  A theoretical approach aimed at the quantum statistical mechanics of a
molecular ensemble coupled to a lossless cavity mode is presented. A canonical
ensemble is considered and an approximate formula is devised for the Helmholtz
free energy correction due to cavity-molecule coupling, which enables the
derivation of experimentally measurable thermodynamic quantities. The frequency
of the cavity mode is assumed to lie in the infrared range. Therefore, the
cavity couples to molecular vibrations and our treatment is restricted to the
electronic ground state of the molecule. The method is tested for an
analytically solvable model system of one-dimensional harmonic oscillators
coupled to the cavity. The performance of the approximation and its range of
validity are discussed in detail. It is shown that the leading-order correction
to the Helmholtz free energy is proportional to the square of the collective
coupling strength. We also demonstrate that the cavity mode does not have a
significant impact on the thermodynamic properties of the system in the
collective ultrastrong coupling regime (the collective coupling strength is
comparable to the frequency of the cavity mode).
