  We derive the quantum thermodynamics of quantum Brownian motion from the
exact solution of its reduced density matrix. We start from the total
equilibrium thermal state between the Brownian particle and its reservoir, and
solve analytically and exactly the reduced density matrix of the system by
taking the partial trace over all the reservoir states. We find that the
reduced Hamiltonian and the reduced partition function of the Brownian motion
must be renormalized significantly, as shown in the general nonperturbative
renormalization theory of quantum thermodynamics for open quantum systems we
developed recently [Phys. Rev. Res. 4, 023141 (2022)]. The reduced Hamiltonian
contains not only a frequency shift but also a squeezing pairing interaction,
where a momentum-dependent potential is generated naturally from the strong
coupling between the Brownian particle and the reservoir, after traced over all
the reservoir states. The resulting exact reduced density matrix of the
Brownian motion is given by a squeezing thermal state. Moreover, beyond the
weak coupling limit, in order to obtain correctly the reduced partition
function of the Brownian motion, one must take into account the non-negligible
changes of the reservoir state induced by the system-reservoir coupling. Using
the exact solutions of the reduced density matrix, the reduced Hamiltonian as
well as the reduced partition function of the Brownian motion, we show that the
controversial results obtained from the different definitions of internal
energy and the issue of the negative heat capacity in the previous studies of
strong-coupling quantum thermodynamics are resolved.
