Maximum entropy analysis of dynamic parameters via the Laplace transform

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Abstract
Research in the field of Brownian motion often raises the problem of inverting the Laplace transform. Thus, the measured multi-exponential relaxations are the Laplace transform of the spectrum of decay times. However, the inversion of the Laplace transform is very ill-conditioned. The maximum entropy (maxent) method is shown to be a very powerful technique for such data analysis, capable of handling single peaks, multiple peaks and broad distributions in the spectra.After a theoretical demonstration of why entropy chooses the optimal reconstruction, the power of the technique is illustrated with two types of experimental examples related to Brownian dynamics. In the first example, photon correlation spectroscopy (PCS) is applied to concentrated colloidal dispersions. The results, analysed by maxent, present a detailed and coherent picture of the dynamics of a concentrated colloidal solution, leading to a test of the many-body hydrodynamic theories. In the second example, the maxent analysis of the time-resolved fluorescent decay of L-tryptophan is presented. Maxent provides some evidence that one of the features is broadened, suggestive of translative motion in the structure and/or heterogeneity of the fluorophore's environment.

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