Different entropies

Suppose we have a random variable \( X \), which may take on \( n \) distinct values \( 0, 1, 2, \dots n-1 \) with probability \( p_i \) for taking value \( i \). If we don't care about the ordering of the values, we may as well assume they are sorted in descending order by probability: \( p_i \gt p_{i+1} \). The Shannon entropy of this system, \[ E = -\sum p_i \log_2 p_i, \] gives a lower bound on the expected path length of a binary code for \( X \), that is, a binary tree having \( X \)'s values at its leaves (in any order).

But there is another formula we can use that defines an entropy based on ranks:

It turns out that \( E \gt R \gt E/2 \). The first inequality follows trivially from the fact that \( 1/(i+1) \gt p_i \); the second, from the entropy lower bound on path length together with the existence of a binary code with expected path length bounded by \( 2R \):

The same inequalities also hold with the same proof if one takes floors of the logs in the definition of \( R \). I imagine that this must all be well known to the statistical physicists: the rank entropy must have some kind of name, and the inequality between rank entropy and Shannon entropy must be in the literature somewhere and probably also has a name. But, not being a statistical physicist myself, I'm having trouble finding any of this literature. Google searches for combinations of "rank", "entropy", and "inequality" lead to lots of interesting material but not (yet) anything describing this. Is anyone else familiar with this kind of rank entropy, and, if so, where do I find out about it?

ETA: Why I care about this.

Comments: