[2] Interpretation of the directional correlation function

In post [1] on random walks with directional persistence, we have defined the correlation function of the direction vectors as

C_{mn}=C_{m-n}=\left\langle \vec{e}_m  \vec{e}_n \right\rangle = \left\langle \cos(\phi_m-\phi_n) \right\rangle,

where the brackets denote the ensemble average. We note that

 C_1 = \left\langle \cos(\phi_{m+1}-\phi_m) \right\rangle. 

The quantity

 \Delta \Phi_1=\phi_{m+1}-\phi_m 

is the so-called turning angle, i.e. the angle between the directions of two successive moves of the random walker. Therefore, the value of correlation function at lagtime 1 is just the average of the cosine of the turning angle. If the probability distribution of the turning angle is denoted by

 P(\Delta \Phi_1), 

we thus have

 C_1 = \int_{-\pi}^{\pi} d\Delta \Phi_1 \;P(\Delta \Phi_1) \; \cos(\Delta \Phi_1).  

In earlier publications, researchers have defined an index of persistence by

 p_{\Phi_1} = 2 \left( \int_{-\pi/2}^{\pi/2} d\Delta \Phi_1 \;P(\Delta \Phi_1)\right) -1. 

This can be rewritten in the form

  p_{\Phi_1} = \int_{-\pi}^{\pi} d\Delta \Phi_1 \;P(\Delta \Phi_1)\; g(\Delta \Phi_1),

where the weight function g() has the constant value +1 in the central interval [-pi/2...+pi/2] and the value -1 in the remaining intervals [-pi...-pi/2] and [+pi/2..+pi]. It is clear that this weight function can be equally well be replaced by the smooth cosine function above. Therefore, we learn that the index of persistence is basically the value of the directional correlation function at lagtime 1:

 p_{\Phi_1} \approx C_1.

In an analogous way, the correlation function at higher lagtimes N>1 is just the index of persistence at the corresponding lagtime N:

 p_{\Phi_N} \approx C_N, 

provided that the turning angles are properly defined as

 \Delta \Phi_N = \phi_{m+N}-\phi_m.