The shapes and sizes of two-dimensional pressurized self-intersecting rings, as models for two-dimensional vesicles

Two-dimensional vesicles are modelled as pressurized, unrestricted closed random walks. The model, though suffering from undesirable physical characteristics, has the decided advantage of yielding exact analytical statistical analysis of the size and shapes of these objects. Closed-form expressions for the average of the square of the radius of gyration and the asphericity are derived. In addition, exact expressions for the full probability distribution are also obtained. Numerical computations of the probability function for several pressures are presented. The authors arrive at a new, exact formula for the distribution function, P(R 1 2 , R 2 2 ), of the principal components of the square of the radius of gyration for two-dimensional rings.