1. 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, –

2. 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.

3. 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
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 V. 3. No. 2. P. 80.

4. Hardware, software, implementation
NVIDIA Tesla K40 (for calculation)
Intel® Core™ i (up to 3.60 GHz) (for comparison)
Heterogeneous Cluster ( (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

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

6. 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

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

8. 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

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

10. 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
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: 

11. 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

12. 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

13. 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

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

15. 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.

16. 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

17. 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 points
Statistics, n
Intel Core i5 3470
(1 thread), estimation
Tesla K40
106
~ 177 days
≈ 11 hours
Method enables calculations impossible before

18. 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.

19. Thank You