k. vantournhout h. feldmeier nustar seminar...nustar seminar april 25, 2012 klaas vantournhout (gsi...
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Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamicsCan it cook nuclear pasta?
K. Vantournhout H. Feldmeier
GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany
NuSTAR seminarApril 25, 2012
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 1 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Outline
1 Introduction
2 Recipes for pasta
3 Molecular dynamics for fermions
4 Periodic boundary conditions and FMD
5 Conclusion and outlook
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 2 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Eta Carinae and thebipolar HomunculusNebula which surroundsthe star.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
March 16, 2012SN2012aw in M95
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : preface
Crab pulsar
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 3 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : composition of a neutron star
Outer Core
Inner Crust
Outer Crust
Inner Core
pasta phasesMantel
?K-condensate?-condensate?
quark matter?
hyperons?
n+p+e+µ
Coulomb CrystalRelativistic electron gasdripped neutrons
(e Z)
(e Z n)
Coulomb CrystalRelativistic electron gas
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 4 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : crustal matter and nuclear pastas
spaghettimeatballs lasagne ziti Swiss cheese
sauce
OuterCore
InnerCrust
Starting from the inner crust, with increasing density
three-dimensional spherical nuclear clusters (meatballs)
two-dimensional cylindrical tubes of dense matter (spaghetti)
one-dimensional slabs interlaid with planar voids (lasagne)
two-dimensional cylindrical neutron liquids within the nuclear matter(bucatini)
three-dimensional spherical neutron-liquid-bubbles embedded in thenuclear matter (Swiss cheese)
transition to the neutron star core, uniform neutron matter (sauce)
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 5 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : what is bucatini?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 6 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : crustal matter and nuclear pastas
spaghettimeatballs lasagne ziti Swiss cheese
sauce
OuterCore
InnerCrust
Where does the crustal and subnuclear density matter play a role?
Supernova physics
Neutron star cooling
The r-process
Neutron star glitches
Gravitational waves
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 7 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : crustal matter and nuclear pastas
spaghettimeatballs lasagne ziti Swiss cheese
sauce
OuterCore
InnerCrust
Where does the crustal and subnuclear density matter play a role?
Supernova physics
Neutron star cooling
The r-process
Neutron star glitches
Gravitational waves
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 7 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : crustal matter and nuclear pastas
spaghettimeatballs lasagne ziti Swiss cheese
sauce
OuterCore
InnerCrust
Where does the crustal and subnuclear density matter play a role?
Supernova physics
Neutron star cooling
The r-process
Neutron star glitches
Gravitational waves
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 7 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : crustal matter and nuclear pastas
spaghettimeatballs lasagne ziti Swiss cheese
sauce
OuterCore
InnerCrust
Where does the crustal and subnuclear density matter play a role?
Supernova physics
Neutron star cooling
The r-process
Neutron star glitches
Gravitational waves
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 7 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
pastas : is there a relation to Turing instabilities?
T. Jr. Bánsági et al., Science 331, pp. 1309–1312 (2011)
Tomographic reconstruction ofthe Belousov-Zhabotinskyreaction incorporated in anwater-in-oil aerosolmicro-emulsion.
Turing patterns form in areaction-diffusion mediumunder the condition of along-ranged reaction inhibitionand a short-ranged activation.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 8 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
pasta recipe : pasta alla liquid drop
The liquid-drop model
K. Nakazato et al., Phys. Rev. Lett. 103, 132501 (2009)
Semi-empirical model
mainly studies the canonical pasta phases
does not allow dynamics
the hard-coded geometry is studied
computationally fairly simple
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 9 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
pasta recipe : pasta in scatola alla Hartree-Fock
The Hartree-Fock model
W.G. Newton et al., Phys. Rev. C 79, 055801 (2009)
mean-field model
studies the canonical pasta phases
does not allow dynamics
the studied geometry strongly depends on the box-geometry
computationally very intensive (large configuration space)
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 10 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
pasta recipe : pasta classica con dinamica molecolare
The molecular-dynamics technique
G. Watanabe et al., Phys. Rev. Lett. 103, 121101 (2009)
a quantum-dressed classical many-body technique
studies the canonical and intermediate pasta phases
allows dynamics
the studied geometry is independent of the box-geometry
computationally fairly intensive
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 11 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
pasta recipe : crosta croccante con dinamica molecolare
The molecular-dynamics technique is awesome!
