On the simulation of indistinguishable fermions in the manybody Wigner formalism
Abstract
The simulation of quantum systems consisting of interacting, indistinguishable fermions is an incredible mathematical problem which poses formidable numerical challenges. Many sophisticated methods addressing this problem are available which are based on the manybody Schrödinger formalism. Recently a Monte Carlo technique for the resolution of the manybody Wigner equation has been introduced and successfully applied to the simulation of distinguishable, spinless particles. This numerical approach presents several advantages over other methods. Indeed, it is based on an intuitive formalism in which quantum systems are described in terms of a quasidistribution function, and highly scalable due to its Monte Carlo nature. In this work, we extend the manybody Wigner Monte Carlo method to the simulation of indistinguishable fermions. To this end, we first show how fermions are incorporated into the Wigner formalism. Then we demonstrate that the Pauli exclusion principle is intrinsic to the formalism. As a matter of fact, a numerical simulation of two strongly interacting fermions (electrons) is performed which clearly shows the appearance of a Fermi (or exchange–correlation) hole in the phasespace, a clear signature of the presence of the Pauli principle. To conclude, we simulate 4, 8 and 16 noninteracting fermions, isolated in a closed box, andmore »
 Authors:
 Publication Date:
 OSTI Identifier:
 22382159
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 280; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPUTERIZED SIMULATION; CORRELATIONS; DISTRIBUTION FUNCTIONS; ELECTRONS; MANYBODY PROBLEM; MONTE CARLO METHOD; PAULI PRINCIPLE; PHASE SPACE; QUANTUM MECHANICS; QUANTUM SYSTEMS; SCHROEDINGER EQUATION; STATISTICS
Citation Formats
Sellier, J.M., Email: jeanmichel.sellier@gmail.com, and Dimov, I. On the simulation of indistinguishable fermions in the manybody Wigner formalism. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.09.026.
Sellier, J.M., Email: jeanmichel.sellier@gmail.com, & Dimov, I. On the simulation of indistinguishable fermions in the manybody Wigner formalism. United States. doi:10.1016/J.JCP.2014.09.026.
Sellier, J.M., Email: jeanmichel.sellier@gmail.com, and Dimov, I. 2015.
"On the simulation of indistinguishable fermions in the manybody Wigner formalism". United States.
doi:10.1016/J.JCP.2014.09.026.
@article{osti_22382159,
title = {On the simulation of indistinguishable fermions in the manybody Wigner formalism},
author = {Sellier, J.M., Email: jeanmichel.sellier@gmail.com and Dimov, I.},
abstractNote = {The simulation of quantum systems consisting of interacting, indistinguishable fermions is an incredible mathematical problem which poses formidable numerical challenges. Many sophisticated methods addressing this problem are available which are based on the manybody Schrödinger formalism. Recently a Monte Carlo technique for the resolution of the manybody Wigner equation has been introduced and successfully applied to the simulation of distinguishable, spinless particles. This numerical approach presents several advantages over other methods. Indeed, it is based on an intuitive formalism in which quantum systems are described in terms of a quasidistribution function, and highly scalable due to its Monte Carlo nature. In this work, we extend the manybody Wigner Monte Carlo method to the simulation of indistinguishable fermions. To this end, we first show how fermions are incorporated into the Wigner formalism. Then we demonstrate that the Pauli exclusion principle is intrinsic to the formalism. As a matter of fact, a numerical simulation of two strongly interacting fermions (electrons) is performed which clearly shows the appearance of a Fermi (or exchange–correlation) hole in the phasespace, a clear signature of the presence of the Pauli principle. To conclude, we simulate 4, 8 and 16 noninteracting fermions, isolated in a closed box, and show that, as the number of fermions increases, we gradually recover the Fermi–Dirac statistics, a clear proof of the reliability of our proposed method for the treatment of indistinguishable particles.},
doi = {10.1016/J.JCP.2014.09.026},
journal = {Journal of Computational Physics},
number = ,
volume = 280,
place = {United States},
year = 2015,
month = 1
}

We present examples of manybody Wigner quantum systems. The position and the momentum operators {bold R}{sub A} and {bold P}{sub A}, A=1,{hor_ellipsis},n+1, of the particles are noncanonical and are chosen so that the Heisenberg and the Hamiltonian equations are identical. The spectrum of the energy with respect to the center of mass is equidistant and has finite number of energy levels. The composite system is spread in a small volume around the center of mass and within it the geometry is noncommutative. The underlying statistics is an exclusion statistics. {copyright} {ital 1997 American Institute of Physics.}

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