# On the simulation of indistinguishable fermions in the many-body 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 many-body Schrödinger formalism. Recently a Monte Carlo technique for the resolution of the many-body 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 quasi-distribution function, and highly scalable due to its Monte Carlo nature. In this work, we extend the many-body 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 phase-space, a clear signature of the presence of the Pauli principle. To conclude, we simulate 4, 8 and 16 non-interacting fermions, isolated in a closed box, andmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22382159

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- 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); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPUTERIZED SIMULATION; CORRELATIONS; DISTRIBUTION FUNCTIONS; ELECTRONS; MANY-BODY PROBLEM; MONTE CARLO METHOD; PAULI PRINCIPLE; PHASE SPACE; QUANTUM MECHANICS; QUANTUM SYSTEMS; SCHROEDINGER EQUATION; STATISTICS

### Citation Formats

```
Sellier, J.M., E-mail: jeanmichel.sellier@gmail.com, and Dimov, I.
```*On the simulation of indistinguishable fermions in the many-body Wigner formalism*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.09.026.

```
Sellier, J.M., E-mail: jeanmichel.sellier@gmail.com, & Dimov, I.
```*On the simulation of indistinguishable fermions in the many-body Wigner formalism*. United States. doi:10.1016/J.JCP.2014.09.026.

```
Sellier, J.M., E-mail: jeanmichel.sellier@gmail.com, and Dimov, I. Thu .
"On the simulation of indistinguishable fermions in the many-body Wigner formalism". United States. doi:10.1016/J.JCP.2014.09.026.
```

```
@article{osti_22382159,
```

title = {On the simulation of indistinguishable fermions in the many-body Wigner formalism},

author = {Sellier, J.M., E-mail: 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 many-body Schrödinger formalism. Recently a Monte Carlo technique for the resolution of the many-body 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 quasi-distribution function, and highly scalable due to its Monte Carlo nature. In this work, we extend the many-body 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 phase-space, a clear signature of the presence of the Pauli principle. To conclude, we simulate 4, 8 and 16 non-interacting 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},

issn = {0021-9991},

number = ,

volume = 280,

place = {United States},

year = {2015},

month = {1}

}