Strong coupling and non-Markovian effects in the statistical notion of temperature

We investigate the emergence of temperature T in the system-plus-reservoir paradigm starting from the fundamental microcanonical scenario at total fixed energy E where, contrary to the canonical approach, T = T (E) is not a control parameter but a derived auxiliary concept. As shown by Schwinger for the regime of weak coupling gamma between subsystems, T (E) emerges from the saddle-point analysis leading to the ensemble equivalence up to corrections O(1/root N) in the number of particles N that defines the thermodynamic limit. By extending these ideas for finite gamma, while keeping N -> infinity, we provide a consistent generalization of temperature T (E, gamma) in strongly coupled systems, and we illustrate its main features for the specific model of quantum Brownian motion where it leads to consistent microcanonical thermodynamics. Interestingly, while this T (E, gamma) is a monotonically increasing function of the total energy E, its dependence with gamma is a purely quantum effect notably visible near the ground-state energy and for large energies differs for Markovian and non-Markovian regimes