>[!warning] >This content has not been peer reviewed. # The minimum cost to erase a bit **Landauer (1961):** Erasing one bit of information in a system at temperature $T$ dissipates at least $k_B T \ln 2$ of energy. The bound is fundamental: any physical process that resets a two-state system to a standard state must pay this cost. It links information and thermodynamics and has been confirmed in experiments (Berut et al. 2012). --- ## What it is - The **Landauer bound** states that the minimum heat produced when erasing one bit is $Q \geq k_B T \ln 2$. - It follows from the Second Law and the link between entropy and information (Szilard 1929; Brillouin 1956). Reversible computation can avoid the cost; erasure cannot. --- ## How RRT/RST uses it - **A2 (Resolution), [[Energy Floor]]:** The cost of *maintaining* a bit against thermal noise is at least this rate. Persistence is dissipative. - **[[Fidelity Derivation]]:** At low signal-to-noise ($\eta \ll 1$), the substrate’s response must be **linear in the noise** — Landauer forces $\mu \to \eta$ (not $\eta^2$ or $\sqrt{\eta}$). That boundary condition, with scale-free exhaustion, fixes the form $\mu(\eta,n) = \eta/(1+\eta^n)^{1/n}$. - **Relational Landauer Principle:** RRT extends the bound to a *relational* setting: the minimum power to maintain resolution over a network involves distance, refresh interval, and bit-count ([[Relational Resolution Theory (RRT)]]). --- ## Links | Concept | Note | |:---|:---| | Energy floor, cost per bit | **[[Energy Floor]]** | | Fidelity at low and high η | **[[Fidelity]]**, **[[Fidelity Derivation]]** | | RRT axioms and principle | **[[Relational Resolution Theory (RRT)]]** | --- ## References - Landauer, R. (1961). *Irreversibility and heat generation in the computing process.* IBM J. Res. Dev. **5**, 183–191. [DOI](https://doi.org/10.1147/rd.53.0183) - Szilard, L. (1929). *Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen.* Z. Phys. **53**, 840–856. (On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings.) - Brillouin, L. (1956). *Science and Information Theory.* Academic Press, New York. - Berut, A. et al. (2012). *Experimental verification of Landauer’s principle linking information and thermodynamics.* Nature **483**, 187–189. [DOI](https://doi.org/10.1038/nature10872)