPC Caen), P. Lipavsky (Charles University Prague)  
The experimental finding of enhancement of midrapidity matter shows the necessity to include correlations beyond BoltzmannUehlingUhlenbeck (BUU) which was performed in the framework of nonlocal kinetic theory. Different theoretical improvements are discussed. While the inmedium cross section changes the number of collisions, it leaves the transferred energy almost unchanged. In contrast the nonlocal scenario changes the energy transferred during collisions and leads to an enhancement of midrapidity matter. The renormalization of quasiparticle energies can be included in nonlocal scenarios and leads to a further enhancement of midrapidity matter distribution. This renormalization is accompanied by a dynamical softening of the equation of state seen in longer oscillation periods of the excited compressional collective mode. We propose to include quasiparticle renormalization by using the Paulirejected collisions which circumvent the problem of backflows in Landau theory. Using the maximum relative velocity of projectile and targetlike fragments we associate experimental events with impact parameters of the simulations. For peripheral collisions we find reasonable agreement between experiment and theory. For more central collisions, the velocity damping is higher in onebody simulations than observed experimentally, because of missing cluster formations in the kinetic theory used.  


Asymmetric BetheSalpeter equation for pairing and condensation
P. Lipavsky (Charles University Prague)  
The MartinSchwinger hierarchy of correlations are reexamined and the threepart icle correlations are investigated under various partial summations. Besides the known approximations of screened, ladder and maximally crossed diagrams the pairpair correlations are considered. It is shown that the recently proposed asymmetric BetheSalpeter equation to avoid unphysical repeated collisions is derived as a result of the hierarchical dependencies of correlations. Exceeding the parquet approximation we show that an asymmetry appears in the selfconsistent propagators. This form is superior over the symmetric selfconsistent one since it provides the NambuGorkov equations and gap equation for fermions and the Beliaev equations for bosons while from the symmetric form no gap equation results. The selfenergy diagrams which account for the subtraction of unphysical repeated collisions are derived from the pairpair correlation in the threeparticle Greenfunction. It is suggested to di stinguish between two types of selfconsistency, the channeldressed propagators and the completely dressed propagators, with the help of which the asymmetric ex pansion completes the Ward identity and is phiderivable. The resulting Tmatrix theory provides a consistent approach to superconductivity above and below the critical temprature.  


Birth of quasiparticles  Femtosecond formation of collective modes
P. Lipavsky (Charles University Prague)  
Starting from a quantum kinetic equation including the mean field and a conserving relaxationtime approximation we derive an analytic formula which describes the time dependence of the dielectric function in a plasma created by a short intense laser pulse. This formula reproduces universal features of the formation of collective modes seen in recent experimental data of femtosecond spectroscopy. The presented formula offers a tremendous simplification for the description of the formation of quasiparticle features in interacting systems. Numerical demanding treatments can now be focused on effects beyond these gross features found here to be describable analytically.  


Universal shorttime response and formation of correlations
after quantum quenches
 
The shorttime evolution of two distinct systems, the pump and probe experiments with
semiconductor and the sudden quench of cold atoms in an optical lattice, is found to be
described by the same universal response function. This analytic formula at short time scales
is derived from the quantum kinetic theory approach observing that correlations need time
to be formed. The influence of finite trapping potential is derived and discussed as well as
SingwiSjølander local field corrections. The quantum kinetic equation allows to understand
how twoparticle correlations are formed and the screening and collective modes are build up. K. Morawetz, Phys. Rev. B 90 (2014) 075303: Universal shorttime response and formation of correlations after quantum quenches; K. Morawetz, P. Lipavsky, M. Schreiber, Phys. Rev. B 72 (2005) 233203: Femtosecond formation of collective modes due to meanfield fluctuations; K. Morawetz, Phys. Rev. E 66 (2002) 022103: On a correspondence between classical and quantum particle systems; K. Morawetz, M. Bonitz, V. G. Morozov, G. Röpke, D. Kremp, Phys. Rev. E 63 (2001) 20102: Short time dynamics with initial correlations; K. Morawetz, V. Spicka, P. Lipavsky: Phys. Lett. A 246 (1998) 311: Formation of binary correlations in strongly coupled plasmas; 



