  We study a microscopic Hamiltonian model describing an N-level quantum system
S coupled to an infinitely extended thermal reservoir R. Initially, the system
S is in an arbitrary state while the reservoir is in thermal equilibrium at
temperature T. Assuming that the coupled system S+R is mixing with respect to
the joint thermal equilibrium state, we study the Full Counting Statistics
(FCS) of the energy transfers S->R and R->S in the process of return to
equilibrium. The first FCS describes the increase of the energy of the system
S. It is an atomic probability measure, denoted $P_{S,\lambda,t}$, concentrated
on the set of energy differences $\sigma(H_S)-\sigma(H_S)$ ($\sigma(H_S)$ is
the spectrum of the Hamiltonian of S, $t$ is the length of the time interval
during which the measurement of the energy transfer is performed, and $\lambda$
is the strength of the interaction between S and R). The second FCS,
$P_{R,\lambda,t}$, describes the decrease of the energy of the reservoir R and
is typically a continuous probability measure whose support is the whole real
line. We study the large time limit $t\rightarrow\infty$ of these two measures
followed by the weak coupling limit $\lambda\rightarrow 0$ and prove that the
limiting measures coincide. This result strengthens the first law of
thermodynamics for open quantum systems. The proofs are based on modular theory
of operator algebras and on a representation of $P_{R,\lambda,t}$ by quantum
transfer operators.
