The degradation of fibrils under the influence of thermal fluctuations was studied experimentally by various groups around the world. In the first set of experiments, it was shown that the decay of fibril content, which can be measured by the ThT fluorescence assay, obeys a bi-exponential function. In the second series of experiments, it was demonstrated that when the monomers separated from the aggregate are not recyclable, the time dependence of the number of monomers belonging to the dominant cluster is described by a single-exponential function if the fraction of bound chains becomes less than a certain threshold. Note that the time dependence of the fraction of bound chains can be measured by tryptophan fluorescence. To understand these interesting experimental results, we developed a phenomenological theory and performed molecular simulation. According to our theory and simulations using the lattice and all-atom models, the time dependence of bound chains is described by a logistic function, which slowly decreases at short time scales but becomes a single exponential function at large time scales. The results, obtained by using lattice and all-atom simulations, ascertained that the time dependence of the fibril content can be described by a bi-exponential function that decays faster than the logistic function on short time scales. We have uncovered the molecular mechanism for the distinction between the logistic and bi-exponential behavior. Since the dissociation of the chain from the fibrils requires the breaking of a greater number of inter-chain contacts as compared to the breaking of the beta sheet structure, the decrease in the number of connected chains is slower than the fibril content. Therefore, the time dependence of the aggregate size is logistic, while the two-exponential behavior is preserved for the content of fibrils. Our results are in agreement with the results obtained in both sets of experiments.