Large time analysis of the rate function associated to the Boltzmann equation: dynamical phase transitions
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912174681620480 |
|---|---|
| author | Basile, Giada Benedetto, Dario Bertini, Lorenzo Heydecker, Daniel |
| author_facet | Basile, Giada Benedetto, Dario Bertini, Lorenzo Heydecker, Daniel |
| contents | We analyse the large time behaviour of the rate function that describes the probability of large fluctuations of an underlying microscopic model associated to the homogeneous Boltzmann equation, such as the Kac walk. We consider in particular the asymptotic of the number of collisions, per particle and per unit of time, and show it exhibits a phase transition in the joint limit in which the number of particles N and the time interval [0,T] diverge. More precisely, due to the existence of Lu-Wennberg solutions, the corresponding limiting rate function vanishes for subtypical values of the number of collisions. We also analyse the second order large deviations showing that the probability of subtypical fluctuations is exponentially small in N, independently on T. As a key point, we establish the controllability of the homogeneous Boltzmann equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00402 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large time analysis of the rate function associated to the Boltzmann equation: dynamical phase transitions Basile, Giada Benedetto, Dario Bertini, Lorenzo Heydecker, Daniel Probability Mathematical Physics 35Q20, 60F10, 82C40 We analyse the large time behaviour of the rate function that describes the probability of large fluctuations of an underlying microscopic model associated to the homogeneous Boltzmann equation, such as the Kac walk. We consider in particular the asymptotic of the number of collisions, per particle and per unit of time, and show it exhibits a phase transition in the joint limit in which the number of particles N and the time interval [0,T] diverge. More precisely, due to the existence of Lu-Wennberg solutions, the corresponding limiting rate function vanishes for subtypical values of the number of collisions. We also analyse the second order large deviations showing that the probability of subtypical fluctuations is exponentially small in N, independently on T. As a key point, we establish the controllability of the homogeneous Boltzmann equation. |
| title | Large time analysis of the rate function associated to the Boltzmann equation: dynamical phase transitions |
| topic | Probability Mathematical Physics 35Q20, 60F10, 82C40 |
| url | https://arxiv.org/abs/2501.00402 |