We know that the Shannon entropy $H(P)=- k_{\mathrm{B}}\sum_i p_i \ln p_i$ is mostly the entropy of the thermodynamic systems. Does the Renyi measure $H_{\alpha}(P)=\frac{1}{1-\alpha}\log \sum p_i^{\alpha}$, $\alpha\neq 1$ also actually measure the entropy of some physical system?

  • $\begingroup$ This question is almost a duplicate from the unanswered question physics.stackexchange.com/questions/16804/… $\endgroup$ – Frédéric Grosshans Jan 5 '12 at 13:26
  • $\begingroup$ The answer to the second question makes a better answer to this question. The second question asks about the physical systems which obey Renyi entropy maximization, and this is not obvious from the answer there. $\endgroup$ – Ron Maimon Jan 5 '12 at 17:36
  • $\begingroup$ This is not really a duplicate of my previous question. There I was wondering whether there was some physical reason for maximizing Renyi entropy. Here I am asking whether the Renyi measure measures the entropy of a system. $\endgroup$ – Ashok Jan 6 '12 at 12:28

The blog post (also arXiv:1102.2098) in your another question already gives pretty good view of it. Recently, I have come across a paper which has talked a bit about an interpretation of Renyi entropy for a physical system and I think it might be interesting for you, though not answering your question directly.

The paper (arXiv:1006.1605) study the scaling behavior of Renyi entropy $R_n(T,L)$ of a ring of length $L$ in an infinite cylinder of Ising model. The probability distribution $p_i$ considered is for each $2^L$ spin configurations along the ring as shown in Fig 1.

Infinite cylindrical Ising model Fig. 1

If my understanding is correct, the Renyi entropy of a ring in this particular system corresponds to the free energy (and so the entropy) of different systems that they called 'Ising book' as shown in Fig. 2. This intrepertation is valid for each of Renyi parameter $n=\frac{1}{2}, 1, \frac{3}{2}, 2, ...$ See cited text below.

Ising book (multiple semi-infinite Ising model glue together) Fig. 2

(Color online) $2n$ Ising models glued together at their boundary ("Ising book"). In our case, each "page" has periodic boundary conditions along the horizontal axis and is semi-infinite in the vertical direction.

We now consider the effect of changing the Rényi parameter $n$. When $2n$ is an integer, $R_n$ has an interpretation in terms of the free energy of $k = 2n$ semi-infinite Ising models which are "glued" together at their boundary.

Using the transfer matrix point of view, it is simple to see that $p_i^{k/2}$ is (proportional to) the probability to observe the spin configuration $i$ on a circle along which $k$ Ising models (defined on semi-infinite cylinder) are forced to coincide.

  • $\begingroup$ Thanks hwlau: It looks very useful. I will take sometime to understand it better. $\endgroup$ – Ashok Dec 27 '12 at 12:22

As you say, the "Shannon Entropy" (which was known to Boltzmann, many years before), is the general entropy definition for any equilibrium thermodynamic system. Entropies, the Renyi, the Tsallis, and other similar ones have been set up to handle non-equilibrium situations and some curious equilibrium ones as well, such as the one described in the previous answer.


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