  Recent experiments have established negative energetic elasticity, the
negative contribution of energy to the elastic modulus, as a universal property
of polymer gels. To reveal the microscopic origin of this phenomenon, Shirai
and Sakumichi investigated a polymer model on a cubic lattice with the energy
effect from the solvent in finite-size calculations [Phys. Rev. Lett. 130,
148101 (2023)]. Motivated by this work, we provide a simple platform to study
the elasticity of polymer chains by considering a one-dimensional random walk
with the energy effect. This model can be mapped onto the classical Ising
chain, leading to an exact form of the free energy in the thermodynamic or
continuous limit. Our analytical results are qualitatively consistent with
Shirai and Sakumichi's work. Our model serves as a fundamental benchmark for
studying negative energetic elasticity.
