Landauer Cost, Information Engines, and Field Reservoirs

Landauer’s principle relates erasing information to entropy production in a specified reservoir. In field settings the memory, coupling region, bath state, conserved charges, and reset error must all be declared. The familiar $k_BT\ln2$ is an ideal single-bit limit, not a universal invoice for every measurement or feedback step.

Required background. Passivity and work supplies cyclic work accounting and the equilibrium reference.

Helpful background. Complete passivity and KMS structure explains why a thermal reservoir is a special resource.

Erasure as a physical channel

Let memory $M$ start in $\rho_M$ and a reservoir $R$ in $\tau_R=e^{-\beta H_R}/Z$. A global unitary implements an approximate reset $\rho_M\mapsto\rho'_M$ while changing the reservoir energy by $Q=\operatorname{tr}[H_R(\rho'_R-\tau_R)]$. Positivity of relative entropy gives

$$\beta Q \ge S(\rho_M)-S(\rho'_M)+I(M{:}R)' ,$$

when the final Hamiltonian and interaction bookkeeping support this form. For a uniformly random classical bit reset to a pure standard state with negligible final correlations, $Q\ge k_BT\ln2$.

The original thermodynamic argument and its computing interpretation are given by Landauer 1961, §§ “The Energy Requirements”–“Logical Versus Physical Irreversibility,” pp. 184–188. Finite reservoirs deviate from this ideal. Their temperature changes, correlations store entropy, and exact reset may be impossible at finite resources. Reeb and Wolf 2014, Theorems 3 and 6, §§ 3–4 derive finite-size corrections without assuming an infinite bath.

A bosonic-reservoir calculation

Model the memory as a two-level detector and the bath as finitely many bosonic modes with cutoff $\Lambda$. Specify:

• the detector Hamiltonian and initial probability distribution;
• the spatial switching and duration;
• the target reset error $\epsilon$;
• bath energy, entropy, and correlation changes;
• controller work used to turn the interaction on and off.

Compare a slow protocol with a finite-time one while holding $\epsilon$ fixed. The quasistatic sequence may approach the Landauer limit as mode density and duration increase. The finite-time protocol generally pays additional dissipation. Increase reservoir size and $\Lambda$ independently; a small gap to $k_BT\ln2$ at one cutoff is not a continuum result.

Landauer erasure uses the equilibrium/passivity branch together with an explicit memory–reservoir operation. It is not derived from a local stress-energy bound. The diagram is schematic.

Information engines and feedback

A feedback engine measures a field or detector, stores outcome $x$, and applies a conditional operation. Mutual information can increase the work extractable during feedback, but closing the cycle requires resetting the record and controller. The net balance must include measurement, feedback, and erasure:

$$W_{\rm net}=W_{\rm extracted} -W_{\rm measure}-W_{\rm feedback}-W_{\rm reset}.$$

A nonthermal or finite reservoir can sometimes outperform a thermal benchmark by consuming its nonequilibrium free energy. That is resource conversion, not a violation of Landauer’s principle.

The principal failure is an open cycle: extracted work is reported while memory reset, switching, bath depletion, or final correlations are omitted. The map is schematic.

Evidence boundary

This account reflects results available through 10 August 2026. Exact finite-reservoir identities and controlled finite models support the principle; extending a particular engine to a continuum field requires cutoff, localization, and finite-time convergence rather than a formal infinite bath.

References

• Landauer, Rolf. “Irreversibility and Heat Generation in the Computing Process.” IBM Journal of Research and Development 5 (1961): 183–191. DOI.
• Reeb, David, and Michael M. Wolf. “An Improved Landauer Principle with Finite-Size Corrections.” New Journal of Physics 16 (2014): 103011. DOI.