  Non-local and non-convex energies represent fundamental interacting effects
regulating the complex behavior of many systems in biophysics and materials
science. We study one dimensional, prototypical schemes able to represent the
behavior of several biomacromolecules and the phase transformation phenomena in
solid mechanics. To elucidate the effects of thermal fluctuations on the
non-convex non-local behavior of such systems, we consider three models of
different complexity relying on thermodynamics and statistical mechanics: (i)
an Ising-type scheme with an arbitrary temperature dependent number of
interfaces between different domains, (ii) a zipper model with a single
interface between two evolving domains, and (iii) an approximation based on the
stationary phase method. In all three cases, we study the system under both
isometric condition (prescribed extension, matching with the Helmholtz ensemble
of the statistical mechanics) and isotensional condition (applied force,
matching with the Gibbs ensemble). Interestingly, in the Helmholtz ensemble the
analysis shows the possibility of interpreting the experimentally observed
thermal effects with the theoretical force-extension relation characterized by
a temperature dependent force plateau (Maxwell stress) and a force peak
(nucleation stress). We obtain explicit relations for the configurational
properties of the system as well (expected values of the phase fractions and
number of interfaces). Moreover, we are able to prove the equivalence of the
two thermodynamic ensembles in the thermodynamic limit. We finally discuss the
comparison with data from the literature showing the efficiency of the proposed
model in describing known experimental effects.
