Path Integral Monte Carlo Simulations on the Blue Waters System. Burkhard Militzer University of California, Berkeley

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1 Path Integral Monte Carlo Simulations on the Blue Waters System Burkhard Militzer University of California, Berkeley

2 Outline 1. Path integral Monte Carlo simulation method 2. Application of CH plastic ablator materials 3. Application to warm, dense silicon Kevin Driver, Francois Soubiran, Shuai Zhang Lorin Benedict (LLNL) and Suxing Hu (LLE) Supported by NSF-DOE partnership for plasma science. PRAC computer time on Bluewaters machine at Urbana-Champaign, IL.

3 Phase Diagram of Hydrogen Solar GP: Jupiter, Saturn n=10 18 cm -3 n=10 26 cm -3 Density

4 Juno Mission Now in orbit around Jupiter Interior model to match gravity data Equation of state calculations for hydrogen-helium mixtures Thermodynamics of heavier elements (Z) Mission Timeline: Launch - August 2011 Earth flyby gravity assist - October 2013 Jupiter arrived in July 2016 End of mission much past October 2017

5 1) Path integral Monte Carlo for T>5000K n=10 18 cm -3 n=10 26 cm -3

6 1) Path integral Monte Carlo for T>5000K 2) Density functional molecular dynamics below Born-Oppenheimer approx. MD with classical nuclei: F = m a Forces derived DFT with electrons in the instantaneous ground state.

7 I. Path Integral Monte Carlo

8 Canonical Ensembles: Classical Quantum Boltzmann factor e E / k BT Density matrix ˆρ = e β Ĥ ρ(r, R #,β) = R e β H ˆ R' ρ(r, R #,β) = Thermodynamic averages: S e β E S Ψ S * (R) Ψ S (R') Z Cl = e β E S S Z Q = Tr[ ˆρ] = Ô = Tr[Ô ˆρ] Tr[ ˆρ] dr R e β Ĥ R

9 Step 1 towards the path integral Matrix squaring property of the density matrix Matrix squaring in operator notation: ˆ ρ = e β H ˆ = ( (β / 2) H ˆ e ) ( / 2) H ˆ ) e (β, β = 1 k B T Matrix squaring in real-space notation: R ˆ ρ R $ = dr 1 (β / 2) H ˆ (β / 2) H ˆ R e R 1 R 1 e R $ Matrix squaring in matrix notation: " R' % $ ' $ R ' # $ &' = " R 1 % $ ' $ R ' # $ &' * " R ( % $ ' $ R 1 ' # $ &'

10 Repeat the matrix squaring step Matrix squaring in operator notation: ˆ ρ = e β H ˆ = ( (β / 4) H ˆ e ) 4, β = 1 k B T Matrix squaring in real-space notation: R ˆ ρ R $ = dr 1 dr 2 dr 3 (β / 4) H ˆ R e R 1 R 1 e (β / 4 ) H ˆ (β / 4) H ˆ R 2 R 2 e R 3 R 3 e (β / 4 ) H ˆ R $

11 Path Integrals in Imaginary Time Every particle is represented by a path, a ring polymer. Density matrix: ˆ ρ =e β H ˆ & = e τh ˆ ) M ( + ( + ˆ ρ = e β H ˆ = ( ( ', β=1/k H 1 T,τ =β/m B O ˆ = Tr[ O ˆ ˆ ρ ] k B T, τ = β M Tr[ ˆ ρ ] ( τ ˆ e ) M, β = + * + Trotter break-up: R ˆ ρ R $ = R (e τ H ˆ ) M R $ = dr 1... dr M 1 R e τ ˆ H R 1 R 1 e τ ˆ H R 2... R M 1 e τ ˆ H $ R

12 Path Integrals in Imaginary Time Simplest form for the paths action: primitive approx. Density matrix: ˆ ρ =e β H ˆ & = e τh ˆ ) M ( + ( + ˆ ρ = e β H ˆ = ( ( ', β=1/k H 1 T,τ =β/m B O ˆ = Tr[ O ˆ ˆ ρ ] k B T, τ = β M Tr[ ˆ ρ ] ( τ ˆ e ) M, β = + * + Trotter break-up: R ˆ ρ R $ = R (e τ H ˆ ) M R $ = dr 1... dr M 1 R e τ ˆ H R 1 R 1 e τ ˆ H R 2... R M 1 e τ ˆ H $ R Path integral and primitive action S : R ˆ ρ $ R = dr t e S[R t ] R R' M (R S[R t ] = i+1 R i ) 2 + τ i=1 4λτ 2 V (R i ) + V (R i +1 ) [ ] Pair action: Militzer, Comp. Phys. Comm. (2016)

13 Douglas Adams: Infinite Improbability Drive of spaceship Heart of Gold

14 Restricted PIMC for fermions: How is the restriction applied? Construct a fermionic trial density matrix in form of a Slater determinant of single-particle density matrices: Imaginary time x ρ T (R, R #,β) = ρ(r 1, r 1 #,β) ρ(r 1, r N #,β) ρ(r N, r 1 #,β) ρ(r N, r N #,β) Enforce the following nodal condition for all time slices along the paths: ρ T [ R(t),R(0),t] > 0 This 3N-dimensional conditions eliminates all negative and some positive contribution to the path Solves the fermion sign problem approx. Free-particle nodes:

