Anomalous diffusion in time transformation of voter model

Last time we have discussed the first non-linear transformation of the noisy voter model considered by me and R. Kazakevicius [1]. This time let us talk about the second non-linear transformation: transformation of the time scale.


Once again we consider the standard noisy voter model (which is equivalent to Kirman's ant colony model we have considered previously on Physics of Risk), but this time we will build upon so-called \( \tau \)-scenario, which introduces additional feedback of the macroscopic state on the rate of micro-dynamics. This scenario was also introduced when we modeled long-range memory in the financial market scenario (also see [2]):

\begin{equation} \tau(x) = x^{1-2 \eta}. \end{equation}

Yet unlike previously we will not consider dynamics of \( y \), but we will stick with dynamics of \( x \). With this \( \tau \)-scenario in place stochastic differential equation for \( x \) in the limit of small \( x \) behaves as a transformation of the Bessel process. Thus we can once again use the results obtained for the Bessel process to find the expressions for the temporal evolution of the moments of \( x \). The most important thing these expressions contain is the power-law dependence of the moments on time:

\begin{equation} \langle x^k \rangle \sim t^{\frac{k}{2 (1 - \eta)}} . \end{equation}

Thus the mean will scale as power-law with exponent \( \frac{1}{2 (1 - \eta)} \), while the variance will scale with exponent \( \frac{1}{1 - \eta} \). Note that these theoretical results work well only for very small \( x \).

Interactive app

This app is identical to the one from the previous post. The only difference is here is the source of non-linearity and the related non-linearity parameter (\( \eta \) here). Normal diffusion is observed for \( \eta = 0 \), smaller \( \eta \) will result in sub-diffusion, while larger - in super diffusion.