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Maxwell's Demon, 'a being whose faculties are so sharpened that he can followevery molecule in its course', has been the centre of much debate about itsabilities to violate the second law of thermodynamics. Landauer's hypothesis,that the Demon must erase its memory and incur a thermodynamic cost, has becomethe standard response to Maxwell's dilemma, and its implications for thethermodynamics of computation reach into many areas of quantum and classicalcomputing. It remains, however, still a hypothesis. Debate has often centredaround simple toy models of a single particle in a box. Despite theirsimplicity, the ability of these systems to accurately represent thermodynamics(specifically to satisfy the second law) and whether or not they displayLandauer Erasure, has been a matter of ongoing argument. The recentNorton-Ladyman controversy is one such example.  In this paper we introduce a programming language to describe these simplethermodynamic processes, and give a formal operational semantics and programlogic as a basis for formal reasoning about thermodynamic systems. We formalisethe basic single-particle operations as statements in the language, and thenshow that the second law must be satisfied by any composition of these basicoperations. This is done by finding a computational invariant of the system. Weshow, furthermore, that this invariant requires an erasure cost to exist withinthe system, equal to kTln2 for a bit of information: Landauer Erasure becomes atheorem of the formal system. The Norton-Ladyman controversy can therefore beresolved in a rigorous fashion, and moreover the formalism we introduce gives aset of reasoning tools for further analysis of Landauer erasure, which areprovably consistent with the second law of thermodynamics.

DEMONIC programming: a computational language for single-particle equilibrium thermodynamics, and its formal semantics