A stochastic approach to time-dependent BEC
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arXiv
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| Format: | Preprint |
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2025
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| author | Borasi, Luigi De Vecchi, Francesco C. Ugolini, Stefania |
| author_facet | Borasi, Luigi De Vecchi, Francesco C. Ugolini, Stefania |
| contents | We propose a stochastic description of the dynamics of a Bose-Einstein condensate within the context of Nelson stochastic mechanics. We start from the $N$ interacting conservative diffusions, associated with the $N$ Bose particles, and take an infinite particle limit. We address several aspects of this formulation. First, we consider the problem of extending to a system with self-interaction the variational formulation of Nelson stochastic mechanics due to Guerra and Morato. In this regard we discuss two possible extensions, one based on a doubling procedure and another based on a constraint Eulerian type variational principle. Then we consider the infinite particle limit from the point of view of the $N$-particles Madelung equations. Since conservative diffusions can be identified with proper infinitesimal characteristics pairs $(ρ_N(t), v_N(t))$, a time marginal probability density and a current velocity field, respectively, we consider a finite Madelung hierarchy for the marginals pairs $(ρ_{N,n}(t), v_{N,n}(t))$, obtained by properly conditioning the processes. The infinite Madelung hierarchy arises from the finite one by performing, for each fixed $n$, a mean-field scaling limit in $N$. Finally, we introduce a $n$-particle conditioned diffusions which naturally parallels the quantum mechanical approach and is a new approach within the context of Nelson stochastic mechanics. We then prove the convergence, in the infinite particle limit, of the law of such a conditioned process to the law of a self-interacting diffusion which describes the condensate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A stochastic approach to time-dependent BEC Borasi, Luigi De Vecchi, Francesco C. Ugolini, Stefania Probability Mathematical Physics 60H10, 60J60, 60K35, 81S20, 94A17, 60H07, 60H30 We propose a stochastic description of the dynamics of a Bose-Einstein condensate within the context of Nelson stochastic mechanics. We start from the $N$ interacting conservative diffusions, associated with the $N$ Bose particles, and take an infinite particle limit. We address several aspects of this formulation. First, we consider the problem of extending to a system with self-interaction the variational formulation of Nelson stochastic mechanics due to Guerra and Morato. In this regard we discuss two possible extensions, one based on a doubling procedure and another based on a constraint Eulerian type variational principle. Then we consider the infinite particle limit from the point of view of the $N$-particles Madelung equations. Since conservative diffusions can be identified with proper infinitesimal characteristics pairs $(ρ_N(t), v_N(t))$, a time marginal probability density and a current velocity field, respectively, we consider a finite Madelung hierarchy for the marginals pairs $(ρ_{N,n}(t), v_{N,n}(t))$, obtained by properly conditioning the processes. The infinite Madelung hierarchy arises from the finite one by performing, for each fixed $n$, a mean-field scaling limit in $N$. Finally, we introduce a $n$-particle conditioned diffusions which naturally parallels the quantum mechanical approach and is a new approach within the context of Nelson stochastic mechanics. We then prove the convergence, in the infinite particle limit, of the law of such a conditioned process to the law of a self-interacting diffusion which describes the condensate. |
| title | A stochastic approach to time-dependent BEC |
| topic | Probability Mathematical Physics 60H10, 60J60, 60K35, 81S20, 94A17, 60H07, 60H30 |
| url | https://arxiv.org/abs/2506.20012 |