9 PHQMD—A Microscopic Transport Approach for Heavy-Ion …
107
the collision integrals of the PHSD approach [10–14] and density dependent 2-body
potential interactions of QMD type models [7, 15, 16]. The model uses as well the
dynamical quasi particle description of the QGP of PHSD. The original PHSD meanfield propagation (realized within the parallel ensemble method) is kept as an option,
too, which will allow to investigate the differences between both the approaches.
In contradistinction to statistical models or coalescence models, in PHQMD the
clusters are formed dynamically. This means that at the end of the heavy-ion reaction
the same nucleon-nucleon potential interaction, which is present during the whole
time evolution, forms bound clusters of nucleons which are well distinct in phase
space from other clusters and free nucleons.
9.2 The PHQMD Approach
The propagation of the Wigner density is determined by a generalized Ritz variational
principle [17], which has been developed for the Time Dependent Hartree–Fock
approach,
δ
t 2
t 1
dt < ψ(t)|i
d
dt
− H |ψ(t) >= 0.
(9.1)
In our approach [6] we assume that the n-body Wigner density is the direct product of
the single-particle Wigner densities. There are also QMD versions which use a Slater
determinant, FMD [17] and AMD [18], but due to the difficulty to formulate collision
terms these approaches have only been applied to low energy heavy-ion collisions.
Assuming that the wave functions have a Gaussian form and that the width of the
wave function is time independent one obtains for the time evolution of the centroids
of the Gaussian single particle wave functions two equations which resemble the
equation of motion of a classical particle with the phase space coordinates r i0 , p i0
[7].
The difference is that here the expectation value of the quantal Hamiltonian is
used and not a classical Hamiltonian:
˙
r i0 =
∂H
∂ p i0
˙
p i0 = −
∂H
∂r i0
.
(9.2)
These time evolution equations are specific for Gaussian wave functions. For other
choices of wave functions the time evolution equations would be different. The
Hamiltonian of the nucleus is the sum of the Hamiltonians of the nucleons, composed
of kinetic and two-body potential energy,
H =
i
H i =
i
(T i + V i ) =
i
⎛
⎝ T i +
j =i
V i, j
⎞
⎠ .
(9.3)
107
the collision integrals of the PHSD approach [10–14] and density dependent 2-body
potential interactions of QMD type models [7, 15, 16]. The model uses as well the
dynamical quasi particle description of the QGP of PHSD. The original PHSD meanfield propagation (realized within the parallel ensemble method) is kept as an option,
too, which will allow to investigate the differences between both the approaches.
In contradistinction to statistical models or coalescence models, in PHQMD the
clusters are formed dynamically. This means that at the end of the heavy-ion reaction
the same nucleon-nucleon potential interaction, which is present during the whole
time evolution, forms bound clusters of nucleons which are well distinct in phase
space from other clusters and free nucleons.
9.2 The PHQMD Approach
The propagation of the Wigner density is determined by a generalized Ritz variational
principle [17], which has been developed for the Time Dependent Hartree–Fock
approach,
δ
t 2
t 1
dt < ψ(t)|i
d
dt
− H |ψ(t) >= 0.
(9.1)
In our approach [6] we assume that the n-body Wigner density is the direct product of
the single-particle Wigner densities. There are also QMD versions which use a Slater
determinant, FMD [17] and AMD [18], but due to the difficulty to formulate collision
terms these approaches have only been applied to low energy heavy-ion collisions.
Assuming that the wave functions have a Gaussian form and that the width of the
wave function is time independent one obtains for the time evolution of the centroids
of the Gaussian single particle wave functions two equations which resemble the
equation of motion of a classical particle with the phase space coordinates r i0 , p i0
[7].
The difference is that here the expectation value of the quantal Hamiltonian is
used and not a classical Hamiltonian:
˙
r i0 =
∂H
∂ p i0
˙
p i0 = −
∂H
∂r i0
.
(9.2)
These time evolution equations are specific for Gaussian wave functions. For other
choices of wave functions the time evolution equations would be different. The
Hamiltonian of the nucleus is the sum of the Hamiltonians of the nucleons, composed
of kinetic and two-body potential energy,
H =
i
H i =
i
(T i + V i ) =
i
⎛
⎝ T i +
j =i
V i, j
⎞
⎠ .
(9.3)