Optimal strategies for controlled growth in metastable Kawasaki dynamics
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908901327241216 |
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| author | Baldassarri, Simone de Jongh, Maike C. |
| author_facet | Baldassarri, Simone de Jongh, Maike C. |
| contents | In this paper, we develop a Markov decision process (MDP) formulation for the low--temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two--dimensional square lattice. We analyze how an external controller can guide the system to the all--occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action--dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency--based criterion promotes the growth from the boundary centers of the cluster, an energy--based reward function favours the growth at the corners of the cluster. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18903 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal strategies for controlled growth in metastable Kawasaki dynamics Baldassarri, Simone de Jongh, Maike C. Optimization and Control Mathematical Physics Probability 90C40, 60K35, 82C26 In this paper, we develop a Markov decision process (MDP) formulation for the low--temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two--dimensional square lattice. We analyze how an external controller can guide the system to the all--occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action--dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency--based criterion promotes the growth from the boundary centers of the cluster, an energy--based reward function favours the growth at the corners of the cluster. |
| title | Optimal strategies for controlled growth in metastable Kawasaki dynamics |
| topic | Optimization and Control Mathematical Physics Probability 90C40, 60K35, 82C26 |
| url | https://arxiv.org/abs/2603.18903 |