On the Rate of Convergence to the Landauer Limit

Every computation that destroys information must dissipate a minimum amount of energy, set by the Landauer limit. Bennett showed in 1973 that this limit is achievable in principle, but three questions remained open: how fast does convergence occur in practice, whether Bennett's construction is optimal, and how work fluctuations behave at finite ensemble sizes. No prior work answered any of these.

We resolve all three. The Fluctuation-Dissipation Compilation (FDC) Theorem proves that the average energy dissipated by any reversible compilation equals the Landauer limit exactly, and that the deviation of finite-sample estimates shrinks at an inverse-square-root rate in the number of independent runs. We prove this rate is tight: Bennett's method achieves it, and no method can converge faster, as established through Berry--Esseen analysis with the best known universal constant.

The proof reveals a structural equivalence between Shannon's channel coding theorem and the thermodynamics of computation. Ensemble size plays the role of blocklength, the Landauer limit plays the role of channel capacity, and a newly derived work variance serves as the thermodynamic analogue of channel dispersion. This identification imports the full toolkit of finite-blocklength information theory into thermodynamic compiler design.

Three further results follow. First, a Pareto theorem proves that no compilation can simultaneously minimise energy, time, and memory. Second, a separation theorem shows that quantum error correction overhead is completely independent of the fundamental thermodynamic cost—fault tolerance never inflates the Landauer floor. Third, batch compilation violates the steady-state thermodynamic uncertainty relation by a factor of six, identifying a regime inaccessible to any continuous-flow machine.

Numerical validation on eight algorithms confirms the predicted convergence rate within one standard deviation across all cases.