A mathematical study finds that a thermodynamic change carried out through a finite number of sudden changes and relaxations has a best possible set of intermediate states. In the model, those states minimize total dissipated work, and even one optimized intermediary lowers dissipation unless the starting and ending distributions coincide.
The work, supplied as arXiv:2608.25207v1, examines work extraction when a system's energy landscape is changed suddenly, a step known as a quench, then allowed to equilibrate through a finite sequence of intermediary Hamiltonians, or energy landscapes. Each quench is followed by complete equilibration, so the central question is how to choose the distributions between the endpoints when the protocol cannot use an unlimited number of intermediate stages.
How the optimum is defined
The cost is represented as a sum of Kullback-Leibler, or KL, divergences between successive equilibrium distributions. A KL divergence is a mathematical penalty for a mismatch between two probability distributions, so the total cost adds up the penalties at each handoff. In the finite-outcome formulation, the minimizer is unique.
The optimality equations use the principal branch of the Lambert W function, a specialized function for equations in which a variable appears both on its own and inside an exponential. The analysis characterizes the optimum through those equations but does not provide a closed-form list of every intermediary. For numerical work, positivity and normalization are imposed and a SciPy nonlinear least-squares solver is used; the reported residual is zero to machine precision.
Many steps, one geometric path
As N, the number of intermediary distributions, becomes large, the optimum approaches a Fisher-Rao geodesic. In ordinary language, it approaches the shortest route defined by the Fisher-Rao distance between probability distributions. With the number of possible outcomes held fixed, mean dissipated work follows the inverse of N and saturates the O(N^-1) Fisher-Rao lower bound.
More intermediate stages also improve the modeled work-extraction objective: optimized extracted work rises monotonically with N and approaches the initial excess free energy available to the protocol. The reported Fisher-Rao lower bound carries an O(N^-2) remainder, meaning the correction shrinks with the square of the number of intermediaries in the stated large-N analysis.
A numerical check and a strange trap
To see how the optimization behaves across many possible endpoints, the numerical study generated 900,000 independent pairs of categorical distributions from the Jeffreys prior, a rule for drawing probability distributions. At fixed N, the mean dissipated work approaches a finite limit as M, the number of generated pairs, becomes large. The result is a model calculation over a selected random-distribution scheme, rather than an experimental measurement.
An optical-trap model produces a less intuitive result. Even when the endpoint densities are unimodal, or single-peaked, the optimal intermediary can correspond to a double-well potential rather than a simple translation of the trap. In the N=1 case at large separation, the last-step KL divergence approaches 1, while the first-step divergence and the total dissipation grow indefinitely. The initial step is identified as the dominant source of loss in that limit.
When speed changes the answer
The enzyme example asks what happens when waiting for equilibration carries a cost. It is a minimal three-state model with bound-state energy 1, unbound-state energy 0 and intermediate-state energy above 1. Direct bound-to-unbound transitions are forbidden, control is restricted to a one-dimensional barrier shift called delta, and the setup uses only an initial quench, with N=0 intermediary steps.
When dissipation is the only objective, the model selects delta* < 0, a barrier-raising shift. The associated equilibration time grows exponentially with -delta*, and the paper interprets this as reaction inhibition. When the cost instead combines dissipated work and equilibration time, C(delta) = Wdiss + lambda Tequi, a sufficiently large time weight is predicted to select delta* > 0, on the barrier-lowering side. The threshold for that preference falls as the intermediate energy rises and tends to zero as that energy becomes very large.
What remains model-dependent
These conclusions stay within the stated assumptions. The optical-trap finding is a one-dimensional model prediction, not a measured optical-trap result, and the enzyme calculation is a minimal three-state model with a one-parameter control family. The supplied document is labeled arXiv:2608.25207v1, a version-one preprint, so the findings are analytical and numerical model results rather than empirical validation.
Paper data and sources
Original title: Finite relaxation protocols with minimal dissipation
Authors: Ben Ansbacher, Harrison Hartle, Abhishek Yadav et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text