Spin dependent transport in quantum dot systems Fabricio Macedo de Souza (Univeristy Uberlandia)  
Spindependent transport in quantum dots is a subject of intense study nowdays due to its relevance to the new generation of proposed spintronic devices that encompasses, for instance, the DattaDas transistor, memory devices and as an ultimate goal quantum computers. In particular, the recent progress in the coherent control of electron spins in quantum dots has stimulated even further the research in this field, for possible applications in quantum computation and quantum information processing. In addition to these fascinating technological applications, quantum dots constitute a unique wellcontrollable system to study fundamental physical aspects of transport in the strong Coulombcorrelated regime, and its interplay with spindependent effects. Our group deals with both ac and dc transport of charge and spin in quantum dots coupled to nonmagnetic of ferromagnetic leads.  
Terahertz outofplane resonances due to spinorbit coupling  
microscopic kinetic theory is developed which allows to investi
gate nonAbelian SU
(2) systems interacting with mean fields and spinorbit coupling under magneti
c fields in one,
two, and three dimensions. The coupled kinetic equations for the scalar and s
pin components are
presented and linearized with respect to an external electric field. The
dynamical classical and
quantum Hall effect are described in this way as well as the anomalous Ha
ll effect for which a
new symmetric dynamical contribution to the conductivity is presented. The coupled density and
spin response functions to an electric field are derived including arbitrary magnet
ic fields. The
magnetic field induces a staircase structure at frequencies of the Landau levels
. It is found that for
linear Dresselhaus and Rashba spinorbit coupling a dynamical outofplane spin res
ponse appears
at these Landau level frequencies establishing terahertz resonances. EPL, 104 (2013) 27005 

Spin and density response with magentic fields and spinorbit coupling J. Kailasvuori (MPIPKS Dresden)  
The quantum kinetic equation for SU(2) symmetric systems is derived with special consideration
of spinorbit coupling in magnetic and electric fields. The theory is applicable for
linear and nonlinear intrinsic and extrinsic spinorbit coupling as well as graphene. The RPA
response functions to an electric field are derived for arbitrary magnetic fields and spinorbit
coupling. The coupled density and spin response functions allow to describe dynamical classical,
quantum, and anomalous Hall effect as well as spinHall effects and its inverse. The
collective modes show a splitting due to polarization and/or spinorbit coupling for neutral
impurity scattering. The longrange Coulomb potential of charged impurities are considered
and the spinorbit coupling leads to characteristic modifications of the screening parameter.
New highfrequency modes outofplane are found. Explicit expressions for the dynamical response
and conductivity for relativistic Fermions, Dirac particles and graphene are presented. Quantum kinetic theory of spinpolarized systems in electric and magnetic fields with spinorbit coupling: I. Kinetic equation and anomalous Hall and spinHall effects, PRB 92 (2015) 245425, II. RPA response functions and collective modes, PRB 92 (2015) 245426 Terahertz outofplane resonances due to spinorbit coupling, Europhysics Letters, 104 (2013) 27005 

Regularizationfree transport properties of graphene  
Based on the quantum kinetic equations for systems with SU(2) symmetry, the density and spin currents are calculated in graphene realized here as the infinite masslimit of electrons with quadratic dispersion and a proper spinorbit coupling. Correspondingly the currents possess no quasiparticle part but only anomalous parts. The intraband and interband conductivities are discussed with respect to magnetic fields and magnetic domain puddles. It is found that the magnetic field and meanfield of domains can be represented by an effective Zeeman field. For large Zeeman fields the dynamical conductivities become independent of the density and are universal in this sense. The different limits of vanishing density, relaxation, frequency, and Zeeman field are not interchangeable. The optical conductivity agrees well with the experimental values using screened impurity scattering and an effective Zeeman field. The universal value of Hall conductivity is shown to be modified due to the Zeeman field. The spin current shows a subtlety in that a quasiparticle part appears though it vanishes for particle currents. The density and spin response functions to an external electric field are calculated and the dielectric function is discussed with respect to collective excitations. A frequency and wavevector range is identified where the dielectric function changes sign and the repulsive Coulomb potential becomes effectively attractive allowing Cooper pairing. 
Real optical conductivity versus frequency for different applied voltages compared with the experimental data (dashed).}  
Transport and noise in organic field effect devices L. M. Eng (IaPP Dresden), S. Gemming (HFZD DresdenRossendorf), T. Kunze (HFZD Dresden Rossendorf), P. Lipavsky (Charles University Prague), R. Luschtinetz (TU Dresden), P. Milde (IaPP Dresden), G. Seifert (TU Dresden)  
The goal to develop low cost storage and microelectronic devices has triggered an enormous activity in the research of organic field effect transistors (OFETs) based on different polymers. It is desireable that the molecular material possesses a high structural ordering to reach high charge carrier mobilities and low resistive losses. One of the most promising materials are Oligothiophenes and their derivatives. In general, the charge transport is largest in the direction perpendicular to the plane of the thiophene rings. This suggest to construct devices with parallely ordered molecular rings. Just recently an OFET structure has been built from ultrathin selfassembly films made up from
oligothiophenes, which are arranged in a highorder lamellar stacking perpendicular to the substrate surface. The transport properties are calculated with the help of surface Green functions.
movies: shuttling transport of molecules(red +, green ,grey 0 charged) 


