Recurrences in multiple-particle Billiard systems

We investigate recurrence statistics of two-particle circular billiards. Dynamics of this billiard is generated by free-motion of two particles (with non-zero radius) inside a closed surface with piecewise smooth circular boundary in 2−D Euclidean space. Particles collide elastically against the boundary and each other, with angle of incidence equal to angle of reflection. We are interested in recurrences of time interval needed for both particles to come back to certain area. We used time series, recurrence plots and Poincaré sections in order to investigate this time recurrences. We compare our results with ones obtained from [3, 5, 13]. Although more investigation and statistics (as well as simulation running time) is needed, we were still able to make conclusions about correlation decay of certain variables in our system, as well as of frequencies and periodicity of appearance of integrable regions in our phase space. Correlation decay for position and velocity obeys a power low with coefficient between -0.2 and -0.8, whilst decay of probability of first return time obeys power low with coefficient approximately -1. We make conjecture about dependence of this coefficient on rate of inner and outer circle radius.