  We propose the construction of thermodynamic ensembles that minimize the
R\'enyi free energy, as an alternative to Gibbs states. For large systems, the
local properties of these R\'enyi ensembles coincide with those of thermal
equilibrium, and they can be used as approximations to thermal states. We
provide algorithms to find tensor network approximations to the 2-R\'enyi
ensemble. In particular, a matrix-product-state representation can be found by
using gradient-based optimization on Riemannian manifolds, or via a non-linear
evolution which yields the desired state as a fixed point. We analyze the
performance of the algorithms and the properties of the ensembles on
one-dimensional spin chains.
