  To what extent do thermodynamic resource theories capture physically relevant
constraints? Inspired by quantum computation, we define a set of elementary
thermodynamic gates that only act on 2 energy levels of a system at a time. We
show that this theory is well reproduced by a Jaynes-Cummings interaction in
rotating wave approximation and draw a connection to standard descriptions of
thermalisation. We then prove that elementary thermal operations present
tighter constraints on the allowed transformations than thermal operations.
Mathematically, this illustrates the failure at finite temperature of
fundamental theorems by Birkhoff and Muirhead-Hardy-Littlewood-Polya concerning
stochastic maps. Physically, this implies that stronger constraints than those
imposed by single-shot quantities can be given if we tailor a thermodynamic
resource theory to the relevant experimental scenario. We provide new tools to
do so, including necessary and sufficient conditions for a given change of the
population to be possible. As an example, we describe the resource theory of
the Jaynes-Cummings model. Finally, we initiate an investigation into how our
resource theories can be applied to Heat Bath Algorithmic Cooling protocols.
