Tobias Grafke

Tobias Grafke

Abstract

We present an algorithm to sample stochastic differential equations conditioned on rather general constraints, including integral constraints, endpoint constraints, and stochastic integral constraints. The algorithm is a pathspace Metropolis-adjusted manifold sampling scheme, which samples stochastic paths on the submanifold of realizations that adhere to the conditioning constraint. We demonstrate the effectiveness of the algorithm by sampling a dynamical condensation phase transition, conditioning a random walk on a fixed Levy stochastic area, conditioning a stochastic nonlinear wave equation on high amplitude waves, and sampling a stochastic partial differential equation model of turbulent pipe flow conditioned on relaminarization events.

arXiv

Jelle Soons, Tobias Grafke, and Henk A. Dijkstra, J. Fluid Mech. 1009 (2025), A53

Abstract

There is a reasonable possibility that the present-day Atlantic Meridional Overturning Circulation is in a bi-stable regime and hence it is relevant to compute probabilities and pathways of noise-induced transitions between the stable equilibrium states. Here, the most probable transition pathway of a noise-induced collapse of the northern overturning circulation in a spatially-continuous two-dimensional model with surface temperature and stochastic salinity forcings is directly computed using Large Deviation Theory (LDT). This pathway reveals the fluid dynamical mechanisms of such a collapse. Paradoxically it starts off with a strengthening of the northern overturning circulation before a short but strong salinity pulse induces a second overturning cell. The increased atmospheric energy input of this two-cell configuration cannot be mixed away quickly enough, leading to the collapse of the northern overturning cell and finally resulting in a southern overturning circulation. Additionally, the approach allows us to compare the probability of this collapse under different parameters in the deterministic part of the salinity surface forcing, which quantifies the increase in collapse probability as the bifurcation point of the system is approached.

doi:10.1017/jfm.2025.248

arXiv

Summary: Earth's climate is a highly complex, non-equilibrium and chaotic stochastic system. In this project, we attempt to classify its chaotic attractors with methods from non-equilibrium statistical mechanics, large deviation theory and manifold learning. For example, due to the ice albedo feedback, the climate is known to exist in two locally stable states, the current (warm) climate, and a "snowball" state, where the globe is covered in ice. Some models even suggest additional metastable climate states, such as the slushball Earth. Similarly, the currently active Atlantic Meridional Overturning Circulation (AMOC, colloquially known as Gulf Steram), transports hot water to northern Europe, but is suspected to be only metastable: If perturbed the right way, the climate would persist with the AMOC non-existent. Transitions between these climate states, and their local stability, can in principle be analyzed in light of the non-equilibrium quasipotential, characterizing the expected transition times and most likely escape paths out of the current climate state.

Relevant publications

1. G. Margazoglou, T. Grafke, A. Laio, and V. Lucarini, "Dynamical Landscape and Multistability of a Climate Model", Proc. R. Soc. A 447 (2021), 2250 (link)

2. Jelle Soons, Tobias Grafke, Henk A. Dijkstra, "Optimal Transition Paths for AMOC Collapse and Recovery in a Stochastic Box Model", J Physical Oceanography 54 (2024), 2537 (link)

3. Jelle Soons, Tobias Grafke, Henk A. Dijkstra, "Most Likely Noise-Induced Overturning Circulation Collapse in a 2D Boussinesq Fluid Model", J Fluid Mech 1009 (2025), A53 (link)

4. Jelle Soons, Tobias Grafke, René M. van Westen, and Henk A. Dijkstra, "Physics of an AMOC Overshoot in a Box Model", ArXiv (2025) (link)

Jelle Soons, Tobias Grafke, Henk A. Dijkstra

Abstract

The present-day Atlantic Meridional Overturning Circulation (AMOC) is considered to be in a bi-stable regime and hence it is important to determine probabilities and pathways for noise-induced transitions between its equilibrium states. Here, using Large Deviation Theory (LDT), the most probable transition pathways for the collapse and recovery of the AMOC are computed in a stochastic box model of the World Ocean. This allows us to determine the physical mechanisms of noise-induced AMOC transitions. We show that the most likely path of an AMOC collapse starts paradoxically with a strengthening of the AMOC followed by an immediate drop within a couple of years due to a short but relatively strong freshwater pulse. The recovery on the other hand is a slow process, where the North Atlantic needs to be gradually salinified over a course of 20 years. The proposed method provides several benefits, including an estimate of probability ratios of collapse between various freshwater noise scenarios, showing that the AMOC is most vulnerable to freshwater forcing into the Atlantic thermocline region.

doi:10.1175/JPO-D-23-0234.1

arXiv

Reyk Börner, Ryan Deeley, Raphael Römer, Tobias Grafke, Valerio Lucarini, Ulrike Feudel

Abstract

In multistable dynamical systems driven by weak Gaussian noise, transitions between competing states are often assumed to pass via a saddle on the separating basin boundary. In contrast, we show that timescale separation can cause saddle avoidance in non-gradient systems. Using toy models from neuroscience and ecology, we study cases where sample transitions deviate strongly from the instanton predicted by Large Deviation Theory, even for weak finite noise. We attribute this to a flat quasipotential and propose an approach based on the Onsager-Machlup action to aptly predict transition paths.

doi:10.1103/PhysRevResearch.6.L042053

arXiv

J. Liu, J. E. Sprittles, T. Grafke, J. Stat. Mech (2024) 103206

Abstract

Thermally activated phenomena in physics and chemistry, such as conformational changes in biomolecules, liquid film rupture, or ferromagnetic field reversal, are often associated with exponentially long transition times described by Arrhenius' law. The associated subexponential prefactor, given by the Eyring-Kramers formula, has recently been rigorously derived for systems in detailed balance, resulting in a sharp limiting estimate for transition times and reaction rates. Unfortunately, this formula does not trivially apply to systems with conserved quantities, which are ubiquitous in the sciences: The associated zeromodes lead to divergences in the prefactor. We demonstrate how a generalised formula can be derived, and show its applicability to a wide range of systems, including stochastic partial differential equations from fluctuating hydrodynamics, with applications in rupture of nanofilm coatings and social segregation in socioeconomics.

doi:10.1088/1742-5468/ad8075

arXiv

T. Grafke, T. Schäfer, and E. Vanden-Eijnden, Commun. Pure Appl. Math. 77 (2024) 2268

Abstract

Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimizer of the Freidlin-Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction-advection-diffusion type.

arXiv

doi

T. Grafke, A. Laio, Multiscale Modeling & Simulation, Vol. 22, Iss. 1 (2024)

Abstract

Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows to estimate rigorously the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection.

doi:10.1137/23M1563025

arXiv

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