Closure of superstatistics

Plasmas and other systems with long-range interactions are commonly found in non-equilibrium steady states that are outside traditional Boltzmann-Gibbs statistics, but can be described using generalized statistical mechanics frameworks such as superstatistics, where steady states are treated as superpositions of canonical ensembles under a temperature distribution. In this work we solve the problem of inferring the possible steady states of a composite system $AB$ where subsystem $A$ is described by superstatistics and $E_{AB} = E_A + E_B$. Our result establishes a closure property of superstatistics, namely that $A$ is described by superstatistics if and only if $AB$ and $B$ are also superstatistical with the same temperature distribution. Some consequences of this result are discussed, such as the impossibility of local thermal equilibrium (LTE) for additive subsystems in non-canonical steady states.