On the law of entropy increasing of a one-dimensional infinite system (Q1116370)

We consider a one-dimensional hard-points system with several colors whose particles have integral positions and velocities of unit magnitude. The states of the system are represented by the probability measures on the phase space. We define Boltzmann type entropy, and prove that for the ``locally equilibrium states'', that is, states with no spatial correlation this entropy increases monotonically with time evolution. Note that our entropy is invariant under the velocity reversal mapping and microscopic dynamics is time reversible, so entropy cannot increases for all the states. We define also Kolmogorov-Sinai type entropy. This entropy is invariant w.r.t. time evolution.