Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction

An infinite system of Newton's equation of motion is considered for one-dimensional particles interacting by a finite-range hard-core potential of singularity like an inverse power of distance between the hard cores. Existence of limiting solutions is proved for initial configurations of finite specific energy and the semigroup of motion is constructed if energy fluctuations near infinity increase only as a small power of distance from the origin. In this case uniqueness of solutions is also proved and the solution is a weakly continuous function of initial data. The allowed set of initial configurations carries a wide class of probability measures including Gibbsian fields with different potentials. In the absence of hard cores limiting solutions are constructed for initial configurations with a logarithmic order of energy and density fluctuations.