wolfgang(bauer( · 2013. 10. 28. · credit:bauer&wesall2013. october28,2013 7...
TRANSCRIPT
Wolfgang Bauer Department of Physics and Astronomy & Ins6tute for Cyber-‐Enabled Research
• Irina Sagert, MSU PHY (Lynen Fellow) • Dirk Colbry, MSU iCER
• Terrance Strother, LANL (former MSU Ph.D.)
• Tobias Bollenbach, MSU M.S. (Studiens6Nung)
• Rodney Picket, MSU CSE undergraduate
• James Howell, MSU CSE undergraduate
• Alec Staber, MSU AST undergraduate
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• … according to fluid dynamics experts
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• Conserva6on Laws – Linear momentum (Newton’s 2nd Law) – Energy (including Mass)
• Navier-‐Stokes Equa6on
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ρ ∂v∂t
+ (vi∇)v⎛
⎝⎜⎞⎠⎟ = −
∇p +
∇T +
F
v = flow velocityp = pressureρ = fluid density
T = stress tensorF = external force Credit: Thierry Dugnolle (Wikipedia)
• = ra6o of iner6al forces to viscous (fric6on) forces
•
• Rule of thumb: Re > 5,000 turbulent flow Re < 2,000 laminar flow
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Credit: Horns Rev 1 owned by Vaeenfall. Photographer Chris6an Steiness;
Re = ρvLη
ρ = fluid densityv = typical flow speedL = characteristic length scaleη = dynamic viscosity
Re = ρvLη
ρ = fluid densityv = typical flow speedL = characteristic length scaleη = dynamic viscosity
• = ra6o of mean free path to characteris6c length scale
• Needed for hydro to be valid • Example ideal gas:
– N2 molecules at STP: Kn ~ 30
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Kn = λL
Knideal gas =kBT
2(4πr2 )pL
Kn→ 0
Credit: Bauer & Wesjall 2013
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arXiv.org > cond-‐mat > arXiv:1105.6256
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Tfit = 419 MeV Tfit = 239 MeV
Rela6vis6c Heavy Ion Collider (RHIC). Data: STAR Collabora6on
100 AGev Au + Au 100 AGev
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Credit: McLerran 2013
Cartoon of the 6me evolu6on of an ultra-‐rela6vis6c heavy ion collision
• Rela6vis6c heavy ion collisions
• Scale 10-‐15 m
• Shock wave (?) • Successful @RHIC
– v2 ✓
– η/s small ✓
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H. Stöcker, J.A. Maruhn, and W. Greiner, PRL 44, 725 (1980)"
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coordinate space
anisotropy
momentum space
anisotropy
( )( )⎟⎠⎞⎜
⎝⎛ Ψ−+
=
∑∞
=1
2
3
3
cos),(21
ddd
21
pdd
nrTn
TT
nypv
yppNNE
φ
π
Ø Azimuthal correlaBon with the reacBon plane.
Ø Built up in the early stage, therefore supplies the early informaBon of maHer generated in the collision.
Credit: Na Li, 25th WWND, 2009
( )( )rn nv Ψ−= φcos
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PRL 98, 162301
Strong indica6on for hydrodynamic flow!
• Type II core collapse supernovae
• Scale 107 m
• Neutrino-‐driven dynamics
• Stalled shock wave
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Fryer & Warren, ApJ 574, L65 (2002 )
• Na6onal Igni6on Facility • Most powerful laser in the world: 0.5 PW (1.8 MJ/4 ns)
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• ICF capsule • Scale 10-‐5 m
• Livermore hydro codes fail – Igni6on predicted, but not achieved
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• Start from many-‐body Hamiltonian
• Construct density matrix for many par6cle wave func6on
• BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) Hierarchy
• Truncate at some level n: – Here: truncate at 3-‐body level; 3-‐body ma6x = product of 2-‐body density matrices
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i∂t ΨN = H ΨN
ρN = ΨN ΨN ⇒ i∂tρN = [H,ρN ]
∂t ρn = F(ρn ,ρn+1)
ρn+1 =G(ρn )
• Introduce Wigner transform:
• Final result: 6me evolu6on equa6on for 1-‐body Wigner-‐transform, which contains two-‐body correla6ons
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f (x, p,t) ≡ 1
πψ (x + y)∫ ψ (x − y) e2ipy/dy
• Approximate f by a sum of delta func6ons in phase space:
• Insert this into integral transport equa6on to obtain equa6ons of mo6on for 6 coordinates of each test par6cle
October 28, 2013 18 Nuclear EOS
Coulomb
2-‐body scaeering
• Energy func6onal as a func6on of density temperature, momentum, isospin, …
• Not easy!
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• Two-‐body cross sec6ons from experiment • Most accurate method: Distance of closest approach
– CPU 6me O(N2) – Arbitrarily precise shock wave localiza6on [J. Cugnon et al. NPA352, 505 (1981)]
• Fastest method: Direct SimulaBon Monte Carlo – Scaeering grid – CPU 6me O(N log N) – Causality viola6ons and shock wave diffusion unavoidable [F.J. Alexander, A.L. Garcia, B.J. Alder, PRL 74, 5212 (1995), G. Kortemeyer, F. Daffin,WB, PLB 374, 25 (1996)]
• Best of both Worlds? [I. Sagert et al, sub. Physics of Fluids (2012)] October 28, 2013 20
• Point 1: Kine6c theory without collisions (= Vlasov) reproduces mean field theory (= TDHF)
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JOSEPH J. MOLITORIS, DETLEV HAHN, HORST STÖCKER, Prog. Nuc. Part. Phys. 15, 239 (1985)
• Point 2: Kine6c theory with collisions (= VUU, BUU, …) reproduces hydro!
