Is there a meaning to $\mathrm{e}^{H(p_{i})}$ or $2^{H(p_{i})}$? In my research I find an equation featuring the "exponential entropy" term $\mathrm{e}^{H(p_{i})}$ and I wonder if it has a specific meaning. I have only found rare references to that term (usually in terms of dispersion or "spread of the distribution") so I'm looking for more insights. I work with natural logarithms and in my case the entropy is Shannon's: $H(p_{i})=-\sum{ p_{i}\ln p_{i}}$... My question is: what is $\mathrm{e}^{H(p_{i})}$ ?
Note: I assume that the same question would arise if I were to work in log-base 2... So  is there a meaning to $2^{H(p_{i})}$ when entropy is now defined by $H(p_{i})=-\sum{p_{i}\log_{2} p_{i}}$ ?
 A: "Spread" is a good name. First, notice that the value is indendent of the base of the log:
$$2^{\sum -p_i\log_2(p_i)}=\exp\left(\log(2)*-\sum p_i\frac{\log(p_i)}{\log(2)}\right)=\exp\left(-\sum p_i\log(p_i)\right)$$
I call $p=(p_i)$ a distribution as in probability theory. Entropy is also called quantity of information.
For discrete entropy, $exp(H)$ is the essential number of possible values (possible values of $i$) of the distribution. It is the number of possible values of a (discrete) uniform distribution that would have the same entropy. $p$ is like an histogram. If you want to draw an histogram for a disbribution containing the same information but with bars having constant height, you will draw $exp(H)$ bars. Of course, $exp(H)$ is not always an integer, but that's the idea.
For example, consider a signal sending messages, each message is made of one character in the infinite alphabet 1,2,3,... Character $i$ has probability $p_i=\frac{1}{2^i}$ (it is a geometric distribution). $exp(H)=4$. Hence, the minimum size of an alphabet required to send the same amount of information is $4$.
For differential entropy (relative to the Lebesgue measure for example), $exp(H)$ is the essential volume. $exp(H)$ is the volume of the support of a uniform distribution having the same entropy. Think of a distribution as a "fuzzy" object with shades of grey. $exp(H)$ is somehow the object's essential volume. Mathematically, it generalizes the notion of measure of a set to a distribution.
I find it useful to understand Liouville's theorem. Liouville theorems says "the Lebesgues mesure in the phase space is preserved by motion": assuming the system is in a certain subset of the phase space at time $0$, it will be in a subset with the same Lebesgue volume at time $t$. It is natural to generalize this to (exp of) entropy as a generalized volume: if the distribution at time $0$ has a certain (exp of) entropy, it has the same (exp of) entropy at time $t$. It's easy to prove. This can be the first step for proving the second law.
