C Appendix
Researchers with a professional background that is different from (statistical)
physics cannot easily spot the correspondence between the two basic equa-
tions that were crucial in the formulation of Tsallis’ entropy: The standard
Boltzmann-Gibbs-Shannon (BGS) entropy formula and the generalized nonex-
tensive entropy formula. It is therefore our intention here to mathematically
clarify this relationship. More specifically, it will be shown that the Tsallis’
1 1~V Pi
entropy Sq = —q—i i (q ∈ R) reduces to BGS entropy SBGS = — ɪɪiPi lnPi
as the nonextensive entropic index q approaches unity. The proof is based on
(Bernoulli-)L’HOpital’s rule.
The rule named after the French mathematician Guillaume de l’Hopital em-
ploys derivatives to calculate limits with indeterminate forms. In this sense,
using this rule, one can convert an indeterminate form (e.g. 0 or ∞) into a
determinate form with an easy computation of the limit.
Let
Sq =
1 — ∑i Pq
q - 1
When q → 1, the numerator of Sq tends to 1 — i Pi = 0 ( i Pi = 1). Since
the denominator also tends to 0, Sq has the indeterminate behavior 0 as q → 1.
Therefore, L'H^pital's Rule can be applied to the limit limq→1 Sq:
lim Sq = lim
q→1 q q→1
(1—⅛qτ
(q — 1)'
(1)
where ' indicates the derivative with respect to q, i.e. ' = dqq. Since we can
differentiate term by term, we obtain
б—∑ ι>q∖ '=(i)'—∑ (pq )'=
— Piq ln Pi
i
and
(q — 1)' =(q)' — (1)' = 1.
We use above the following differentiation rules: (1)' = 0, (q)' = 1, and the rule
for differentiating the exponential function aq (the base a is a constant and the
variable q is in the exponent), which reads (aq)' = aqlna (in our case a = pi).
Therefore, from (1) we get
lim Sq = lim ∑ipi lnpi = — ∑pi lnpi.
q→1 q→1 1