content_image/332077.jpg content_image/332079.jpg content_image/332078.jpg content_image/332080.jpg
X$_{t}$ . Moreover, since at long times X$_{t}$ can be accurately evaluated as a total length of the intervals ( nl, nl + l ) ( n = 0 , 1 , . . . ) which a particle passes, many of the details of the particle dynamics described by Eq. (2.1) are needless for this purpose. 

It is therefore reasonable to consider, instead of the model based on Eq. (2.1), the uni- directional CTRW of a particle on a semi-infinite one-dimensional lattice with the period l . Introduced more than four decades ago [15], the CTRW model has become one of the most effective and powerful tools in the theory of anomalous diffusion (see, e.g., Refs. [8–10]). Within this model, we describe the particle position by a discrete variable Y$_{t}$ = lN ( t ), where N ( t ) is the random number of jumps up to time t . In order to guarantee that the long-time behavior of Y$_{t}$ and X$_{t}$ are the same, we assume that for all n the waiting time τ ( n $^{)}$, i.e., the time of occupation of the site nl , is equal to the time that a particle spends moving from the site nl to the site nl + l . If the inertial effects can be neglected then Eq. (2.1) yields τ ( n ) = νl/ ( f + g ( n $^{)}$), where g ( n ) = g ( x ) and x belongs to the n th interval, i.e., x ∈ [ nl, nl + l ). Since the random forces g$_{n}$ are statistically independent and distributed with the same prob- ability density u ( g ), the waiting times τ ( n ) are also statistically independent variables whose probability density is given by 

where 

It is important to emphasize that the inertial effects, at least in the underdamped regime characterized by the condition ντ$_{min}$/µ ≫ 1 (weakly underdamped regime), can also be incorporated into the CTRW framework. In order to illustrate this, let us first write the particle velocity v ( n $^{)}$(Δ t ) [Δ t ∈ (0 , τ ( n $^{)}$)] on the n th ( n ≥ 1) interval. The straightforward integration of Eq. (2.1) yields 

<img src='content_image/332077.jpg'>

<img src='content_image/332079.jpg'>

<img src='content_image/332078.jpg'>

where κ = ν/µ and v ( n ) − = v ( n $^{)}$(0) is the particle velocity at the left end of the n th interval. Introducing also the particle velocity at the right end of this interval, v ( n ) + = v ( n $^{)}$( τ ( n $^{)}$), from Eq. (2.5) we obtain 

<img src='content_image/332080.jpg'>

