Stochastic Counterdiabatic Driving via Biorthogonal Liouvillian Eigenmodes
2026-07-27 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how speeding up changes in systems with randomness usually causes extra energy loss, making estimates of free energy less accurate. They propose a new method that uses special mathematical transforms based on the exact solutions of equations describing the system's evolution, to perfectly cancel out this extra loss. Their approach is similar to techniques in quantum physics that avoid unwanted transitions. They test their method on particles moving in changing environments and show it greatly improves accuracy, almost eliminating the usual lag and wasted energy in simulations.
stochastic systemsJarzynski equalityfree energy estimationFokker-Planck equationcounterdiabatic drivingnon-adiabatic lagspectral decompositionshortcuts to adiabaticitydissipated workbiorthogonal decomposition
Authors
Sandeep Suresh Cranganore, Sebastian Lehner, Johannes Brandstetter, Max Welling
Abstract
Finite-time driving of stochastic systems generates excess dissipation, causing the evolving probability distribution to lag behind the instantaneous equilibrium, and consequently degrading the convergence of nonequilibrium free energy estimators based on the Jarzynski equality. Escorted free energy simulations address the non-adiabatic lag by engineering control fields $\mathbf{u}$ that eliminate the lag, enforcing the trajectory-wise equality $\mathcal{W}_\mathbf{u} = Δ\mathcal{F}$, and yielding zero-variance estimators. However, constructing the escorting field in closed form remains a challenge, approached variously through flow-field methods, targeted free energy perturbation, or learned diffeomorphisms. In this work, we construct a complementary numerical framework based on gauge-type transforms instead of generalized coordinate transforms for perfect escorting based on the exact spectral decomposition of the time-dependent Fokker-Planck generator. The biorthogonal decomposition of the Liouville operator directly yields a counterdiabatic correction whose action on the instantaneous equilibrium distribution exactly cancels the non-adiabatic lag at arbitrary driving speed in formal analogy with shortcuts-to-adiabaticity techniques such as Berry's transitionless driving for quantum systems. Numerical verification for simulations of an overdamped particle in a time-varying double-well potential and harmonic traps confirms that the counterdiabatic condition is satisfied to machine precision, with the non-adiabatic lag suppressed by roughly twelve orders of magnitude in total variation distance and sixteen orders in KL divergence relative to the unescorted dynamics. As a diagnostic, we demonstrate vanishing dissipated work $\mathcal{W}_{\text{diss}}(t) \approx 0$ for the deterministically propagated Fokker-Planck density across all protocol speeds.