Joint Institute for Nuclear Research, Dubna, Russia Parallel implementation of numerical solution of few-body problem using Feynman’s continual integrals M.A. Naumenko, V.V. Samarin Joint Institute for Nuclear Research, Dubna, Russia Dubna, 03.07 – 07.07.2017

Motivation High interest in structure and reactions with few-body nuclei from both theoreticians and experimentalists (e.g., 2H, 3H, 3He, 6He, etc.) Feynman's continual integrals method [1] provides a mathematically more simple possibility for calculating the energy and the probability density of the ground states of N-particle systems compared to other approaches (e.g., expansion into hyperspherical harmonics [2]). The method allows application of modern parallel computing solutions to speed up the calculations [3]. [1] R. P. Feynman, A. R. Hibbs. Quantum Mechanics and Path Integrals. New York, McGraw-Hill, 1965. [2] R. I. Dzhibuti, K. V. Shitikova. Method of Hyperspherical Functions in Atomic and Nuclear Physics [in Russian]. Moscow, Energoatomizdat, 1993. [3] J. Sanders, E. Kandrot. CUDA by Example: An Introduction to General-Purpose GPU Programming. New York, Addison-Wesley, 2011.

Feynman’s continual integrals (FCI) method Feynman’s continual integral [1] is propagator - probability amplitude for a particle to travel from one point to another in a given time t Euclidean time t=iτ Example of three random trajectories Parallel calculations by Monte Carlo method [3] using NVIDIA CUDA technology were performed on cluster http://hybrilit.jinr.ru averaging over random trajectories with distribution in form of multidimensional Gaussian distribution [4] Euclidean propagator of free particle [2] Number of time steps [1] R. P. Feynman, A. R. Hibbs. Quantum Mechanics and Path Integrals. New York, McGraw-Hill, 1965. [2] D. I. Blokhintsev. Principles of Quantum Mechanics [in Russian]. Moscow, Nauka, 1976. [3] S. M. Ermakov. Monte Carlo Method in Computational Mathematics [in Russian]. St. Petersburg, Nevskiy Dialekt, 2009. [4] M. A. Naumenko, V. V. Samarin. Supercomp. Front. Innov. 2016. V. 3. No. 2. P. 80.

Hardware, software, implementation NVIDIA Tesla K40 (for calculation) Intel® Core™ i5-3470 (up to 3.60 GHz) (for comparison) Heterogeneous Cluster (http://hybrilit.jinr.ru/) (LIT, Joint Institute for Nuclear Research) Implemented in C++ language (single precision) Code compiled for architecture 3.5, CUDA version 7.5 cuRAND random number generator 1 thread calculates 1 trajectory

Scheme of calculation of ground state energy E0 Number of time steps number of trajectories

Scheme of calculation of ground state probability density |Ψ0|2 Number of time steps number of trajectories Benefits: Accumulation of data points 2) Low memory consumption (no grid) 6

Jacobi coordinates (q) for 2, 3, 4 - body systems

Calculation for exactly solvable 4-body oscillator model Four particles with masses m1=m2=m3=m4=m interact with each other by oscillator potentials: Normalized Jacobi coordinates: Energies: For example: U0=0, E0=9 U0=15, E0=15+9=6 Linear dependence of E0 value on U0 value demonstrates the correctness of technique. Feynman’s continual integrals U0=0, E0=9.05±0.1; U0=15, E0=5.98±0.02 Monte-Carlo calculation with statistics N=7107. Checking correctness of calculations: method reproduces exact result. 8

Unified set of effective two-body central potentials Nucleon-nucleon Nucleon-alpha-cluster i=1,2,3 i=1,2 i=1 Rα = 1.6755 fm; Model parameters U1 = -76 MeV, R1 = 2.05 fm, а1 = 0.3 fm; U2 = 62 MeV, R2 = 1.32 fm, а2 = 0.3 fm; U3 = 112 MeV, R3 = 1 fm, а3 = 0.5 fm.

Good agreement with experimental data Results of calculation of binding energies for nuclei 2H, 3,4,6He, 6Li, 9Be Comparison of theoretical and experimental binding energies in unified set of potentials (*for alpha-cluster nuclei 6He and 6Li energy of separation into alpha particles and nucleons is given). slope coefficient gives ground state energy Atomic nucleus Theoretical value, MeV Experimental value [1], MeV 2H 2.22 ± 0.15 2.225 3H 8.21 ± 0.3 8.482 3He 7.37 ± 0.3 7.718 4He 30.60 ± 1.0 28.296 6He* 0.96 ± 0.05 0.97542 6Li* 3.87 ± 0.2 3.637 9Be* 1.573  1.7 ± 0.1  4He 2H 3H 3He 6Li 6He 9Be Good agreement with experimental data [1] NRV web knowledge base on low-energy nuclear physics. URL: http://nrv.jinr.ru/

distance between proton and neutron, fm Checking the correctness of calculations of probability density |Ψ0|2 for ground state of 2H (p + n) Probability density for deuteron 2H Difference scheme (curve) Feynman’s continual integral (dots), Radial Schrödinger equation was solved by difference scheme for 2-body system (p + n) distance between proton and neutron, fm there are no bound subsystems 11

Propagator KE and probability density |Ψ0|2 for ground state of 3He (p + p + n) Potential wall at ri =5 fm for excluding excitation of break-up states. Probability density |Y|2 is consistent with potential landscape Logarithmic scale

Results for ground state of 3He nucleus (p + p + n) Charge density Binding energy, MeV Theory Experiment 7.37 ± 0.3 7.718 Charge radius, fm Theory Experiment 1.94 1.9664 Reasonable results and good agreement with experimental data are obtained

Qualitative comparison with other calculations for 3H nucleus Correlation function Oganessian Yu.Ts., Zagrebaev V.I., Vaagen J.S. Phys. Rev. C 60. 044605 (1999). Approximate calculation using Jastrow functions. Feynman’s continual integrals method

Probability density |Y|2 for ground state of 6He (a + n + n) There is possibility of investigating probability density for small values ​​of x and y Jacobi coordinates Most probable configurations are α-cluster + dineutron (1) and cigar configuration (2). Configuration n + 5He (3) has low probability.

Probability density for ground state of 4He nucleus (p + p + n + n) Jacobi coordinates (R1, r, R2) parallel shift of R2 Logarithmic scale Logarithmic scale 16

Comparison of calculation time for 3He nucleus (p + p + n) . Comparison of calculation time for 3He nucleus (p + p + n) Ground state energy Statistics, n Intel Core i5 3470 (1 thread), sec Tesla K40s, sec Performance gain, times 105 ≈1854 ≈8 ≈241 106 ≈18377 ≈47 ≈389 5·106 ─ ≈221 107 ≈439 Probability density |Y|2 in 43200 points Statistics, n Intel Core i5 3470 (1 thread), estimation Tesla K40 106 ~ 177 days ≈ 11 hours Method enables calculations impossible before

Conclusions The algorithm of calculation of ground states of few-body nuclei by Feynman's continual integrals method allowing us to perform calculations directly on GPU using NVIDIA CUDA technology was developed and implemented on C++ language. The energy and the square modulus of the wave function of the ground states of several few-body nuclei and cluster nuclei have been calculated. Correctness of the calculations was checked by comparison with the wave function obtained using the difference scheme for system (p + n), exact result for 4-body oscillator model, experimental binding energies, charge radii, charge distributions, etc. The results show that the use of GPGPU significantly increases the speed of calculations. This allows us to increase the statistics and accuracy of calculations, reduce the space step in the calculation of the wave functions, simplifies the process of debugging and testing, enables calculations impossible before.

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