Attractive and repulsive interactions in a two-dimensional billiardlike system: The Lennard-Jones stadium

We consider a two-dimensional (2D) square billiard with one particle where the interaction of the particle with an elementary wall segment is the Lennard-Jones (LJ) potential. We name the resulting potential the LJ stadium (LJS). It has three qualitatively different regions and here we study the region with energy E>0 (Sinai’s region). We also find an integrable potential close to that of the LJS and discuss the motion in the LJS in terms of this potential. We draw the Poincaré surface of section for the LJS at various energies and the frequency-frequency curves (FFC) and find that, the lower the energy, the more probable a chaotic motion exists. We identify the corners as the probable source of chaos and discuss the mechanisms of resonance interaction and jumps of tori. We evaluate the Lyapunov exponent and discuss it in terms of qualitative characteristics of the potential. By deleting the attractive terms in the LJ potential we study the square billiard system with a softened repulsion. We find that the motion in this billiard is regular. Therefore we conclude that by softening the hard-sphere potential there are no qualitatively new phenomena, while with an attractive interaction, stochastic motion appears at low energies and global stochastic behavior can be seen.