path integral Monte Carlo

{{Short description|Quantum algorithm to calculate path integrals in quantum multi-body problems}}

{{Use American English|date=January 2019}}

{{Use mdy dates|date=January 2019}}

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.{{Cite journal | last1 = Barker | first1 = J. A. | title = A quantum-statistical Monte Carlo method; path integrals with boundary conditions | doi = 10.1063/1.437829 | journal = The Journal of Chemical Physics | volume = 70 | issue = 6 | pages = 2914–2918 | year = 1979 |bibcode = 1979JChPh..70.2914B }}{{cite journal |last1=Cazorla |first1=Claudio |last2=Boronat |first2=Jordi |title=Simulation and understanding of atomic and molecular quantum crystals |journal=Reviews of Modern Physics |date=2017 |volume=89 |issue=3 |page=035003 |doi=10.1103/RevModPhys.89.035003 |arxiv=1605.05820 |bibcode=2017RvMP...89c5003C |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.89.035003 |access-date=13 May 2022}}

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{{cite journal |last1=Topper |first1=Robert Q. |title=Adaptive path-integral Monte Carlo methods for accurate computation of molecular thermodynamic properties |journal=Advances in Chemical Physics |date=1999 |volume=105 |pages=117–170 |url=https://www.wiley.com/en-gb/Monte+Carlo+Methods+in+Chemical+Physics,+Volume+105-p-9780470141649 |access-date=12 May 2022}} such as the internal energy,{{cite journal|doi=10.1063/1.1460861|title=An improved thermodynamic energy estimator for path integral simulations|year=2002|last1=Glaesemann|first1=Kurt R.|last2=Fried|first2=Laurence E.|journal=The Journal of Chemical Physics|volume=116|issue=14|pages=5951–5955|bibcode = 2002JChPh.116.5951G |url=https://zenodo.org/record/1231880}} heat capacity,{{cite journal|doi=10.1063/1.1493184|title=Improved heat capacity estimator for path integral simulations|year=2002|last1=Glaesemann|first1=Kurt R.|last2=Fried|first2=Laurence E.|journal=The Journal of Chemical Physics|volume=117|issue=7|pages=3020–3026|bibcode = 2002JChPh.117.3020G }} or free energy.{{cite journal | doi = 10.1063/1.1529682 | title = A path integral approach to molecular thermochemistry | year = 2003 | last1 = Glaesemann | first1 = Kurt R. | last2 = Fried | first2 = Laurence E. | journal = The Journal of Chemical Physics | volume = 118 |issue=4| pages = 1596–1602|bibcode = 2003JChPh.118.1596G | url = https://zenodo.org/record/1231864 }}{{cite journal|doi=10.1063/1.1954771|title=Quantitative molecular thermochemistry based on path integrals|year=2005|last1=Glaesemann|first1=Kurt R.|last2=Fried|first2=Laurence E.|journal=The Journal of Chemical Physics|volume=123|issue=3|pages=034103|pmid= 16080726 |bibcode = 2005JChPh.123c4103G |url=https://digital.library.unt.edu/ark:/67531/metadc793257/|type=Submitted manuscript|doi-access=free}} 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),{{cite journal |last1=Doll |first1=J.D. |title=Monte Carlo Fourier path integral methods in chemical dynamics |journal=Journal of Chemical Physics |date=1998 |volume=81 |issue=8 |page=3536 |doi=10.1063/1.448081 |url=https://aip.scitation.org/doi/10.1063/1.448081 |access-date=13 May 2022|url-access=subscription }} 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. 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).