15 Starting from Restricted PIMC Simulations of Hydrogen

16 Water

17 DFT-MD Simulations of a Water Plasma at Temperatures of K and Megabar Pressures (P-P 0 )/P 0 (P-P 0 )/P Water Density=3.18 g/cm 3 24-atom cell Water Density=11.18 g/cm 3 24-atom cell Water DFT-MD Temperature (K)

18 Debye Plasma model added Big Gap in Temperature remains (P vs T for a Water Plasma) (P-P 0 )/P 0 (P-P 0 )/P Water Density=3.18 g/cm 3 24-atom cell Water Density=11.18 g/cm 3 24-atom cell Water DFT-MD Debye model Temperature (K)

19 First Path Integral Monte Carlo Simulations for Heavier Elements Fill this Gap in Temperature (P-P 0 )/P 0 (P-P 0 )/P Water Density=3.18 g/cm 3 24-atom cell Water Density=11.18 g/cm 3 24-atom cell Water PIMC DFT-MD Debye model Temperature (K)

20 Study Structural Properties: Pair Correlation Functions for Water and Carbon Plasmas Carbon Density =12.64 g/cm 3 Water Density=11.18 g/cm 3 T=750,000 K PIMC DFT-MD g(r) Water (H-H) Water (O-H) Carbon Water (O-O) r(å)

21 Path Integral Monte Carlo and DFT-MD are in very good agreement Carbon Density =12.64 g/cm 3 Water Density=11.18 g/cm 3 T=750,000 K PIMC DFT-MD g(r) Water (H-H) Water (O-H) Carbon Water (O-O) Driver, Militzer, Phys. Rev. Lett. (2012) r(å)

22 New Path Integral Monte Carlo Simulations of Heavier Elements Aid Fusion Capsule Design ICF Hohlraum ICF Capsule

23 Study planetary interiors in the laboratory: shock wave experiments Two-stage gas gun (Livermore) 0.2 Mbar Z capacitor bank (Sandia) 2 Mbar Nova laser (Livermore) 3.4 Mbar National Ignition Facility 700 Mbar

24 Shock wave measurements determine the Equation of State on the Hugoniot curve Conservation of mass, momentum and energy yields: ρ = ρ 0 u s u s u p P = P 0 + ρ 0 u s u p E = E (V 0 V )(P + P 0 )

25 CH plastics

26 Inertial confinement fusion experiments with plastic coated spheres of liquid H 2 CH H 2 (liquid) (Graphics: Bachmann et al. LLNL)

27 PIMC and DFT-MD simulations performed for C 2 H, CH, C 2 H 3, CH 3 and CH 4.

28 CH Shock Hugoniot Curves: Comparison of Theory and Experiments

29 Silicon

30 Path Integral Monte Carlo with localized nodal surfaces and application to silicon plasmas How the nodes are enforced: Nodes are a Slater determinant: Before we used only free-particle orbitals (plane waves): New idea: Add Hartree-Fock orbitals Militzer, Driver, Phys. Rev. Lett. (2015)

31 Silicon: Energy and Pressure Comparison of Path Integral Monte Carlo and Kohn-Sham DFT Militzer, Driver, Phys. Rev. Lett. (2015)

32 Silicon Hugoniot Curve: Experiments and Semi-analytical EOS models Hu, Militzer, et al. Rev. B 94 (2016)

33 Silicon Hugoniot Curve: Path Integral Monte Carlo, Orbital-Free DFT Hu, Militzer, et al. Rev. B 94 (2016)

34 Silicon Hugoniot Curve: Average-Atom models Hu, Militzer, et al. Rev. B 94 (2016)

35 Comparison of Hugoniot Curves for Different Materials Driver, Militzer, PRB 93, (2016)

36 Comparison of Hugoniot Curves for Different Materials Driver, Militzer, PRB 93, (2016)

37 Comparison of Hugoniot Curves for Different Materials Driver, Militzer, PRB 93, (2016)

38 Comparison of Hugoniot Curves for Different Materials Driver, Militzer, PRB 93, (2016)

39 Comparison of Hugoniot Curves for Different Materials Driver, Militzer, PRB 93, (2016)

40 Conclusions For elements from hydrogen to through silicon, the energy, pressure, and g(r) functions computed with PIMC and KS-DFT agree well at K. Internal energy agrees to better than 5 Ha/atom. Pressure to 2%. So far, we could not detect any problem with the zerotemperature exchange-correlation functionals. We will provide more EOS data to improve orbit-free methods. Second row elements: More comparison between simulations and experiments Extending PIMC to third row elements in particular iron.

41 Conclusions Blue Waters provided a 1 order of magnitude of increase in productivity. Allowed us to solve more complex problems (heavier elements and more accurate nodal surfaces) Enabled us to study a wider parameter space (vary material s composition in addition to density and temperature)

42 The End