  We develop a rigorous error analysis for coarse-graining of defect-formation
free energy. For a one-dimensional constrained atomistic system, we establish
the thermodynamic limit of the defect-formation free energy and obtain
explicitly the rate of convergence. We then construct a sequence of
coarse-grained energies with the same rate but significantly reduced
computational cost. We illustrate our analytical results through explicit
computations for the case of harmonic potentials and through numerical
simulations.
