--- _id: '7790' abstract: - lang: eng text: "We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density \U0001D70C and inverse temperature \U0001D6FD differs from the one of the noninteracting system by the correction term \U0001D70B\U0001D70C\U0001D70C\U0001D6FD\U0001D6FD . Here, is the scattering length of the interaction potential, and \U0001D6FD is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit \U0001D70C and if \U0001D6FD\U0001D70C ." article_number: e20 article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Andreas full_name: Deuchert, Andreas id: 4DA65CD0-F248-11E8-B48F-1D18A9856A87 last_name: Deuchert orcid: 0000-0003-3146-6746 - first_name: Simon full_name: Mayer, Simon id: 30C4630A-F248-11E8-B48F-1D18A9856A87 last_name: Mayer - first_name: Robert full_name: Seiringer, Robert id: 4AFD0470-F248-11E8-B48F-1D18A9856A87 last_name: Seiringer orcid: 0000-0002-6781-0521 citation: ama: Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. Forum of Mathematics, Sigma. 2020;8. doi:10.1017/fms.2020.17 apa: Deuchert, A., Mayer, S., & Seiringer, R. (2020). The free energy of the two-dimensional dilute Bose gas. I. Lower bound. Forum of Mathematics, Sigma. Cambridge University Press. https://doi.org/10.1017/fms.2020.17 chicago: Deuchert, Andreas, Simon Mayer, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” Forum of Mathematics, Sigma. Cambridge University Press, 2020. https://doi.org/10.1017/fms.2020.17. ieee: A. Deuchert, S. Mayer, and R. Seiringer, “The free energy of the two-dimensional dilute Bose gas. I. Lower bound,” Forum of Mathematics, Sigma, vol. 8. Cambridge University Press, 2020. ista: Deuchert A, Mayer S, Seiringer R. 2020. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. Forum of Mathematics, Sigma. 8, e20. mla: Deuchert, Andreas, et al. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” Forum of Mathematics, Sigma, vol. 8, e20, Cambridge University Press, 2020, doi:10.1017/fms.2020.17. short: A. Deuchert, S. Mayer, R. Seiringer, Forum of Mathematics, Sigma 8 (2020). date_created: 2020-05-03T22:00:48Z date_published: 2020-03-14T00:00:00Z date_updated: 2023-08-21T06:18:49Z day: '14' ddc: - '510' department: - _id: RoSe doi: 10.1017/fms.2020.17 ec_funded: 1 external_id: arxiv: - '1910.03372' isi: - '000527342000001' file: - access_level: open_access checksum: 8a64da99d107686997876d7cad8cfe1e content_type: application/pdf creator: dernst date_created: 2020-05-04T12:02:41Z date_updated: 2020-07-14T12:48:03Z file_id: '7797' file_name: 2020_ForumMath_Deuchert.pdf file_size: 692530 relation: main_file file_date_updated: 2020-07-14T12:48:03Z has_accepted_license: '1' intvolume: ' 8' isi: 1 language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '03' oa: 1 oa_version: Published Version project: - _id: 25C6DC12-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '694227' name: Analysis of quantum many-body systems publication: Forum of Mathematics, Sigma publication_identifier: eissn: - '20505094' publication_status: published publisher: Cambridge University Press quality_controlled: '1' related_material: record: - id: '7524' relation: earlier_version status: public scopus_import: '1' status: public title: The free energy of the two-dimensional dilute Bose gas. I. Lower bound tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 8 year: '2020' ... --- _id: '8134' abstract: - lang: eng text: We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the free energy per unit volume differs from the one of the non-interacting system by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]2+) to leading order, where a is the scattering length of the two-body interaction potential, ρ is the density, β is the inverse temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality in the asymptotic expansion. article_number: '061901' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Simon full_name: Mayer, Simon id: 30C4630A-F248-11E8-B48F-1D18A9856A87 last_name: Mayer - first_name: Robert full_name: Seiringer, Robert id: 4AFD0470-F248-11E8-B48F-1D18A9856A87 last_name: Seiringer orcid: 0000-0002-6781-0521 citation: ama: Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. II. Upper bound. Journal of Mathematical Physics. 2020;61(6). doi:10.1063/5.0005950 apa: Mayer, S., & Seiringer, R. (2020). The free energy of the two-dimensional dilute Bose gas. II. Upper bound. Journal of Mathematical Physics. AIP Publishing. https://doi.org/10.1063/5.0005950 chicago: Mayer, Simon, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. II. Upper Bound.” Journal of Mathematical Physics. AIP Publishing, 2020. https://doi.org/10.1063/5.0005950. ieee: S. Mayer and R. Seiringer, “The free energy of the two-dimensional dilute Bose gas. II. Upper bound,” Journal of Mathematical Physics, vol. 61, no. 6. AIP Publishing, 2020. ista: Mayer S, Seiringer R. 2020. The free energy of the two-dimensional dilute Bose gas. II. Upper bound. Journal of Mathematical Physics. 