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# Counting statistics in finite Fermi systems: Illustrations with the atomic nucleus

## Abstract

Here, we analyze here in details the probability to find a given number of particles in a finite volume inside a normal or super-fluid finite system. This probability, also known as counting statistics, is obtained using projection operator techniques directly linked to the characteristic function of the probability distribution. The method is illustrated in atomic nuclei. The nature of the particle number fluctuations from small to large volumes compared to the system size is carefully analyzed in three cases: normal systems, super-fluid systems and super fluid systems with total number restoration. The transition from Poissonian distribution in the small volume limit to Gaussian fluctuations as the number of particles participating to the fluctuations increases is analyzed both in the interior and at the surface of the system. Lastly, while the restoration of the total number of particles is not necessary for small volume, we show that it affects the counting statistics as soon as more than a very few particles are involved.

- Authors:

- Univ. Paris-Saclay, Orsay Cedex (France)
- Tennessee Technological Univ., Cookeville, TN (United States)

- Publication Date:

- Research Org.:
- Tennessee Technological Univ., Cookeville, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1593774

- Grant/Contract Number:
- SC0015513

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review C

- Additional Journal Information:
- Journal Volume: 101; Journal Issue: 1; Journal ID: ISSN 2469-9985

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Nuclear density functional theory; Nucleon distribution

### Citation Formats

```
Lacroix, Denis, and Ayik, Sakir. Counting statistics in finite Fermi systems: Illustrations with the atomic nucleus. United States: N. p., 2020.
Web. doi:10.1103/PhysRevC.101.014310.
```

```
Lacroix, Denis, & Ayik, Sakir. Counting statistics in finite Fermi systems: Illustrations with the atomic nucleus. United States. doi:10.1103/PhysRevC.101.014310.
```

```
Lacroix, Denis, and Ayik, Sakir. Mon .
"Counting statistics in finite Fermi systems: Illustrations with the atomic nucleus". United States. doi:10.1103/PhysRevC.101.014310.
```

```
@article{osti_1593774,
```

title = {Counting statistics in finite Fermi systems: Illustrations with the atomic nucleus},

author = {Lacroix, Denis and Ayik, Sakir},

abstractNote = {Here, we analyze here in details the probability to find a given number of particles in a finite volume inside a normal or super-fluid finite system. This probability, also known as counting statistics, is obtained using projection operator techniques directly linked to the characteristic function of the probability distribution. The method is illustrated in atomic nuclei. The nature of the particle number fluctuations from small to large volumes compared to the system size is carefully analyzed in three cases: normal systems, super-fluid systems and super fluid systems with total number restoration. The transition from Poissonian distribution in the small volume limit to Gaussian fluctuations as the number of particles participating to the fluctuations increases is analyzed both in the interior and at the surface of the system. Lastly, while the restoration of the total number of particles is not necessary for small volume, we show that it affects the counting statistics as soon as more than a very few particles are involved.},

doi = {10.1103/PhysRevC.101.014310},

journal = {Physical Review C},

number = 1,

volume = 101,

place = {United States},

year = {2020},

month = {1}

}

Works referenced in this record:

##
Gauge angle dependence in time-dependent Hartree-Fock-Bogoliubov calculations of ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ head-on collisions with the Gogny interaction

journal, July 2016

- Hashimoto, Yukio; Scamps, Guillaume
- Physical Review C, Vol. 94, Issue 1

##
Microscopic description of pair transfer between two superfluid Fermi systems: Combining phase-space averaging and combinatorial techniques

journal, March 2018

- Regnier, David; Lacroix, Denis; Scamps, Guillaume
- Physical Review C, Vol. 97, Issue 3

##
Projection of good quantum numbers for reaction fragments

journal, September 2019

- Bulgac, Aurel
- Physical Review C, Vol. 100, Issue 3

##
The canonical form of an antisymmetric tensor and its application to the theory of superconductivity

journal, December 1962

- Bloch, Claude; Messiah, Albert
- Nuclear Physics, Vol. 39

##
Particle-number restoration within the energy density functional formalism: Nonviability of terms depending on noninteger powers of the density matrices

journal, April 2009

- Duguet, T.; Bender, M.; Bennaceur, K.
- Physical Review C, Vol. 79, Issue 4

##
Sign of the overlap of Hartree-Fock-Bogoliubov wave functions

journal, February 2009

- Robledo, L. M.
- Physical Review C, Vol. 79, Issue 2

##
Density correlations in ultracold atomic Fermi gases

journal, June 2007

- Belzig, W.; Schroll, C.; Bruder, C.
- Physical Review A, Vol. 75, Issue 6

##
Short range correlations and the isospin dependence of nuclear correlation functions

journal, October 2018

- Cruz-Torres, R.; Schmidt, A.; Miller, G. A.
- Physics Letters B, Vol. 785

##
Short-range correlations and the charge density

journal, March 2019

- Weiss, Ronen; Schmidt, Axel; Miller, Gerald A.
- Physics Letters B, Vol. 790

##
Particle Transfer Reactions with the Time-Dependent Hartree-Fock Theory Using a Particle Number Projection Technique

journal, November 2010

- Simenel, Cédric
- Physical Review Letters, Vol. 105, Issue 19

##
Time-dependent Hartree-Fock calculations for multinucleon transfer and quasifission processes in the ${}^{64}\text{Ni}+{}^{238}\text{U}$ reaction

journal, May 2016

- Sekizawa, Kazuyuki; Yabana, Kazuhiro
- Physical Review C, Vol. 93, Issue 5

##
Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices

journal, October 2011

- González-Ballestero, C.; Robledo, L. M.; Bertsch, G. F.
- Computer Physics Communications, Vol. 182, Issue 10

