  We consider quantum nonlinear many-body systems with dissipation described
within the Caldeira-Leggett model, i.e., by a nonlocal action in the path
integral for the density matrix. Approximate classical-like formulas for
thermodynamic quantities are derived for the case of many degrees of freedom,
with general kinetic and dissipative quadratic forms. The underlying scheme is
the pure-quantum self-consistent harmonic approximation (PQSCHA), equivalent to
the variational approach by the Feynman-Jensen inequality with a suitable
quadratic nonlocal trial action. A low-coupling approximation permits to get
manageable PQSCHA expressions for quantum thermal averages with a classical
Boltzmann factor involving an effective potential and an inner Gaussian average
that describes the fluctuations originating from the interplay of quanticity
and dissipation. The application of the PQSCHA to a quantum phi4-chain with
Drude-like dissipation shows nontrivial effects of dissipation, depending upon
its strength and bandwidth.
