  The free energy, or equivalently the ground state energy in finite volume,
may be calculated from forward scattering amplitudes using a formula due to
Dashen, Ma, and Bernstein. However a naive treatment leads to singularities
when considering the scattering of three or more particles. It is shown in
detail how the approach can be applied to multi-particle scattering in various
massive scalar theories in 1+1d, with or without integrability. The results for
the sinh-Gordon, Lieb-Liniger, and $O(N)$ non-linear sigma models are compared
to exact results. It is shown how bound states can be considered in this
approach by considering the attractive Lieb-Liniger model.
