---
_id: '7514'
abstract:
- lang: eng
text: "We study the interacting homogeneous Bose gas in two spatial dimensions in
the thermodynamic limit at fixed density. We shall be concerned with some mathematical
aspects of this complicated problem in many-body quantum mechanics. More specifically,
we consider the dilute limit where the scattering length of the interaction potential,
which is a measure for the effective range of the potential, is small compared
to the average distance between the particles. We are interested in a setting
with positive (i.e., non-zero) temperature. After giving a survey of the relevant
literature in the field, we provide some facts and examples to set expectations
for the two-dimensional system. The crucial difference to the three-dimensional
system is that there is no Bose–Einstein condensate at positive temperature due
to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic
formula for the free energy holds similarly to the three-dimensional case.\r\nWe
motivate this formula by considering a toy model with δ interaction potential.
By restricting this model Hamiltonian to certain trial states with a quasi-condensate
we obtain an upper bound for the free energy that still has the quasi-condensate
fraction as a free parameter. When minimizing over the quasi-condensate fraction,
we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity,
which plays an important role in our rigorous contribution. The mathematically
rigorous result that we prove concerns the specific free energy in the dilute
limit. We give upper and lower bounds on the free energy in terms of the free
energy of the non-interacting system and a correction term coming from the interaction.
Both bounds match and thus we obtain the leading term of an asymptotic approximation
in the dilute limit, provided the thermal wavelength of the particles is of the
same order (or larger) than the average distance between the particles. The remarkable
feature of this result is its generality: the correction term depends on the interaction
potential only through its scattering length and it holds for all nonnegative
interaction potentials with finite scattering length that are measurable. In particular,
this allows to model an interaction of hard disks."
accept: '1'
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Simon
full_name: Mayer, Simon
id: 30C4630A-F248-11E8-B48F-1D18A9856A87
last_name: Mayer
cc_license: '''https://creativecommons.org/licenses/by/4.0/'''
citation:
ama: Mayer S. *The Free Energy of a Dilute Two-Dimensional Bose Gas*. IST Austria;
2020. doi:10.15479/AT:ISTA:7514
apa: Mayer, S. (2020). *The free energy of a dilute two-dimensional Bose gas*.
IST Austria. https://doi.org/10.15479/AT:ISTA:7514
chicago: Mayer, Simon. *The Free Energy of a Dilute Two-Dimensional Bose Gas*.
IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:7514.
ieee: S. Mayer, *The free energy of a dilute two-dimensional Bose gas*. IST
Austria, 2020.
ista: Mayer S. 2020. The free energy of a dilute two-dimensional Bose gas, IST Austria,
148p.
mla: Mayer, Simon. *The Free Energy of a Dilute Two-Dimensional Bose Gas*.
IST Austria, 2020, doi:10.15479/AT:ISTA:7514.
short: S. Mayer, The Free Energy of a Dilute Two-Dimensional Bose Gas, IST Austria,
2020.
date_created: 2020-02-24T09:17:27Z
date_published: 2020-02-24T00:00:00Z
date_updated: 2020-02-26T09:11:39Z
day: '24'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.15479/AT:ISTA:7514
file:
- access_level: open_access
content_type: application/pdf
creator: dernst
date_created: 2020-02-24T09:15:06Z
date_updated: 2020-02-24T09:15:06Z
file_id: '7515'
file_name: thesis.pdf
file_size: 1563429
open_access: 1
relation: main_file
success: 1
- access_level: closed
content_type: application/x-zip-compressed
creator: dernst
date_created: 2020-02-24T09:15:16Z
date_updated: 2020-02-24T09:15:16Z
file_id: '7516'
file_name: thesis_source.zip
file_size: 2028038
open_access: 0
relation: source_file
request_a_copy: 0
file_date_updated: 2020-02-24T09:15:16Z
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: '148'
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '7524'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
title: The free energy of a dilute two-dimensional Bose gas
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '7508'
abstract:
- lang: eng
text: In this paper, we introduce a novel method for deriving higher order corrections
to the mean-field description of the dynamics of interacting bosons. More precisely,
we consider the dynamics of N d-dimensional bosons for large N. The bosons initially
form a Bose–Einstein condensate and interact with each other via a pair potential
of the form (N−1)−1Ndβv(Nβ·)forβ∈[0,14d). We derive a sequence of N-body functions
which approximate the true many-body dynamics in L2(RdN)-norm to arbitrary precision
in powers of N−1. The approximating functions are constructed as Duhamel expansions
of finite order in terms of the first quantised analogue of a Bogoliubov time
evolution.
