The grand canonical ensemble lies at the core of statistical mechanics. A small system
thermalizes to this state while exchanging heat and particles with a bath. A quantum system may
exchange quantities, or “charges,” represented by operators that fail to commute. Whether such a
system thermalizes, and what form the thermal state has, concerns truly quantum
I characterize this state in three ways: First, I generalize the system-and-bath microcanonical
ensemble. Tracing out the bath yields the system’s thermal state. Second, this thermal state is
expected to be the fixed point of typical dynamics. Finally, the thermal state is completely
passive (unable to output thermodynamic work) in a resource-theory model for thermodynamics.
This study opens new avenues into equilibrium in the presence of quantum noncommutation.
Yunger Halpern et al.
Nature Communications 7, 12051 (2016).
This work was conducted with Philippe Faist, Jonathan Oppenheim, and Andreas Winter.