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By quenching into the metastable region of the three-dimensional Ising model, we investigate the paths that the magnetization (energy) takes as a function of time. We accumulate the magnetization (energy) paths into time-dependent distributions from which we reconstruct the free energy as a function of the magnetic field, temperature, and system size. From the reconstructed free energy, we obtain the free-energy barrier that is associated with the transition from a metastable state to the stable equilibrium state. Although mean-field theory predicts a sharp transition between the metastable and the unstable region where the free-energy barrier is zero, the results for the nearest-neighbor Ising model show that the free-energy barrier does not go to zero.

Reconstruction of the free energy in the metastable region using the path ensemble