Second-order dichotomous processes: Damped free motion, critical behavior, and anomalous superdiffusion

We study damped inertial processes driven by dichotomous Markov noise in the absence of a potential. We obtain exact differential equations for the joint probability density and for the marginal densities of velocity and position. Several aspects of the critical behavior of the system are examined in detail. The exact equation for the displacement of a free Brownian particle is also found and from this equation we study the damping effects on the mean-square displacement 〈${\mathit{X}}^2$(t)〉, by evaluating the dynamical exponent and showing a crossover from a superdiffusive motion of the form 〈${\mathit{X}}^2$(t)〉∼${\mathit{t}}^3$ to ordinary diffusion where 〈${\mathit{X}}^2$(t)〉∼t.