accept: '1'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Lea
full_name: Bossmann, Lea
id: A2E3BCBE-5FCC-11E9-AA4B-76F3E5697425
last_name: Bossmann
orcid: 0000-0002-6854-1343
- first_name: Nataša
full_name: Pavlović, Nataša
last_name: Pavlović
- first_name: Peter
full_name: Pickl, Peter
last_name: Pickl
- first_name: Avy
full_name: Soffer, Avy
last_name: Soffer
cc_license: '''https://creativecommons.org/licenses/by/4.0/'''
citation:
ama: Bossmann L, Pavlović N, Pickl P, Soffer A. Higher order corrections to the
mean-field description of the dynamics of interacting bosons. *Journal of Statistical
Physics*. 2020. doi:10.1007/s10955-020-02500-8
apa: Bossmann, L., Pavlović, N., Pickl, P., & Soffer, A. (2020). Higher order
corrections to the mean-field description of the dynamics of interacting bosons.
*Journal of Statistical Physics*. https://doi.org/10.1007/s10955-020-02500-8
chicago: Bossmann, Lea, Nataša Pavlović, Peter Pickl, and Avy Soffer. “Higher Order
Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons.”
*Journal of Statistical Physics*, 2020. https://doi.org/10.1007/s10955-020-02500-8.
ieee: L. Bossmann, N. Pavlović, P. Pickl, and A. Soffer, “Higher order corrections
to the mean-field description of the dynamics of interacting bosons,” *Journal
of Statistical Physics*, 2020.
ista: Bossmann L, Pavlović N, Pickl P, Soffer A. 2020. Higher order corrections
to the mean-field description of the dynamics of interacting bosons. Journal of
Statistical Physics.
mla: Bossmann, Lea, et al. “Higher Order Corrections to the Mean-Field Description
of the Dynamics of Interacting Bosons.” *Journal of Statistical Physics*,
Springer Nature, 2020, doi:10.1007/s10955-020-02500-8.
short: L. Bossmann, N. Pavlović, P. Pickl, A. Soffer, Journal of Statistical Physics
(2020).
date_created: 2020-02-23T09:45:51Z
date_published: 2020-02-21T00:00:00Z
date_updated: 2020-02-27T12:38:44Z
day: '21'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1007/s10955-020-02500-8
external_id:
arxiv:
- '1905.06164'
file:
- access_level: open_access
content_type: application/pdf
creator: dernst
date_created: 2020-02-24T14:17:24Z
date_updated: 2020-02-24T14:17:24Z
file_id: '7518'
file_name: 2020_JournStatisticPhysics_Bossmann.pdf
file_size: 587276
open_access: 1
relation: main_file
success: 1
file_date_updated: 2020-02-24T14:17:24Z
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Journal of Statistical Physics
publication_identifier:
eissn:
- 1572-9613
issn:
- 0022-4715
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Higher order corrections to the mean-field description of the dynamics of interacting
bosons
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '7611'
abstract:
- lang: eng
text: We consider a system of N bosons in the limit N→∞, interacting through singular
potentials. For initial data exhibiting Bose–Einstein condensation, the many-body
time evolution is well approximated through a quadratic fluctuation dynamics around
a cubic nonlinear Schrödinger equation of the condensate wave function. We show
that these fluctuations satisfy a (multi-variate) central limit theorem.
article_processing_charge: No
article_type: original
author:
- first_name: Simone Anna Elvira
full_name: Rademacher, Simone Anna Elvira
id: 856966FE-A408-11E9-977E-802DE6697425
last_name: Rademacher
citation:
ama: Rademacher SAE. Central limit theorem for Bose gases interacting through singular
potentials. *Letters in Mathematical Physics*. 2020. doi:10.1007/s11005-020-01286-w
apa: Rademacher, S. A. E. (2020). Central limit theorem for Bose gases interacting
through singular potentials. *Letters in Mathematical Physics*. https://doi.org/10.1007/s11005-020-01286-w
chicago: Rademacher, Simone Anna Elvira. “Central Limit Theorem for Bose Gases Interacting
through Singular Potentials.” *Letters in Mathematical Physics*, 2020. https://doi.org/10.1007/s11005-020-01286-w.
ieee: S. A. E. Rademacher, “Central limit theorem for Bose gases interacting through
singular potentials,” *Letters in Mathematical Physics*, 2020.
ista: Rademacher SAE. 2020. Central limit theorem for Bose gases interacting through
singular potentials. Letters in Mathematical Physics.
