A combinatorial interpretation for Tsallis 2-entropy
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While Shannon entropy is related to the growth rate of multinomial coefficients, we show that Tsallis 2-entropy is connected to their q-version; when q is a prime power, these coefficients count the number of flags in F_q^n with prescribed length and dimensions (F_q denotes the field of order q). In particular, the q-binomial coefficients count vector subspaces of given dimension. We obtain this way a combinatorial explanation for non-additivity. We show that statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the q-binomial distribution, that generates at time n a vector subspace of F_q^n. The concentration of measure on certain "typical subspaces" allows us to extend the asymptotic equipartition property to this setting. We discuss the applications to information theory, particularly to source coding.
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