Generality of optimal-intermediate trends beyond Glauber Markov models

Determine whether the trends identified for optimal intermediate Hamiltonians—particularly endpoint energy jumps, improved mean-squared-error performance, and the relationship between dissipation and estimator convergence—hold for dynamical schemes other than Glauber dynamics and for more complex systems.

Background

The study derives and numerically evaluates mean-squared-error-minimizing intermediate Hamiltonians for finite-state Markov models evolving under discrete-time Glauber dynamics. Across the two-state, double-well, and shifted-well systems, the authors identify recurring features such as jumps at the initial and final times and substantial error reductions relative to standard interpolation schemes.

The authors explicitly leave unresolved whether these observations are artifacts of the chosen Glauber dynamics and simple model systems or whether they extend to other dynamical rules and realistic molecular, colloidal, or many-body systems. They note that answering this question requires studying a broad range of more realistic systems.

References

Whether the identified trends hold for other dynamics and more complex systems remains an open question; answering this question will require studying a broad range of more realistic systems.

— Optimal Intermediate Hamiltonians for Non-Equilibrium Free Energy Calculations: A Numerical Study of Markov Models  (2609.10519 - Beyer et al., 9 Sep 2026) in Section Summary and Outlook

We conjecture that changing the intermediate path cannot systematically eliminate the metastable bottleneck when every stage continues to rely solely on the same local WGF/FODP dynamics. Establishing this conjecture across general annealing schedules, initial distributions, and target densities would require an optimality or lower-bound result beyond the scope of this work.

— Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling  (2610.02081 - McBride et al., 1 Oct 2026) in Section 3.4, “A Log-Linear Annealing Schedule Does Not Alter the Observed Scaling”