mla: Rademacher, Simone Anna Elvira. “Central Limit Theorem for Bose Gases Interacting
through Singular Potentials.” *Letters in Mathematical Physics*, Springer
Nature, 2020, doi:10.1007/s11005-020-01286-w.
short: S.A.E. Rademacher, Letters in Mathematical Physics (2020).
date_created: 2020-03-23T11:11:47Z
date_published: 2020-03-12T00:00:00Z
date_updated: 2020-03-23T14:08:57Z
day: '12'
department:
- _id: RoSe
doi: 10.1007/s11005-020-01286-w
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s11005-020-01286-w
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Letters in Mathematical Physics
publication_identifier:
issn:
- 0377-9017
- 1573-0530
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Central limit theorem for Bose gases interacting through singular potentials
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '80'
abstract:
- lang: eng
text: 'We consider an interacting, dilute Bose gas trapped in a harmonic potential
at a positive temperature. The system is analyzed in a combination of a thermodynamic
and a Gross–Pitaevskii (GP) limit where the trap frequency ω, the temperature
T, and the particle number N are related by N∼ (T/ ω) 3→ ∞ while the scattering
length is so small that the interaction energy per particle around the center
of the trap is of the same order of magnitude as the spectral gap in the trap.
We prove that the difference between the canonical free energy of the interacting
gas and the one of the noninteracting system can be obtained by minimizing the
GP energy functional. We also prove Bose–Einstein condensation in the following
sense: The one-particle density matrix of any approximate minimizer of the canonical
free energy functional is to leading order given by that of the noninteracting
gas but with the free condensate wavefunction replaced by the GP minimizer.'
accept: '1'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Andreas
full_name: Deuchert, Andreas
id: 4DA65CD0-F248-11E8-B48F-1D18A9856A87
last_name: Deuchert
orcid: 0000-0003-3146-6746
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
- first_name: Jakob
full_name: Yngvason, Jakob
last_name: Yngvason
cc_license: '''https://creativecommons.org/licenses/by/4.0/'''
citation:
ama: Deuchert A, Seiringer R, Yngvason J. Bose–Einstein condensation in a dilute,
trapped gas at positive temperature. *Communications in Mathematical Physics*.
2019;368(2):723-776. doi:10.1007/s00220-018-3239-0
apa: Deuchert, A., Seiringer, R., & Yngvason, J. (2019). Bose–Einstein condensation
in a dilute, trapped gas at positive temperature. *Communications in Mathematical
Physics*, *368*(2), 723–776. https://doi.org/10.1007/s00220-018-3239-0
chicago: 'Deuchert, Andreas, Robert Seiringer, and Jakob Yngvason. “Bose–Einstein
Condensation in a Dilute, Trapped Gas at Positive Temperature.” *Communications
in Mathematical Physics* 368, no. 2 (2019): 723–76. https://doi.org/10.1007/s00220-018-3239-0.'
ieee: A. Deuchert, R. Seiringer, and J. Yngvason, “Bose–Einstein condensation in
a dilute, trapped gas at positive temperature,” *Communications in Mathematical
Physics*, vol. 368, no. 2, pp. 723–776, 2019.
ista: Deuchert A, Seiringer R, Yngvason J. 2019. Bose–Einstein condensation in a
dilute, trapped gas at positive temperature. Communications in Mathematical Physics.
368(2), 723–776.
mla: Deuchert, Andreas, et al. “Bose–Einstein Condensation in a Dilute, Trapped
Gas at Positive Temperature.” *Communications in Mathematical Physics*, vol.
368, no. 2, Springer, 2019, pp. 723–76, doi:10.1007/s00220-018-3239-0.
short: A. Deuchert, R. Seiringer, J. Yngvason, Communications in Mathematical Physics
368 (2019) 723–776.
date_created: 2018-12-11T11:44:31Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2020-01-21T13:22:16Z
day: '01'
ddc:
- '530'
department:
- _id: RoSe
doi: 10.1007/s00220-018-3239-0
file:
- access_level: open_access
content_type: application/pdf
creator: dernst
date_created: 2018-12-17T10:34:06Z
date_updated: 2018-12-17T10:34:06Z
file_id: '5688'
file_name: 2018_CommunMathPhys_Deuchert.pdf
file_size: 893902
open_access: 1
relation: main_file
success: 1
file_date_updated: 2018-12-17T10:34:06Z
intvolume: ' 368'
issue: '2'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 723-776
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
- _id: 25C878CE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P27533_N27
name: Structure of the Excitation Spectrum for Many-Body Quantum Systems
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '7974'
quality_controlled: '1'
status: public
title: Bose–Einstein condensation in a dilute, trapped gas at positive temperature
type: journal_article
user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425
volume: 368
year: '2019'
...