61(6), 061901. mla: Mayer, Simon, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. II. Upper Bound.” Journal of Mathematical Physics, vol. 61, no. 6, 061901, AIP Publishing, 2020, doi:10.1063/5.0005950. short: S. Mayer, R. Seiringer, Journal of Mathematical Physics 61 (2020). date_created: 2020-07-19T22:00:59Z date_published: 2020-06-22T00:00:00Z date_updated: 2023-08-22T08:12:40Z day: '22' department: - _id: RoSe doi: 10.1063/5.0005950 ec_funded: 1 external_id: arxiv: - '2002.08281' isi: - '000544595100001' intvolume: ' 61' isi: 1 issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2002.08281 month: '06' oa: 1 oa_version: Preprint project: - _id: 25C6DC12-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '694227' name: Analysis of quantum many-body systems publication: Journal of Mathematical Physics publication_identifier: issn: - '00222488' publication_status: published publisher: AIP Publishing quality_controlled: '1' scopus_import: '1' status: public title: The free energy of the two-dimensional dilute Bose gas. II. Upper bound type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 61 year: '2020' ... --- _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." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Simon full_name: Mayer, Simon id: 30C4630A-F248-11E8-B48F-1D18A9856A87 last_name: Mayer citation: ama: Mayer S. The free energy of a dilute two-dimensional Bose gas. 2020. doi:10.15479/AT:ISTA:7514 apa: Mayer, S. (2020). The free energy of a dilute two-dimensional Bose gas. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7514 chicago: Mayer, Simon. “The Free Energy of a Dilute Two-Dimensional Bose Gas.” Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7514. ieee: S. Mayer, “The free energy of a dilute two-dimensional Bose gas,” Institute of Science and Technology Austria, 2020. ista: Mayer S. 2020. The free energy of a dilute two-dimensional Bose gas. Institute of Science and Technology Austria. mla: Mayer, Simon. The Free Energy of a Dilute Two-Dimensional Bose Gas. Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7514. short: S. Mayer, The Free Energy of a Dilute Two-Dimensional Bose Gas, Institute of Science and Technology Austria, 2020. date_created: 2020-02-24T09:17:27Z date_published: 2020-02-24T00:00:00Z date_updated: 2023-09-07T13:12:42Z day: '24' ddc: - '510' degree_awarded: PhD department: - _id: RoSe - _id: GradSch doi: 10.15479/AT:ISTA:7514 ec_funded: 1 file: - access_level: open_access checksum: b4de7579ddc1dbdd44ff3f17c48395f6 content_type: application/pdf creator: dernst date_created: 2020-02-24T09:15:06Z date_updated: 2020-07-14T12:47:59Z file_id: '7515' file_name: thesis.pdf file_size: 1563429 relation: main_file - access_level: closed checksum: ad7425867b52d7d9e72296e87bc9cb67 content_type: application/x-zip-compressed creator: dernst date_created: 2020-02-24T09:15:16Z date_updated: 2020-07-14T12:47:59Z file_id: '7516' file_name: thesis_source.zip file_size: 2028038 relation: source_file file_date_updated: 2020-07-14T12:47:59Z has_accepted_license: '1' 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: Institute of Science and Technology 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 orcid: 0000-0002-6781-0521 title: The free energy of a dilute two-dimensional Bose gas tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2020' ... --- _id: '7524' abstract: - lang: eng text: "We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\\rho$ and inverse temperature $\\beta$ differs from the one of the non-interacting system by the correction term $4 \\pi \\rho^2 |\\ln a^2 \\rho|^{-1} (2 - [1 - \\beta_{\\mathrm{c}}/\\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\\cdot]_+ = \\max\\{ 0, \\cdot \\}$ and $\\beta_{\\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit\r\n$a^2\\rho \\ll 1$ and if $\\beta \\rho \\gtrsim 1$." article_processing_charge: No author: - first_name: Andreas full_name: Deuchert, Andreas id: 4DA65CD0-F248-11E8-B48F-1D18A9856A87 last_name: Deuchert orcid: 0000-0003-3146-6746 - first_name: Simon full_name: Mayer, Simon id: 30C4630A-F248-11E8-B48F-1D18A9856A87 last_name: Mayer - first_name: Robert full_name: Seiringer, Robert id: 4AFD0470-F248-11E8-B48F-1D18A9856A87 last_name: Seiringer orcid: 0000-0002-6781-0521 citation: ama: Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. arXiv:191003372. apa: Deuchert, A., Mayer, S., & Seiringer, R. (n.d.). The free energy of the two-dimensional dilute Bose gas. I. Lower bound. arXiv:1910.03372. ArXiv. chicago: Deuchert, Andreas, Simon Mayer, and Robert Seiringer. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” ArXiv:1910.03372. ArXiv, n.d. ieee: A. Deuchert, S. Mayer, and R. Seiringer, “The free energy of the two-dimensional dilute Bose gas. I. Lower bound,” arXiv:1910.03372. ArXiv. ista: Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas. I. Lower bound. arXiv:1910.03372, . mla: Deuchert, Andreas, et al. “The Free Energy of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” ArXiv:1910.03372, ArXiv. short: A. Deuchert, S. Mayer, R. Seiringer, ArXiv:1910.03372 (n.d.). date_created: 2020-02-26T08:46:40Z date_published: 2019-10-08T00:00:00Z date_updated: 2023-09-07T13:12:41Z day: '08' department: - _id: RoSe ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1910.03372 month: '10' oa: 1 oa_version: Preprint page: '61' project: - _id: 25C6DC12-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '694227' name: Analysis of quantum many-body systems publication: arXiv:1910.03372 publication_status: draft publisher: ArXiv related_material: record: - id: '7790' relation: later_version status: public - id: '7514' relation: dissertation_contains status: public scopus_import: 1 status: public title: The free energy of the two-dimensional dilute Bose gas. I. Lower bound type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2019' ...