The aim of this work is to explore the connections between chaos and diffusion by examining the properties of particle motion in non-chaotic systems. To this end, particle transport and diffusion are studied for point particles moving in systems with fixed polygonal scatterers of four types: (i) a periodic lattice containing many-sided polygonal scatterers; (ii) a periodic lattice containing few-sided polygonal scatterers; (iii) a periodic lattice containing randomly oriented polygonal scatterers; and (iv) a periodic lattice containing polygonal scatterers with irrational angles. The motion of a point particle in each of these system is non-chaotic, with Lyapunov exponents strictly equal to zero.;For many-sided polygons, greater than 100 sides, we present the results of our study that shows that the systems appear to be diffusive with a transport coefficient nearly equal to that of a periodic Lorentz gas with circular scatterers at the same density. The partial van Hove function for the polygonal system has, numerically, a fractal dimension equal to that of the partial van Hove function for the periodic Lorentz gas with circular scatterers. Further, we show that a non-zero average Lyapunov exponent for the system can be defined, numerically, in spite of the fact that the actual Lyapunov exponent is zero. It is also possible to verify a relationship, valid for chaotic systems, between the diffusion coefficient, the average Lyapunov exponent, and the fractal dimension of the partial van Hove function.;We also report results of a study of the transport properties and dynamical properties of a system with few-sided polygons, of less than 100 sides. These systems always appear to be super-diffusive, and non-chaotic, with a value of zero for the Lyapunov exponent. The partial van Hove function has the same fractal dimension as that for a periodic Lorentz gas with circular scatterers.;For randomly oriented scatterers and scatterers with irrational angles, we construct a simple channel model that allows us to isolate individual features of the polygonal Lorentz gases and study their effects on transport properties. The systems have a value of zero for their Lyapunov exponents, and, depending on the orientation of the scatterers, the systems can appear to be either diffusive or super-diffusive.;Although there does not seem to be a direct link between mathematical chaos and ordinary diffusion in these models, the non-chaotic systems show that if any such connection exists, it must be very subtle. Even a weak form of random walk motion may result in ordinary diffusion.
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