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Using the path integral approach, we obtain the characteristic functions ofthe gyration radius distributions for Gaussian star and Gaussian rosettemacromolecules. We derive the analytical expressions for cumulants of the bothdistributions. Applying the steepest descent method, we estimate theprobability distribution functions of the gyration radius in the limit of alarge number of star and rosette arms in two limiting regimes: for stronglyexpanded and strongly collapsed macromolecules. We show that in both cases, inthe regime of a large gyration radius relative to its mean-square value, theprobability distribution functions can be described by the Gaussian functions.In the shrunk macromolecule regime, both distribution functions tend to zerofaster than any power of the gyration radius. Based on the asymptotic behaviorof the distribution functions and the behavior of statistical dispersions, wedemonstrate that the probability distribution function for the rosette is moredensely localized near its maximum than that for the star polymer. We constructthe interpolation formula for the gyration radius distribution for the Gaussianstar macromolecule which can help to take into account the conformationalentropy of the flexible star macromolecules within the Flory-type mean-fieldtheories

On gyration radius distributions of star-like macromolecules