[GR 2] Quantifying hopping-and-stalling motion

Intuitively, hopping-and-stalling-motion (in the case of a discrete time random walk) is marked by

• long chains of successive steps in which the particle remains bound within a box of small volume,
  
• separated by a single long jump, carrying the particle to a new box.

The characteristic dimensionless numbers seem to be

• The average number of steps within the small boxes
• The ratio r=L/s of the long jump length L to size s of the small box

The problem is how to identify the long jumps without introducing an artificial threshold.

Let us first assume that the critical ratio r is prescribed. We can then start at the beginning of the trajectrory (t=0), go through all steps (t->t+1), and keep track of the box size s(t) covered so far (In 1D, the box size s(t)=x_max-x_min is the distance between the most right point x_max and the most left point x_min visited since the last jump). As soon as a single step exceeds s(t) by a factor of r, we have a jump. Then the N_b is known for this special chain and we continue with the next one. When we reach the end of the trajectory, we can compute the average N_b, L and s.

These averages, however, depend on the prescribed parameter r. If r is small, even not so long steps count as jumps and the typical chains will be short. Consequently, the averages of N_b as well as s should become small, and the same holds for L. Reversely, all averages increase with r.

Is there a way to define the "natural value of r", which is inherent to the trajectory ? If the above averages, when plotted against r, would for example show a sudden change of slope at some r_0, this could serve as a characteristic value, similar to the so-called "ellbow criterion" in cluster analysis.

It is also interesting to consider the special case r=1, where a jump must at least exceed the box size (jumping just out of the box): Starting at the beginning of a new chain, s(t) is increasing or staying constant at every step. Once it reaches a certain size, it becomes difficult for any single step to exceed s(t). Thus, (in particular when many successive steps are heading into the same direction, leading to a steady increase of s(t)) the r=1 criterion might never be met. In such a case there is definitely no hopping-and-stalling motion !

Note that for any trajectory of finite length, there is a maximum jump ratio r_max which is barely met a single time, separating the trajectory into two stalling clusters. We can define the hopping-and-stalling criterion to be fulfilled when r_max>1.

Unfortunately, it is not clear how to extend this idea to the 2D case.