On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular,...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2021
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| _version_ | 1826270300755460096 |
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| author | Delgadino, MG Gvalani, R Pavliotis, G |
| author_facet | Delgadino, MG Gvalani, R Pavliotis, G |
| author_sort | Delgadino, MG |
| collection | OXFORD |
| description | The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. |
| first_indexed | 2024-03-06T21:38:41Z |
| format | Journal article |
| id | oxford-uuid:472532e9-e75a-43ac-911e-5ec59b16131e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:38:41Z |
| publishDate | 2021 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:472532e9-e75a-43ac-911e-5ec59b16131e2022-03-26T15:18:20ZOn the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:472532e9-e75a-43ac-911e-5ec59b16131eEnglishSymplectic ElementsSpringer2021Delgadino, MGGvalani, RPavliotis, GThe objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. |
| spellingShingle | Delgadino, MG Gvalani, R Pavliotis, G On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions |
| title | On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions |
| title_full | On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions |
| title_fullStr | On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions |
| title_full_unstemmed | On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions |
| title_short | On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions |
| title_sort | on the diffusive mean field limit for weakly interacting diffusions exhibiting phase transitions |
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