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Abstract This work proposes a general framework for analyzing noise-driven transitions in spatially extended non-equilibrium systems and explaining the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterization formulas to reduce the complexity of the original equations while preserving the essential dynamical effects of unresolved scales. The approach is flexible and operates for both Gaussian noise and non-Gaussian noise with jumps. Our stochastic parameterization formulas offer two key advantages. First, they can approximate stochastic invariant manifolds when these manifolds exist. Second, even when such manifolds break down, our formulas can be adapted through a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifolds or other time-scale separation methods. The optimized stochastic parameterizations capture then how small-scale noise impacts larger scales through the system’s nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian (memory-dependent) coefficients into the reduced equation. These coefficients account for the noise’s past influence, not just its current value, using a finite memory length that is selected for optimal performance. The specific memory function, which determines how this past influence is weighted, depends on both the strength of the noise and how it interacts with the system’s nonlinearities. Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions. This approach retains indeed good accuracy in predicting noise-induced transitions, including rare events, when tested against a large ensemble of different noise paths. This success stems from our hybrid approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the ‘extrapolation problem.’
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Optimal finite-time processes in weakly driven overdamped Brownian motion
Abstract The complete physical understanding of the optimization of the thermodynamic work still is an important open problem in stochastic thermodynamics. We address this issue using the Hamiltonian approach of linear response theory in finite time and weak processes. We derive the Euler–Lagrange equation associated and discuss its main features, illustrating them using the paradigmatic example of driven Brownian motion in overdamped regime. We show that the optimal protocols obtained either coincide, in the appropriate limit, with the exact solutions by stochastic thermodynamics or can be even identical to them, presenting the well-known jumps. However, our approach reveals that jumps at the extremities of the process are a good optimization strategy in the regime of fast but weak processes for any driven system. Additionally, we show that fast-but-weak optimal protocols are time-reversal symmetric, a property that has until now remained hidden in the exact solutions far from equilibrium.
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- Award ID(s):
- 2010127
- PAR ID:
- 10369700
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Journal of Physics Communications
- Volume:
- 6
- Issue:
- 8
- ISSN:
- 2399-6528
- Page Range / eLocation ID:
- Article No. 083001
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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