Stretched exponentials and barrier distributions Original Research Article

Non exponential relaxation in complex macromolecular systems may be the consequence of dispersedness giving rise to different free energy barriers for different molecules. An approximate analytic formula that relates the time derivative of the decaying function to a probability distribution for the barrier is derived. From this, so called stretched exponentials, e−tβ, are obtained from barrier distributions with width kBT/β in energy and some asymmetry towards low energies. They may be represented as double exponential functions. An exact general formula that relates the Fourier transforms of the barrier height distribution and the time decaying function is also derived. This is gives a much more stable method for the numerical determination of the barrier height distribution than direct inversion of the Laplace transform.