出版社：SISSA, Scuola Internazionale Superiore di Studi Avanzati
摘要：We study two important aspects of the diffusion of a free particle in the presence of a timedependent
control parameter. The latter is represented by a friction coefcient that is a given
function of time. We solve the stochastic Liouville equation (the Fokker-Planck equation) for the
probability density of the particle in phase space, i. e., in both position and velocity. The exact
solution is then used to analyze the behavior of (i) the variance in the position, a global characterizer
of the system; and (ii) the mean rate of crossings of an arbitrary threshold in the position,
a local characterizer. The former is the more conventional descriptor of diffusive processes, but
the latter provides valuable complementary information on the dynamical behavior. Depending
on the long-time behavior of the friction coefcient, the asymptotic behaviors of both these characterizers
vary, and exhibit several cross-overs. This helps elucidate the nature of the interplay
between the destabilizing effects of the noise and the stabilizing tendency of the damping, as the
latter undergoes a controlled variation in time.