##
Microscopic description of pair transfer between two superfluid Fermi systems. II. Quantum mixing of time-dependent Hartree-Fock-Bogolyubov trajectories

journal, June 2019

- Regnier, David; Lacroix, Denis
- Physical Review C, Vol. 99, Issue 6

##
Considerations on the Flow of Superfluid Helium

journal, April 1966

- Anderson, P. W.
- Reviews of Modern Physics, Vol. 38, Issue 2

##
Particle-number projection method in time-dependent Hartree-Fock theory: Properties of reaction products

journal, December 2014

- Sekizawa, Kazuyuki; Yabana, Kazuhiro
- Physical Review C, Vol. 90, Issue 6

##
Projection in the occupation-number space and the canonical transformation

journal, January 1970

- Fomenko, V. N.
- Journal of Physics A: General Physics, Vol. 3, Issue 1

##
Number of particles in fission fragments

journal, August 2019

- Verriere, Marc; Schunck, Nicolas; Kawano, Toshihiko
- Physical Review C, Vol. 100, Issue 2

##
Self-consistent mean-field models for nuclear structure

journal, January 2003

- Bender, Michael; Heenen, Paul-Henri; Reinhard, Paul-Gerhard
- Reviews of Modern Physics, Vol. 75, Issue 1

##
Probing Pair-Correlated Fermionic Atoms through Correlations in Atom Shot Noise

journal, March 2005

- Greiner, M.; Regal, C. A.; Stewart, J. T.
- Physical Review Letters, Vol. 94, Issue 11

##
Effect of pairing on one- and two-nucleon transfer below the Coulomb barrier: A time-dependent microscopic description

journal, January 2013

- Scamps, Guillaume; Lacroix, Denis
- Physical Review C, Vol. 87, Issue 1

##
Symmetry Restoration in Hartree-Fock-Bogoliubov Based Theories

journal, January 2012

- Bertsch, G. F.; Robledo, L. M.
- Physical Review Letters, Vol. 108, Issue 4

##
Generalized virial theorem and pressure relation for a strongly correlated Fermi gas

journal, December 2008

- Tan, Shina
- Annals of Physics, Vol. 323, Issue 12

##
Suppression of Density Fluctuations in a Quantum Degenerate Fermi Gas

journal, July 2010

- Sanner, Christian; Su, Edward J.; Keshet, Aviv
- Physical Review Letters, Vol. 105, Issue 4

##
Local Observation of Antibunching in a Trapped Fermi Gas

journal, July 2010

- Müller, Torben; Zimmermann, Bruno; Meineke, Jakob
- Physical Review Letters, Vol. 105, Issue 4

##
Large momentum part of a strongly correlated Fermi gas

journal, December 2008

- Tan, Shina
- Annals of Physics, Vol. 323, Issue 12

##
Novel Role of Superfluidity in Low-Energy Nuclear Reactions

journal, July 2017

- Magierski, Piotr; Sekizawa, Kazuyuki; Wlazłowski, Gabriel
- Physical Review Letters, Vol. 119, Issue 4

##
Energetics of a strongly correlated Fermi gas

journal, December 2008

- Tan, Shina
- Annals of Physics, Vol. 323, Issue 12

##
Fluctuations of the number of particles within a given volume in cold quantum gases

journal, December 2007

- Astrakharchik, G. E.; Combescot, R.; Pitaevskii, L. P.
- Physical Review A, Vol. 76, Issue 6

##
Normal Forms of Complex Matrices

journal, September 1962

- Zumino, Bruno
- Journal of Mathematical Physics, Vol. 3, Issue 5

##
Examining empirical evidence of the effect of superfluidity on the fusion barrier

journal, April 2018

- Scamps, Guillaume
- Physical Review C, Vol. 97, Issue 4

##
Local atom-number fluctuations in quantum gases at finite temperature

journal, September 2011

- Klawunn, M.; Recati, A.; Pitaevskii, L. P.
- Physical Review A, Vol. 84, Issue 3

##
Particle-number restoration within the energy density functional formalism

journal, April 2009

- Bender, M.; Duguet, T.; Lacroix, D.
- Physical Review C, Vol. 79, Issue 4

##
Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels

journal, April 2009

- Lacroix, D.; Duguet, T.; Bender, M.
- Physical Review C, Vol. 79, Issue 4