---
_id: '6840'
abstract:
- lang: eng
text: "We discuss thermodynamic properties of harmonically trapped\r\nimperfect
quantum gases. The spatial inhomogeneity of these systems imposes\r\na redefinition
of the mean-field interparticle potential energy as compared\r\nto the homogeneous
case. In our approach, it takes the form a\r\n2N2 ωd, where\r\nN is the number
of particles, ω—the harmonic trap frequency, d—system’s\r\ndimensionality, and
a is a parameter characterizing the interparticle interaction.\r\nWe provide arguments
that this model corresponds to the limiting case of\r\na long-ranged interparticle
potential of vanishingly small amplitude. This\r\nconclusion is drawn from a computation
similar to the well-known Kac scaling\r\nprocedure, which is presented here in
a form adapted to the case of an isotropic\r\nharmonic trap. We show that within
the model, the imperfect gas of trapped\r\nrepulsive bosons undergoes the Bose–Einstein
condensation provided d > 1.\r\nThe main result of our analysis is that in d =
1 the gas of attractive imperfect\r\nfermions with a = −aF < 0 is thermodynamically
equivalent to the gas of\r\nrepulsive bosons with a = aB > 0 provided the parameters
aF and aB fulfill\r\nthe relation aB + aF = \x1F. This result supplements similar
recent conclusion\r\nabout thermodynamic equivalence of two-dimensional (2D) uniform
imperfect\r\nrepulsive Bose and attractive Fermi gases."
article_number: '063101'
author:
- first_name: Krzysztof
full_name: Mysliwy, Krzysztof
id: 316457FC-F248-11E8-B48F-1D18A9856A87
last_name: Mysliwy
- first_name: Marek
full_name: Napiórkowski, Marek
last_name: Napiórkowski
citation:
ama: 'Mysliwy K, Napiórkowski M. Thermodynamics of inhomogeneous imperfect quantum
gases in harmonic traps. *Journal of Statistical Mechanics: Theory and Experiment*.
2019;2019(6). doi:10.1088/1742-5468/ab190d'
apa: 'Mysliwy, K., & Napiórkowski, M. (2019). Thermodynamics of inhomogeneous
imperfect quantum gases in harmonic traps. *Journal of Statistical Mechanics:
Theory and Experiment*, *2019*(6). https://doi.org/10.1088/1742-5468/ab190d'
chicago: 'Mysliwy, Krzysztof, and Marek Napiórkowski. “Thermodynamics of Inhomogeneous
Imperfect Quantum Gases in Harmonic Traps.” *Journal of Statistical Mechanics:
Theory and Experiment* 2019, no. 6 (2019). https://doi.org/10.1088/1742-5468/ab190d.'
ieee: 'K. Mysliwy and M. Napiórkowski, “Thermodynamics of inhomogeneous imperfect
quantum gases in harmonic traps,” *Journal of Statistical Mechanics: Theory
and Experiment*, vol. 2019, no. 6, 2019.'
ista: 'Mysliwy K, Napiórkowski M. 2019. Thermodynamics of inhomogeneous imperfect
quantum gases in harmonic traps. Journal of Statistical Mechanics: Theory and
Experiment. 2019(6), 063101.'
mla: 'Mysliwy, Krzysztof, and Marek Napiórkowski. “Thermodynamics of Inhomogeneous
Imperfect Quantum Gases in Harmonic Traps.” *Journal of Statistical Mechanics:
Theory and Experiment*, vol. 2019, no. 6, 063101, IOP Publishing, 2019, doi:10.1088/1742-5468/ab190d.'
short: 'K. Mysliwy, M. Napiórkowski, Journal of Statistical Mechanics: Theory and
Experiment 2019 (2019).'
date_created: 2019-09-01T22:00:59Z
date_published: 2019-06-13T00:00:00Z
date_updated: 2020-01-21T12:04:50Z
day: '13'
department:
- _id: RoSe
doi: 10.1088/1742-5468/ab190d
external_id:
arxiv:
- '1810.02209'
intvolume: ' 2019'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1810.02209
month: '06'
oa: 1
oa_version: Preprint
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication: 'Journal of Statistical Mechanics: Theory and Experiment'
publication_identifier:
eissn:
- 1742-5468
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2019
year: '2019'
...