Path integral Monte Carlo

Path integral Monte Carlo (PIMC) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral formulation. The application of Monte Carlo methods to path integral simulations of condensed matter systems was first pursued in a key paper by John A. Barker.[1][2]

The method is typically (but not necessarily) applied under the assumption that symmetry or antisymmetry under exchange can be neglected, i.e., identical particles are assumed to be quantum Boltzmann particles, as opposed to fermion and boson particles. The method is often applied to calculate thermodynamic properties[3] such as the internal energy,[4] heat capacity,[5] or free energy.[6][7] As with all Monte Carlo method based approaches, a large number of points must be calculated.

In principle, as more path descriptors are used (these can be "replicas", "beads," or "Fourier coefficients," depending on what strategy is used to represent the paths),[8] the more quantum (and the less classical) the result is. However, for some properties the correction may cause model predictions to initially become less accurate than neglecting them if a small number of path descriptors are included. At some point the number of descriptors is sufficiently large and the corrected model begins to converge smoothly to the correct quantum answer.[5] Because it is a statistical sampling method, PIMC can take anharmonicity fully into account, and because it is quantum, it takes into account important quantum effects such as tunneling and zero-point energy (while neglecting the exchange interaction in some cases).[6]

The basic framework was originally formulated within the canonical ensemble,[9] but has since been extended to include the grand canonical ensemble[10] and the microcanonical ensemble.[11] Its use has been extended to fermion systems[12] as well as systems of bosons.[13]

An early application was to the study of liquid helium.[14] Numerous applications have been made to other systems, including liquid water[15] and the hydrated electron.[16] The algorithms and formalism have also been mapped onto non-quantum mechanical problems in the field of financial modeling, including option pricing.[17]

See also

References
  1. Barker, J. A. (1979). "A quantum-statistical Monte Carlo method; path integrals with boundary conditions". The Journal of Chemical Physics. 70 (6): 2914–2918. Bibcode:1979JChPh..70.2914B. doi:10.1063/1.437829.
  2. Cazorla, Claudio; Boronat, Jordi (2017). "Simulation and understanding of atomic and molecular quantum crystals". Reviews of Modern Physics. 89 (3): 035003. arXiv:1605.05820. Bibcode:2017RvMP...89c5003C. doi:10.1103/RevModPhys.89.035003. Retrieved May 13, 2022.
  3. Topper, Robert Q. (1999). "Adaptive path-integral Monte Carlo methods for accurate computation of molecular thermodynamic properties". Advances in Chemical Physics. 105: 117–170. Retrieved May 12, 2022.
  4. Glaesemann, Kurt R.; Fried, Laurence E. (2002). "An improved thermodynamic energy estimator for path integral simulations". The Journal of Chemical Physics. 116 (14): 5951–5955. Bibcode:2002JChPh.116.5951G. doi:10.1063/1.1460861.
  5. Glaesemann, Kurt R.; Fried, Laurence E. (2002). "Improved heat capacity estimator for path integral simulations". The Journal of Chemical Physics. 117 (7): 3020–3026. Bibcode:2002JChPh.117.3020G. doi:10.1063/1.1493184.
  6. Glaesemann, Kurt R.; Fried, Laurence E. (2003). "A path integral approach to molecular thermochemistry". The Journal of Chemical Physics. 118 (4): 1596–1602. Bibcode:2003JChPh.118.1596G. doi:10.1063/1.1529682.
  7. Glaesemann, Kurt R.; Fried, Laurence E. (2005). "Quantitative molecular thermochemistry based on path integrals". The Journal of Chemical Physics (Submitted manuscript). 123 (3): 034103. Bibcode:2005JChPh.123c4103G. doi:10.1063/1.1954771. PMID 16080726.
  8. Doll, J.D. (1998). "Monte Carlo Fourier path integral methods in chemical dynamics". Journal of Chemical Physics. 81 (8): 3536. doi:10.1063/1.448081. Retrieved May 13, 2022.
  9. Feynman, Richard P.; Hibbs, Albert R. (1965). Quantum Mechanics and Path Integrals. New York: McGraw-Hill.
  10. Wang, Q.; Johnson, J. K.; Broughton, J. Q. (1997). "Path integral grand canonical Monte Carlo". The Journal of Chemical Physics. 107 (13): 5108–5117. Bibcode:1997JChPh.107.5108W. doi:10.1063/1.474874.
  11. Freeman, David L; Doll, J. D (1994). "Fourier path integral Monte Carlo method for the calculation of the microcanonical density of states". The Journal of Chemical Physics. 101 (1): 848. arXiv:chem-ph/9403001. Bibcode:1994JChPh.101..848F. CiteSeerX 10.1.1.342.765. doi:10.1063/1.468087. S2CID 15896126.
  12. Shumway, J.; Ceperley, D.M. (2000). "Path integral Monte Carlo simulations for fermion systems : Pairing in the electron-hole plasma". J. Phys. IV France. 10: 3–16. arXiv:cond-mat/9909434. doi:10.1051/jp4:2000501. S2CID 14845299. Retrieved May 13, 2022.
  13. Dornheim, Tobias (2020). "Path-integral Monte Carlo simulations of quantum dipole systems in traps: Superfluidity, quantum statistics, and structural properties". Physical Review A. 102 (2): 023307. arXiv:2005.03881. Bibcode:2020PhRvA.102b3307D. doi:10.1103/PhysRevA.102.023307. S2CID 218570984. Retrieved May 13, 2022.
  14. Ceperley, D. M. (1995). "Path integrals in the theory of condensed helium". Reviews of Modern Physics. 67 (2): 279–355. Bibcode:1995RvMP...67..279C. doi:10.1103/RevModPhys.67.279.
  15. Noya, Eva G.; Sese, Luis M.; Ramierez, Rafael; McBride, Carl; Conde, Maria M.; Vega, Carlos (2011). "Path integral Monte Carlo simulations for rigid rotors and their application to water". Molecular Physics. 109 (1): 149–168. arXiv:1012.2310. Bibcode:2011MolPh.109..149N. doi:10.1080/00268976.2010.528202. S2CID 44166408. Retrieved May 12, 2022.
  16. Wallqvist, A; Thirumalai, D.; Berne, B.J. (1987). "Path integral Monte Carlo study of the hydrated electron". Journal of Chemical Physics. 86 (11): 6404. Bibcode:1987JChPh..86.6404W. doi:10.1063/1.452429. Retrieved May 12, 2022.
  17. Capuozzo, Pietro; Panella, Emanuele; Gherardini, Tancredi Schettini; Vvedensky, Dmitri D. (2021). "Path integral Monte Carlo method for option pricing". Physica A: Statistical Mechanics and Its Applications. 581: 126231. Bibcode:2021PhyA..58126231C. doi:10.1016/j.physa.2021.126231. Retrieved May 13, 2022.


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