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Questions:

  1. If I am given a system, which I might have to describe using a generalized entropy, like the "$q$-deformed" Tsallis entropy, do I have to fit $q$ from experiment or might I know it beforehand?

  2. How do I know the parameter $q$ and/or how can I possibly obtain the degree of non-extensitivity via experiment?

  3. How can I measure the entropy of a part of the system, if the system is non-extensive?


Edit

After some browsing I think the answer might be related to the fact, that for $q$-deformed entropy, the most probable distribution is not the Gaussian, but seems to be the $q$-deformed Gaussian.

Then I played around a bit: [broken link].

Maybe one applies such an entropy concept if one comes across a distribution of such type, but that's only a guess. And I don't see why one would/could conclude non-extensitivity from a distribution?!

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    $\begingroup$ Can you give one example? I am struggling to find a single case which is described by this thing. $\endgroup$
    – Ron Maimon
    Oct 10 '11 at 6:04
  • $\begingroup$ The kappa velocity distribution function is related to the Tsallis q-distribution and has been heavily researched in space plasma physics. $\endgroup$ Mar 7 '16 at 18:00
  • $\begingroup$ There are some papers that derive a value for $q$ in some model or another. If you're interested in the applications of Tsallis $q$-statistics to a particular system, it's worth reading the existing research on it to see whether any such derivation exists. Of course, even if it does empirical data can test that. $\endgroup$
    – J.G.
    May 15 '18 at 20:44
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Tsallis entropy is mathematically defined, and there is no clear relation to physical characteristics of the systems. It is proposed to solve the problem of scaling non-extensive systems. In an extensive system, properties like energy and entropy scale up linearly with the number of constituents. For non-extensive systems, this property is broken, but the scaling could occur in many ways in theory. Tsallis form allows to have a parametric form for scaling these type of systems, with the parameter q, which cannot be determined a priori from some property. Even in theoretical calculations of hypothetical systems, q has to be determined from fitting the scaling to larger numbers.

Theoretically you can hypothesize q beforehand, but that does not give you precise description of your system composition or dynamics. Also, knowing the microscopic description of the system is not sufficient to derive q, unless the scaling of entropy versus number of particles is determined by other means than Tsallis form.

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You have to find it from the experiment. Look at these two papers

  • Simulation: This paper fits a q-Tsallis distribution to the energy states of multi-particle electron-hole-photon system

  • Experiment: This one fits a q-Tsallis distribution to model a meta-stable state in plasma dynamics

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