Fig. 1
Pictorical representation of the equivalence between macrostate operations and microstate operations. Panel a shows macrostate operations and b microstate operation. Closed boxes represent systems from which we only know some partial information, in this case the mean energy. Inside the box there is the actual microstate unknown to us if the box is closed. Scenario a shows the situation where one has an initial system of which only the mean energy e is known and one can use any environment, being again limited to knowledge of its initial average energy e β . The question is whether we can find a unitary U that takes the two systems, regardless of what is actually inside of them, to one box for which we are certain that we will find inside the microstate ρf. The answer to this question is provided by scenario b, where the initial boxes of system and environment are both open (implying that we know what is the microstate) and populated with the maximum-entropy ensemble. U exists if and only if there exists a unitary Umic that implements the transition in b when taking ρ = γ e (H). This shows that a thermodynamic transition is possible if and only if it is also possible under the assignment of ensembles to systems
