Motivation for maximum Renyi/Tsallis entropy

The Conditional limit theorem of Van Campenhout and Cover gives a physical reason for maximizing (Shannon) entropy. Nowadays, in statistical mechanics, people talk about maximum Renyi/Tsallis entropy distributions. Is it just because these distributions are heavy tailed?

Is there any motivation (or physical significance) for maximizing Renyi/Tsallis entropies?

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Warning: I do not work in statistical physics, and I therefore do not now if the following has any link with the reason people use Rényi entropies in statistical physics.

John C. Baez wrote a paper (arXiv:1102.2098), stating that the Rényi entropy $H_\beta$ is proportinal to the free energy at temperature defined by $\beta=T_0/T$. This paper is nicely explained on his blog.

The free energy of a system kept at a constant volume and temperature is minimal, and this corresponds, when $T>T_0$ to a maximum Rényi entropy $H_{T/T_0}$.

Edited after reading the paper to correct some mistakes.

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Although simple, this is an interesting observation. The question to me is whether the free energy calculated with the phony temperature is actually extractable as energy by doing all the work at the fictitious temperature. Does this even have meaning? If you bring an object at a temperature of T/2 into contact with an object at temperature T, how can you extract the work reversibly without matching the temperature? There might be a crazy coupling which allows this--- for example take two copies of the system and only allow them to interact only when the two copies are in the exact same state. –  Ron Maimon Jan 5 '12 at 14:55
Thanks for the response @Frédéric Grosshans. Later I too came to know about John C. Baez's arxiv paper and his blog. –  Ashok Jan 6 '12 at 12:37
Thanks to @Ron Maimon. I am not a physicist. So I can understand your question to certain extent only. –  Ashok Jan 6 '12 at 12:40