Metalinsulator transition in heterojunctions
E. Nakhmedov (TU Chemnitz), C. Olbricht (TU Chemnitz)  
The conductivity in quasi twodimensional systems is calculated using the quantu m kinetic equation as well as molecular dynamical simulations. The system of qua si twodimensional electrons in heterojunctions which interact with charged and neutral impurities and the low temperature correction to the conductivity is ca lculated analytically. It turns out that the dynamical screening due to charged impurities leads to a linear temperature dependence, while the scattering from n eutral impurities leads to the usual Fermiliquid behavior. The experimental met alinsulator transition at low temperatures are reproduced. The effective ma ss of quasiparticle excitations in quasi twodimensional systems is calculated analytically. It is shown that the effective mass increases sharply when the density approaches the critical one of metalinsulator transition. This suggests a Motttype of transition rather than an Andersonlike transition.  
Anomalous 0.7 conductance and analogs in 1D systems Marcelo Rosenau da Costa (Uberlandia)  
We analyze the fractional quantization of the ballistic conductance associated with the light and heavy hole bands in Si, Ge, and GaAs systems. It is shown that the formation of a localized hole state in the region of the quantum point contact connecting two quasionedimensional hole leads drastically modifies the conductance pattern. The exchange interaction between localized and propagating holes results in fractional quantization of the ballistic conductance unlike that in electronic systems. The value of the conductance at the additional plateaus depends on the offset between the bands of the light and heavy holes and the sign of the exchange interaction constant.  
Equation of state for interacting Bose/Fermi systems and the BCSBEC transition
P. Lipavsky (Charles University Prague), M. Männel (FH Münster)  
Near Feshbach resonances it is possible to tune the interaction in ultracold Fermi and Bose gases. One can therefore vary the interaction in a BoseEinstein condensate (BEC) or drive a Fermi gas through a transition from Cooper pairs in a BCS state to twoparticle bound states which can also form a BEC. The main interest lies on the conditions for the formation of bound states, Cooper pairs or a BEC. Especially the dependence of these conditions on the interaction is investigated. Calculations are done mainly analytically with a manybody Green functions technique for idealized potentials. Therefore also medium effects such as Bose enhancement and Pauli repulsion can be included.  
Enhancement of pairing in correlated electron systems with nanostructures P. Lipavsky (Charles University Prague), M. Männel (TU Chemnitz), M. Schreiber (TU Chemnitz)  
A correlated fermion system is considered surrounding a finite cavity with virtual levels. The pairing properties are calculated and the influence of the cavity is demonstrated. To this end the GellMann and Goldberger formula is generalized to manybody systems. We find a possible enhancement of pairing temperature if the Fermi momentum times the cavity radius fulfills a certain resonance condition which suggests an experimental realization.  
Bernoulli potential at superconducting surfaces and deformation E. H. Brandt (MPI Stuttgart), J. Kolacek (Charles University Prague), P. Lipavsky (Charles University Prague)  
In superconductors penetrated by Abrikosov vortices the magnetic
pressure and the inhomogeneous condensate density induce a
deformation of the ionic lattice. Forces evoked
by vortices create a tension which modifies the total volume of
the sample and which was observed as magnetostriction.
The force which holds
electrons inward naturally pulls the oppositely charged ions
outwards and also corrugates the surface. The general relation
between the surface dipole and the surface tension with the
contribution of the superconducting condensate has been worked out for
anisotropic materials.
movie: direction dependent effective vortex mass of YBCO The NMR and NQR spectra of 63Cu in the CuO2 plane of YBa2Cu3O7 in the superconducting state are discussed in terms of a phenomenological theory of GinzburgLandau type extended to lower temperatures. We show that the observed spectra, can be explained by a standard theory of the Bernoulli potential with the charge transfer between CuO2 planes and CuO chains assumed. Book: P. Lipavský, J. Kolacek, K. Morawetz, E. H. Brandt, T. Yang; Bernoulli potential in superconductors  how electric fields help to understand superconductivity Springer, Lecture Notes in Physics, Vol. 733, Berlin (2008) 
 
