On the law of entropy increasing of a one-dimensional infinite system. II (Q1894175)

We consider a one-dimensional infinite hard-points system on \(\mathbf Z\) whose particles have several colors and velocities with unit magnitude, and interact mutually by collisions. We show that Boltzmann type entropy increases monotonically to the future (and therefore decreases monotonically to the past by the time reversibility) for the initial states which have initially no spatial correlation. It is shown that, in general, Boltzmann type entropy of a single color can decrease. We also derive master equations from our particle model [see part I, \textit{T. Niwa}, ibid. 27, No. 4, 621-633 (1987; Zbl 0665.70019)] by taking various scaling limits. Namely, we show that in certain hydrodynamic limits the density of particles of a one colour obeys some diffusion type equation, and in other types of limits it obeys some kind of wave equation.