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 γ between subsystems, T(E) emerges from the saddle-point analysis leading to the ensemble equivalence up to corrections O(1/sqrt[N]) in the number of particles N that defines the thermodynamic limit. By extending these ideas for finite γ, while keeping N→∞, we provide a consistent generalization of temperature T(E,γ) 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,γ) is a monotonically increasing function of the total energy E, its dependence with γ is a purely quantum effect notably visible near the ground-state energy and for large energies differs for Markovian and non-Markovian regimes.