The basic framework was originally formulated within the canonical ensemble,{{cite book |last1=Feynman |first1=Richard P. |last2=Hibbs |first2=Albert R. |title=Quantum Mechanics and Path Integrals |date=1965 |publisher=McGraw-Hill |location=New York}} but has since been extended to include the grand canonical ensemble{{Cite journal | last1 = Wang | first1 = Q. | last2 = Johnson | first2 = J. K. | last3 = Broughton | first3 = J. Q. | doi = 10.1063/1.474874 | title = Path integral grand canonical Monte Carlo | journal = The Journal of Chemical Physics | volume = 107 | issue = 13 | pages = 5108–5117| year = 1997 |bibcode = 1997JChPh.107.5108W }} and the microcanonical ensemble.{{Cite journal|doi=10.1063/1.468087|title=Fourier path integral Monte Carlo method for the calculation of the microcanonical density of states|journal=The Journal of Chemical Physics|volume=101|issue=1|pages=848|year=1994|last1=Freeman|first1=David L|last2=Doll|first2=J. D|arxiv=chem-ph/9403001|bibcode=1994JChPh.101..848F|citeseerx=10.1.1.342.765|s2cid=15896126}} Its use has been extended to fermion systems{{cite journal |last1=Shumway |first1=J. |last2=Ceperley |first2=D.M. |title=Path integral Monte Carlo simulations for fermion systems : Pairing in the electron-hole plasma |journal=J. Phys. IV France |date=2000 |volume=10 |pages=3–16 |doi=10.1051/jp4:2000501 |arxiv=cond-mat/9909434 |s2cid=14845299 |url=https://jp4.journaldephysique.org/articles/jp4/abs/2000/05/jp4200010PR501/jp4200010PR501.html |access-date=13 May 2022}} as well as systems of bosons.{{cite journal |last1=Dornheim |first1=Tobias |title=Path-integral Monte Carlo simulations of quantum dipole systems in traps: Superfluidity, quantum statistics, and structural properties |journal=Physical Review A |date=2020 |volume=102 |issue=2 |page=023307 |doi=10.1103/PhysRevA.102.023307 |arxiv=2005.03881 |bibcode=2020PhRvA.102b3307D |s2cid=218570984 |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.023307 |access-date=13 May 2022}}

An early application was to the study of liquid helium.{{Cite journal | last1 = Ceperley | first1 = D. M. | title = Path integrals in the theory of condensed helium | doi = 10.1103/RevModPhys.67.279 | journal = Reviews of Modern Physics | volume = 67 | issue = 2 | pages = 279–355| year = 1995 |bibcode = 1995RvMP...67..279C }} Numerous applications have been made to other systems, including liquid water{{cite journal |last1=Noya |first1=Eva G. |last2=Sese |first2=Luis M. |last3=Ramierez |first3=Rafael |last4=McBride |first4=Carl |last5=Conde |first5=Maria M. |last6=Vega |first6=Carlos |title=Path integral Monte Carlo simulations for rigid rotors and their application to water |journal=Molecular Physics |date=2011 |volume=109 |issue=1 |pages=149–168 |doi=10.1080/00268976.2010.528202 |arxiv=1012.2310 |bibcode=2011MolPh.109..149N |s2cid=44166408 |url=https://www.tandfonline.com/doi/abs/10.1080/00268976.2010.528202 |access-date=12 May 2022}} and the hydrated electron.{{cite journal |last1=Wallqvist |first1=A |last2=Thirumalai |first2=D. |last3=Berne |first3=B.J. |title=Path integral Monte Carlo study of the hydrated electron |journal=Journal of Chemical Physics |date=1987 |volume=86 |issue=11 |page=6404 |doi=10.1063/1.452429 |bibcode=1987JChPh..86.6404W |url=https://aip.scitation.org/doi/10.1063/1.452429 |access-date=12 May 2022|url-access=subscription }} The algorithms and formalism have also been mapped onto non-quantum mechanical problems in the field of financial modeling, including option pricing.{{cite journal |last1=Capuozzo |first1=Pietro |last2=Panella |first2=Emanuele |last3=Gherardini |first3=Tancredi Schettini |last4=Vvedensky |first4=Dmitri D. |title=Path integral Monte Carlo method for option pricing |journal=Physica A: Statistical Mechanics and Its Applications |date=2021 |volume=581 |page=126231 |doi=10.1016/j.physa.2021.126231 |bibcode=2021PhyA..58126231C |url=https://www.sciencedirect.com/science/article/abs/pii/S0378437121005045 |access-date=13 May 2022|url-access=subscription }}