C. Horowitz et al., Phys. Rev. Lett. 102, 191102 (2009)
allows laboratory like simulations (e.g. viscosity or stress tensors)allows to study the effect of external probestime-dependent, out-of-equilibrium, thermodynamics, phasetransitions, . . .lacks quantum features (mimics fermion behaviour by a Paulipotential)
How can we use MD with correct quantum features?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 12 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
What is classical molecular dynamics
I. Newton
Molecular dynamics studies systems ofinteracting particles by numerically solvingthe equations of motion. These particles canbe point-like, spherical or non-spherical rigidbodies with extra internal degrees of freedom.
Time-dependent or static studies are done bysolving Newton’s or Hamilton’s equations ofmotion (here for point-particles) :
q =∂ H
∂ p, p =−∂ H
∂ q.
W.R. Hamilton
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 13 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the wave function
J.C.F. Gauß The single particle fermion wave function is aparametrised quantum-dressed Gaussian wavepacket
|qp⟩=∑
k
ckp|Akpbkp⟩ ⊗ |χkp⟩ ⊗ |ζkp⟩,
⟨x |Ab⟩= exp�
−1
2(x − b) ·A−1 · (x − b)
�
The antisymmetrised many-body fermionwave function |Q⟩ is a Slater determinant ofnon-orthogonal fermion wave-packets :
⟨x1, . . . , xA|Q⟩=1pA!
det
⟨x1|q1⟩ . . . ⟨x1|qA⟩...
. . ....
⟨xA|q1⟩ . . . ⟨xA|qA⟩
J.C. Slater
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 14 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the math
Adam Ries FMD has a matrix formalism. All relevantfermion physics is contained in the overlapmatrix n= [⟨qp|qq⟩]Apq=1 and its inverseo= n−1. Expectation values are evaluated as :
BI =⟨Q|BI |Q⟩⟨Q|Q⟩ =
A∑
pq=1
⟨qp|BI |qq⟩oqp
Time-dependent or static studies are done bymeans of variational principles on thevariables z of trail state |Q(z)⟩ :
iC·z = ∂
∂ z?⟨Q|H|Q⟩⟨Q|Q⟩ ,
∂
∂ z?⟨Q|H− TC M |Q⟩
⟨Q|Q⟩ = 0.
H. Feldmeier et al., Rev. Mod. Phys. 72, pp. 655–688 (2000)
Lord Rayleigh
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 15 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : its fascinating!
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 16 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
How to simulate bulk systems?
Finite-sized simulations introduceunwanted surface effects.
The more particles, the heavierthe computation (N2(4))
The less particles, the largerthe influence of surface effects.(488/1000 particles)
Can we use a small number ofparticles and avoid surfaceeffects?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 17 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
How to simulate bulk systems?
Finite-sized simulations introduceunwanted surface effects.
The more particles, the heavierthe computation (N2(4))
The less particles, the largerthe influence of surface effects.(488/1000 particles)
Can we use a small number ofparticles and avoid surfaceeffects?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 17 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
How to simulate bulk systems?
Finite-sized simulations introduceunwanted surface effects.
The more particles, the heavierthe computation (N2(4))
The less particles, the largerthe influence of surface effects.(488/1000 particles)
Can we use a small number ofparticles and avoid surfaceeffects?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 17 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : bulk systems
Periodic boundary conditions
One unit cell contains A singleparticle states
The unit cell is periodicallyreplicated in all directions
Which shape of unit cell?What kind of periodicity?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 18 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Periodic boundary conditions : Bravais lattices
a1
a2
A Bravais lattice is an infinite arrayof points periodically placedthroughout space.
The lattice appears identicalfrom any point.
Each lattice point is referred tothrough a Bravais latticevector :
R = n1a1 + n2a2 + · · · .
