DEMONIC Programming: A Computational Language for Single-particle Equilibrium Thermodynamics, and its Formal Semantics

摘要

Maxwell's Demon, 'a being whose faculties are so sharpened that he can follow every molecule in its course', has been the centre of much debate about his abilities to violate the second law of thermodynamics. Landauer's hypothesis, that the Demon must erase its memory and incur a thermodynamic cost, has become the standard response to Maxwell's dilemma, and its implications for the thermodynamics of computation reach into many areas of quantum and classical computing. It remains, however, still a hypothesis. Debate over the existence of an erasure cost for information has often centred around simple toy models of a single particle in a box. Despite their simplicity, the ability of these systems to accurately represent thermodynamics (specifically to satisfy the second law) and whether or not they display Landauer Erasure, has been a matter of ongoing argument. The recent Norton-Ladyman controversy is one such example. In this paper we give a computational language for formal reasoning about thermodynamic systems. We formalise the basic single-particle operations as statements in the language, and then show that the second law must be satisfied by any composition of these basic operations. This is done by finding a computational invariant of the system. We show, furthermore, that this invariant requires an erasure cost to exist within the system, equal to kT In 2 for a bit of information: Landauer Erasure becomes a theorem of the formal system. The Norton-Ladyman controversy can therefore be resolved in a rigorous fashion, and moreover the formalism we introduce gives a set of reasoning tools for further analysis of Landauer erasure, which are provably consistent with the second law of thermodynamics.