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| Formato: | Recurso digital |
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Zenodo
2025
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| Acceso en línea: | https://doi.org/10.5281/zenodo.14897617 |
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- <p>In Part I, we suggested that the Tsallis family of entropy, {1- Sum over i pi power q} / (1-q) may be derived from the idea of assigning a very simple uncertainty expression 1-1/n to a uniform distribution, where n is the number of outcomes. In other words, 1/n is the probability associated with a guessed number for one die toss and 1-1/n is the probability this result is not obtained. </p> <p> We argued that one could consider guessing two numbers associated with two successive die tosses etc and obtained the general result: 1- sum over i pi power q where pi=1/n for a uniform distribution and q-1, the number of numbers guessed or die tosses. We further argued that one may divide this value by q-1 to put all q’s on the same footing so to speak and showed that for q=1, one obtained Sum over i pi ln(pi), Shannon’s entropy (multiplied by -1). In other words, the Tsallis family of entropies {1 - Sum over i pi power q } / (q-1) follows from trying to find a simple uncertainty expression for a uniform distribution. Here we note that if one maximizes this entropy subject to the constraint: Sum over n n pi(n) power q, then pi(n) is no longer a uniform distribution which is already well-known. We suggest that the uniform distribution gives equal probability to all n values of the die. In other words, n is simply a marking and does not map to a physical quantity which would weight different n values differently. The notion of maximization of entropy is to remove as much information as possible (i.e. create as much uncertainty as possible), but a constraint such as Sum over i i pi imposes a second piece of information, namely that i does carry weight in the constraint. This is a physical consideration we argue, not a mathematical one.</p> <p> Thus, maximization of the entropy expression (Tsallis form), obtained from considerations of the uniform distribution, does not yield the uniform distribution under a constraint which introduces information, i.e. special treatment of n values. We further note that the pi ln(pi) Shannon’s form which results from q=1, even in the uniform distribution scenario, is specially linked to the notion of n being quantity because if this is the case, then n1+n2 = n total, and ln(p1) + ln(p2) = ln(p1+p2) implies that p(n) = Cexp(-n/T) for p(n) to drop with n.</p> <p> </p>