  Classical thermodynamics is built with the concept of equilibrium states.
However, it is less clear how equilibrium thermodynamics emerges through the
dynamics that follows the principle of quantum mechanics. In this paper, we
develop a theory to study the exact nonequilibrium thermodynamics of quantum
systems that is applicable to arbitrary small systems, even for single particle
systems, in contact with a reservoir. We generalize the concept of temperature
into nonequilibrium regime that depends on the detailed dynamics of quantum
states. When we apply the theory to the cavity system and the two-level atomic
system interacting with a heat reservoir, the exact nonequilibrium theory
unravels unambiguously (1) the emergence of classical thermodynamics from
quantum dynamics in the weak system-reservoir coupling regime, without
introducing equilibrium hypothesis; (2) the breakdown of classical
thermodynamics in the strong coupling regime, which is induced by non-Markovian
memory dynamics; and (3) the occurrence of dynamical quantum phase transition
characterized by inflationary dynamics associated with a negative
nonequilibrium temperature, from which the third law of thermodynamics,
allocated in the deep quantum realm, is naturally proved. The corresponding
dynamical criticality provides the border separating the classical and quantum
thermodynamics. The inflationary dynamics may also provide a simple picture for
the origin of big bang and universe inflation.