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JOSEPH J. MOLITORIS, DETLEV HAHN, HORST STÖCKER, Prog. Nuc. Part. Phys. 15, 239 (1985)
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JOSEPH J. MOLITORIS, DETLEV HAHN, HORST STÖCKER, Prog. Nuc. Part. Phys. 15, 239 (1985)
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Perform scattering
Update particles'
positions and velocities
Generate particle distriubtion
Loop over all bins
Collision partner search in
neighborhood bins
Collision partner search in
neighborhood binsCollision partner search in
neighborhood bins
Populate bins
[000]
[010]
[020]
[030]
[040]
[100] [200] [300] [400]
1 1 1 2 2 2 5 5 5
1 1 1 2 2 2 5 5 5
1 1 1 2 2 2 5 5 5
3 3 3 4 4 4 6 6 6
3 3 3 4 4 4 6 6 6
Parallel neighborhood search by six CPUs
(1)
(2)
(a) (b) (c)
0
0.02
0.04
0.06
0.08
0 0.02 0.04 0.06 0.08
y
x
Collision test 1Collision test 2Collision test 3
Particle of interest
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0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
Sp
eed
-up
S=
T#C
PU
/T1
Number of CPUs
(d) SimulationIdeal speed-up
• Observables – Bulk velocity
– Pressure (= average of diagonal elements of stress tensor per volume)
– Density
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Ini6al condi6ons:
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Ini6al condi6ons:
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Homogeneous gas with uniform radial inward speed vin
( dof)
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1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Den
sity
n/n
in
x
Nbin=100Nbin=150Nbin=711Analytic
2.8
2.9
3
3.1
0 0.2 0.4 0.6 0.8
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• No wall hea6ng • No causality viola6ons
• No shock wave diffusion • No “running ahead” of shock front
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Turbulent flow
• Code passes all standard hydrodynamic tests • (Slow) convergence to analy6c results with increasing test par6cle number – Typical number of test par6cles used in 3d tests: 10 million – 100 million
• No physical limit to precision of shock wave localiza6on
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• Large mean free path, large Knudsen number • Sod shock test:
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λ = mean free path dx = box size Fs = “free streaming”, λ infinite
• Large mean free path, large Knudsen number • 2d Noh shock test:
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λ = mean free path dx = box size Fs = “free streaming”, λ infinite
• Supernova explosion driven by neutrino shock (?) • Neutrinos cannot be modeled by hydro
– Extremely small cross sec6ons – Very large Knudsen number
• Kine6c theory: no problem – Can be calculated in the same framework
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• 2-‐body collision terms structurally iden6cal to BUU source term – Couples transport equa6ons of baryons and neutrinos – Essen6al input: neutrino-‐nucleus cross sec6ons (Nakamura et al, ApJ 1999; K. Sumiyoshi et al, NPA 2001, Fröhlich et al, PRL 2006, B.A. Brown, …)
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3 free neutrons
• Explicitly represent all nuclei – Many hundreds of isotopes – Lots of work: reac6on network, weak interac6on cross sec6ons
– All Z,A between drip lines – Ensemble propaga6on
– “Coupled channels” in reac6on network
– Free baryons
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• Start with Woosley & Weaver’s 15 M¤ progenitor
Use for the first 107 years Concentrate on last 0.3 seconds
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• Mcore = 1.33 M¤ • Spherically symmetric
• Radius ≈ 1000 km
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• Single processor (spherical symmetry) • 1 million maeer test par6cles
– 385 nuclei + free baryons • Cold soN BKD nuclear EOS • Weak interac6on network
– Electron capture (reduced FFN rates) – Neutrino-‐maeer interac6ons
– Neutrino oscilla6ons a la “MSW”
• No fusion or photo-‐disintegra6on channels included
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Interplay of macro-‐ and micro-‐scales forces very large number of compara6vely small 6me steps (c ~ 1 N/ns) Δt = 10-‐5 s => Mostly boring ini6al 6me evolu6on (take 1000 steps between frames)
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3% c
Boring first 9000 6me steps are done Now: Movie
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~0.002ρ0
~0.2ρ0
58 km
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Remnant • M=0.25 M¤ • R=7.3 km • ρcentral = 1.7ρ0 • ηcentral = 0.27
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• Electron frac6on spike “cuts” the core in two – Proto-‐remnant “gently” assumes ideal configura6on
– Role of nuclear EOS totally different • How does the spike form?
– ρ(rexp) ~ 0.002ρ0 – Study neutrino-‐maeer interac6on probabili6es
• Nuclear structure • Rela6vis6c electron gas sta6s6cal mechanics
• Essen6al input: neutrino cross sec6ons & nuclear structure (weak neutral current ~ A2)
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t = 0.01000 s
Average neutrino interac6on probability
Isotope composi6on
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t = 0.03000 s
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t = 0.05000 s
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t = 0.07000 s
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nuclear structure
t = 0.09000 s
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Free neutrons
t = 0.10000 s
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electron gas degeneracy
t = 0.11265 s
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t = 0.12000 s
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P. Sorensen (this mee6ng)
• Angular momentum conserva6on
• Baryons fall in on equator; neutrinos escape along poles
• Macroscopic parity viola6on
• Finite recoil of neutron start
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• New solu6on method for supernova dynamics – Test par6cle method
– Link between nuclear dynamics and astrophysics – Passes all standard hydrodynamic verifica6on tests
• New explosion mechanism – Shockwave originates ~ 50 km above neutron star surface
– Due to neutrino hea6ng / opacity change – VERY dependent on nuclear structure and neutrino cross sec6ons
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Mike Lisa:
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NOT YET!
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RANP 4 Advisory Board, August 1995
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