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 19 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Periodic boundary conditions : Primitive cell
A primitive cell tessellates spaceperfectly without overlap or voids.
A primitive cell contains only asingle lattice point.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 19 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Periodic boundary conditions : Wigner-Seitz cell
A Wigner-Seitz cell is a primitivecell preserving the lattice symmetry.
It is a region of space that is closerto one lattice point than to anyother.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 19 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Periodic boundary conditions : Reciprocal lattice
The reciprocal lattice is a duallattice in the Fourier space of theBravais lattice.
Its lattice vectors K satisfy
eiK ·R = 1.
Its Wigner-Seitz cell is referredto as the first Brillouin zone.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 19 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Periodic Boundary Conditions : Common lattices
Square lattice Hexagonal lattice
Simple cubic Face-centred cubic Body-centred cubic
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 20 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : bulk systems
Periodic boundary conditions
One unit cell contains A singleparticle states
The unit cell is periodicallyreplicated in all directions
Which Bravais lattice?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 21 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : bulk systems
What about FMD and PBC?
Fermions requireantisymmetrisation
Antisymmetry does not stop atthe border of the unit cell
FMD must be applied to theinfinite periodic system
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 21 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
N=
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
N=
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
N=
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
N=
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
N=
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Fermionic molecular dynamics : the bulk overlap matrix
The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.
N=
= nP−Q,pq = ⟨qp|T (Q− P)|qq⟩
Expectation values becomefinite-dimensional
Bρ,I =1
VBZ
∫
BZ
A∑
pq=1
B I ,pq(k)O qp(k)dk,
where a volume-integral reflects theinfinite system.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 22 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : the one-dimensional free Fermi gas
−2 −1 0 1 2
0
1
2
3
4
ρx
a/ = 0.2(a)
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
ρkk
F
a/ = 0.2(b)
−2 −1 0 1 2scaled position x/
0.0
0.5
1.0
1.5
2.0
ρx
a/ = 1.0(c)
0.0 0.5 1.0 1.5 2.0scaled momentum |k|/ kF
0.0
0.5
1.0
1.5
2.0
ρkk
F
a/ = 1.0kF = π/
(d)
` `
` `
`
`
``
For small overlap : the particles behave as distinguishable particles
For large overlap : the particles reproduce free Fermi gas behaviour
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 23 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : understanding antisymmetry and PBC
−0.5
0.0
0.5
y/`
−0.5 0.0 0.5x/`
Coordinate density ρVW Z
0.0
1.0
2.0
−2
−1
0
1
2
k y/k `
−2 −1 0 1 2kx/k`
Momentum density ρVBZ
0.0
5.0
10.0
Gaussian width :p
a = `/5without periodicity
without antisymmetryDensities in momentum and coordinatespace are summed.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 24 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : understanding antisymmetry and PBC
−0.5
0.0
0.5
y/`
−0.5 0.0 0.5x/`
Coordinate density ρVW Z
0.0
0.5
1.0
−2
−1
0
1
2
k y/k `
−2 −1 0 1 2kx/k`
Momentum density ρVBZ
0.0
0.5
1.0
1.5
2.0
2.5
Gaussian width :p
a = `/5without periodicity
with antisymmetry
Antisymmetry enables the Pauli-exclusion principle. It seems likeparticles are pushed away.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 24 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : understanding antisymmetry and PBC
−0.5
0.0
0.5
y/`
−0.5 0.0 0.5x/`
Coordinate density ρVW Z
0.0
1.0
2.0
−2
−1
0
1
2
k y/k `
−2 −1 0 1 2kx/k`
Momentum density ρVBZ
0.0
5.0
10.0
Gaussian width :p
a = `/5without periodicity
without antisymmetryDensities in momentum and coordinatespace are summed.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 24 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : understanding antisymmetry and PBC
−0.5
0.0
0.5
y/`
−0.5 0.0 0.5x/`
Coordinate density ρVW Z
0.0
1.0
2.0
−2
−1
0
1
2
k y/k `
−2 −1 0 1 2kx/k`
Momentum density ρVBZ
0.0
5.0
10.0
Gaussian width :p
a = `/5with periodicity
without antisymmetry
Densities in momentum and coordinatespace are summed and periodic correc-tions added in the spatial density.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 24 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : understanding antisymmetry and PBC
−0.5
0.0
0.5
y/`
−0.5 0.0 0.5x/`
Coordinate density ρVW Z
0.0
0.5
1.0
−2
−1
0
1
2
k y/k `
−2 −1 0 1 2kx/k`
Momentum density ρVBZ
0.0
0.5
1.0
1.5
2.0
2.5
Gaussian width :p
a = `/5with periodicity
with antisymmetry
Antisymmetry is a long-range many-body correlation. Both densities tend tothat of a free fermion system.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 24 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : the two-dimensional free Fermi gas
With increasing overlap, the system moves towardsthe results of the free Fermi gas, however there is astrong particle dependence.