Huddling of penguins as collective motion due to farfield cor
relations F. Morawetz (Kreuzgymnasium Dresden), D. Zitterbart (University of ErlangenNuremberg)  
Emperor penguins (Aptenodytes forsteri) have developed a specific tactics to sur
vive the Antarctic winter where they breed during 3 months basically living on t
heir reserves. They stand together so tightly that the temperature is more than
20 degrees higher than surrounding. At the same time the huddle is in continuous
move allowing a reshuffling of the outer and inner positions. Within an observe
d time of 3060 seconds, all penguins make small steps that travel as a wave thr
ough the entire huddle. We simulate the huddling behavior by a cellular automate
and show that nearest neighbor interactions are not sufficient to create this c
ollective behavior. Instead the next nearest neighbor interaction is necessary t
o describe basic properties of the collective motion, the rotation and mixing as
well as regular outbursts.
movie: Simulation of 113 pinguins, individuals colored, periodic boundaries, start as cluster movie: Simulation of 1239 pinguins, individuals colored, periodic boundaries, start as random movie: Simulation of 709 pinguins, individuals colored, with panic between times 1020, 310320, 610, 620 

Formation
of brine channels in sea ice Bernd Kutschan (FH Münster), Silke Thoms (AWI Bremerhafen), Sibylle Gemming (HFZD DresdenRossendorf)  
Sea ice is a very variable biotope with respect to extension,thickness, porosity or texture. Therefore the basic understanding of brine channel formation in sea ice is important for the interplay between the microbial colonization and their natural habitat. The early phase of brine channel formation in sea ice is considered. The first structures emerging during seaice formation are determined by the phase instability of the icewater system in the presence of salt. We apply a GinzburgLandau type approach to describe the phase separation in the twocomponent system (ice, salt). The free energy density involves two order parameters: one for the hexagonal ice phase with low salinity, and one for the liquid water with high salinity. A gradient dynamics minimizes the free energy with respect to the conservation of the salinity. The resulting model equations are solved numerically in one and two dimensions. The numerical solution shows a shorttime behavior of structure formation where the freezing is assumed and a largetime broadening of the structure. A stability analysis provides the phase diagram where brine channels can be formed. In thermodynamics the parameters determine the supercooling or superheating region and the specific heat respectively. The size of the brine channels depends on the salinity and the temperature. With the help of realistic parameters the brine channel distribution is calculated and found in agreement with the measured samples. 
Turing vs phase field structure 

Dynamical mechanism of antifreeze proteins Bernd Kutschan (FH Münster), Silke Thoms (AWI Bremerhafen), Sibylle Gemming (HFZD DresdenRossendorf)  
The fascinating ability of algae, insects and fishes to survive at temperatures below normal freezing is realized by antifreeze proteins (AFPs). Antifreeze proteins (AFPs) are surfaceactive molecules and interact with the diffusive water/ice interface preventing a complete solidification. A new dynamical mechanism is proposed how these proteins inhibit the freezing of water. We apply a GinzburgLandau type approach to describe the phase separation in the twocomponent system (ice, AFP). The free energy density involves two fields: one for the ice phase with low AFP concentration, and one for the liquid water with high AFP concentration. The time evolution of the ice reveals microstructures as a result of phase separation in the presence of AFPs. We observe a faster clustering of preice structure connected with a locking of grain size by the action of AFP which is an essentially dynamical process. The adsorption of additional water molecules are inhibited and the further growth of ice grains are stopped. The interfacial energy between ice and water is lowered by which the AFPs allow only smaller critical ice nucleus to be formed. Analogously to hysteresis in magnetic materials we observe a thermodynamic hysteresis leading to a nonlinear density dependence of the freezing point depression [1] in agreement with the experiments [2]. [1] B. Kutschan, K. Morawetz, S. Thoms, Phys. Rev. E 90, 022711 (2014). [2] H. Nada, Y. Furukawa, Polymer Journal 44, 690 
The freezing temperature depression of four different classes of AFP structures and insects with the fits compared to the experimental data (points) of [2]. 

Theory of water and charged liquid bridges  
The phenomenon of liquid bridge formation due to an applied electric field is investigated. A solution of a
charged catenary is presented, which allows one to determine the static and dynamical stability conditions where
charged liquid bridges are possible. The creeping height, the bridge radius and length, as well as the shape of
the bridge are calculated showing an asymmetric profile, in agreement with observations. The flow profile is
calculated from the NavierStokes equation leading to a mean velocity, which combines charge transport with
neutral mass flow and which describes recent experiments on water bridges.
movie: Experiment by Woisetschläger, Fuchs,..(http://www.youtube.com/ElmarCFuchs) 