Thermodynamics of quantum oscillators

Abstract

In this work, we present a compact analytical approximation for the quantum partition function of systems composed of quantum oscillators. The proposed formula is general and applicable to an arbitrary number of oscillators described by a rather general class of potential energy functions (not necessarily polynomials). Starting from the exact path-integral expression of the partition function, we introduce a temperature-dependent Gaussian approximation for the high-temperature propagator and then invoke a principle of minimal sensitivity to minimize the error. This leads to a system of coupled nonlinear equations whose solution yields the optimal parameters of the Gaussian approximation. The resulting approximate partition function accurately reproduces thermodynamic quantities such as the free energy, average energy, and specific heat—even at zero temperature—with typical relative errors in the range of about 1%–5%. The accuracy deteriorates only moderately when the anharmonicity and coupling strengths are increased. We illustrate the performance of our analytical formula with numerical results for systems of up to ten coupled anharmonic oscillators. These results are compared with “exact” numerical results obtained via Hamiltonian diagonalization for small systems and path-integral Monte Carlo simulations for larger ones.