Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given 
by $$H(\mathcal{A},m)=-\sum_{i=0}^mp_i\log_2p_i$$  Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution $m\rightarrow\infty$ and $H(\mathcal{A})$ be limiting entropy.

$H(\mathcal{A})$ has interesting properties. Example: discussion in http://mathoverflow.net/questions/191678/limiting-entropy-of-deterministic-sequences-1 shows following holds if $k>0$:


$$a_{i+1}=a_i+\theta(\log^ka_i)\implies H(\mathcal{A})\rightarrow\infty$$
$$a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})\implies H(\mathcal{A})<\infty$$

Is there a physical interpretation of application of Shannon entropy in this situation?