A $Γ$-convergence of level-two large deviation for metastable systems: The case of zero-range processes
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909206742827008 |
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| author | Choi, Kyuhyeon |
| author_facet | Choi, Kyuhyeon |
| contents | This study explores the relationship between the precise asymptotics of the level-two large deviation rate function and the behavior of metastable stochastic systems. Initially identified for overdamped Langevin dynamics (Ges{ù} et al., SIAM J Math Anal 49(4), 3048-3072, 2017), this connection has been validated across various models, including random walks in a potential field. We extend this connection to condensing zero-range processes, a complex interacting particle system.
Specifically, we investigate a certain class of zero-range processes on a fixed graph $G$ with $N > 0$ particles and interaction parameter $α> 1$. On the time scale $N^2$, this process behaves like an absorbing-type diffusion and converges to a condensed state where all particles occupy a single vertex of $G$ as $N$ approaches infinity. Once condensed, on the time scale $N^{1+α}$, the condensed site moves according to a Markov chain on $G$, showing metastable behavior among condensed states. The time scales $N^2$ and $N^{1+α}$ are called the pre-metastable and metastable time scales. It is conjectured that this behavior is encapsulated in the level-two large deviation rate function $\mathcal{I}_N$ of the zero-range process. Specifically, it is expected that the $Γ$-expansion of $\mathcal{I}_N$ can be expressed as:$$\mathcal{I}_N = \frac{1}{N^2} \mathcal{K} + \frac{1}{N^{1+α}} \mathcal{J},$$ where $\mathcal{K}$ and $\mathcal{J}$ are the level-two large deviation rate functions of the absorbing diffusion processes and the Markov chain on $G$. We rigorously prove this $Γ$-expansion by developing a methodology for $Γ$-convergence in the pre-metastable time scale and establishing a link between the resolvent approach to metastability (Landim et al., J Eur Math Soc, 2023. arXiv:2102.00998) and the $Γ$-expansion in the metastable time scale. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A $Γ$-convergence of level-two large deviation for metastable systems: The case of zero-range processes Choi, Kyuhyeon Probability Mathematical Physics 60F10, 60G53, 60K35 This study explores the relationship between the precise asymptotics of the level-two large deviation rate function and the behavior of metastable stochastic systems. Initially identified for overdamped Langevin dynamics (Ges{ù} et al., SIAM J Math Anal 49(4), 3048-3072, 2017), this connection has been validated across various models, including random walks in a potential field. We extend this connection to condensing zero-range processes, a complex interacting particle system. Specifically, we investigate a certain class of zero-range processes on a fixed graph $G$ with $N > 0$ particles and interaction parameter $α> 1$. On the time scale $N^2$, this process behaves like an absorbing-type diffusion and converges to a condensed state where all particles occupy a single vertex of $G$ as $N$ approaches infinity. Once condensed, on the time scale $N^{1+α}$, the condensed site moves according to a Markov chain on $G$, showing metastable behavior among condensed states. The time scales $N^2$ and $N^{1+α}$ are called the pre-metastable and metastable time scales. It is conjectured that this behavior is encapsulated in the level-two large deviation rate function $\mathcal{I}_N$ of the zero-range process. Specifically, it is expected that the $Γ$-expansion of $\mathcal{I}_N$ can be expressed as:$$\mathcal{I}_N = \frac{1}{N^2} \mathcal{K} + \frac{1}{N^{1+α}} \mathcal{J},$$ where $\mathcal{K}$ and $\mathcal{J}$ are the level-two large deviation rate functions of the absorbing diffusion processes and the Markov chain on $G$. We rigorously prove this $Γ$-expansion by developing a methodology for $Γ$-convergence in the pre-metastable time scale and establishing a link between the resolvent approach to metastability (Landim et al., J Eur Math Soc, 2023. arXiv:2102.00998) and the $Γ$-expansion in the metastable time scale. |
| title | A $Γ$-convergence of level-two large deviation for metastable systems: The case of zero-range processes |
| topic | Probability Mathematical Physics 60F10, 60G53, 60K35 |
| url | https://arxiv.org/abs/2405.10631 |