Minimal kinetic energyfor N fermions perunit-cell.ρ = 1/fm2.
What is the origin ofthis seemingly erraticbehaviour?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 25 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Where is Wally?
Simulations often introduce constraints to overcome certain problems.But how do these constraints influence the physics of the system under
study? Periodic boundary conditions are such a restriction.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 26 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Where is Wally?
Where is my box?
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 26 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Coordinate space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
The effect of periodic boundary conditions
Momentum space
Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.
To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 27 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : the two-dimensional free Fermi gas
Spatial and momentum distribution for minimal kinetic energy
−0.4
−0.2
0.0
0.2
0.4
y/`
−0.4
−0.2
0.0
0.2
0.4
y/`
−0.4
−0.2
0.0
0.2
0.4
y/`
−0.4−0.2 0.0 0.2 0.4x/`
−0.4−0.2 0.0 0.2 0.4x/`
0.8 0.9 1.0 1.1 1.2 1.3
−2
−1
0
1
2
k y/k `
−2
−1
0
1
2
k y/k `
−2
−1
0
1
2k y/k `
−2 −1 0 1 2kx/k`
−2 −1 0 1 2kx/k`
0.0 0.5 1.0
ρ = 1/fm2, square latticeKlaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 28 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : the two-dimensional free Fermi gas
Spatial and momentum distribution for minimal kinetic energy
−0.4
−0.2
0.0
0.2
0.4
y/`
−0.4
−0.2
0.0
0.2
0.4
y/`
−0.4
−0.2
0.0
0.2
0.4
y/`
−0.4−0.20.0 0.2 0.4x/`
−0.4−0.20.0 0.2 0.4x/`
0.7 0.8 0.9 1.0 1.1 1.2
−2
−1
0
1
2
k y/k `
−2
−1
0
1
2
k y/k `
−2
−1
0
1
2k y/k `
−2 −1 0 1 2kx/k`
−2 −1 0 1 2kx/k`
0.0 0.5 1.0
ρ = 1/fm2, hexagonal latticeKlaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 28 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Results : the two-dimensional free Fermi gas
Brillouin zones are the limit for large overlap, and become morespherical with increasing N .
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 29 / 33
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Conclusion and outlook
Conclusion :
FMD can study bulk matter usingPBC
It reproduces free fermionbehaviour
The effect of PBC is wellunderstood
Outlook : We are finally on the stagewhere we can investigate the neutronstar’s crust.
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 30 / 33
THE ANTISYMMETRY OF A WAVE FUNCTION IS A LONG-RANGE EFFECT, TREAT IT
WITH RESPECT!
THE GEOMETRY OF THE LATTICE INFLUENCES THE PHYSICS AND SHOULD BE
TREATED AS AN ACTIVE PARAMETER IN SIMULATIONS.
Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook
Introduction : composition of a neutron star
Outer Core
Inner Crust
Outer Crust
Inner Core
pasta phasesMantel
?K-condensate?-condensate?
quark matter?
hyperons?
n+p+e+µ
Coulomb CrystalRelativistic electron gasdripped neutrons
(e Z)
(e Z n)
Coulomb CrystalRelativistic electron gas
Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 32 / 33
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