For a probability distribution $P$, Renyi fractal dimension is defined as

$$D_q = \lim_{\epsilon\rightarrow 0} \frac{R_q(P_\epsilon)}{\log(1/\epsilon)},$$
where $R_q$ is Renyi entropy of order $q$ and $P_\epsilon$ is the coarse-grained probability distribution (i.e. put in boxes of linear size $\epsilon$).

The question is if there are any phenomena, for which using non-trivial $q$ (i.e. $q\neq0,1,2,\infty$) is beneficial or naturally preferred?

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