A geometric formulation of GENERIC stochastic differential equations
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914087317798912 |
|---|---|
| author | Peletier, Mark A. Seri, Marcello |
| author_facet | Peletier, Mark A. Seri, Marcello |
| contents | We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A geometric formulation of GENERIC stochastic differential equations Peletier, Mark A. Seri, Marcello Dynamical Systems Mathematical Physics Differential Geometry Probability 37J55, 37K05, 60H10, 60H30, 53D17, 53C17, 80A05, 82C31, 82C35 We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems. |
| title | A geometric formulation of GENERIC stochastic differential equations |
| topic | Dynamical Systems Mathematical Physics Differential Geometry Probability 37J55, 37K05, 60H10, 60H30, 53D17, 53C17, 80A05, 82C31, 82C35 |
| url | https://arxiv.org/abs/2509.09566 |