Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Istituto dei Sistemi Complessi (ISC-CNR) c/o Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Phys Rev E. 2017 May;95(5-1):052112. doi: 10.1103/PhysRevE.95.052112. Epub 2017 May 8.
The intriguing and still open question concerning the composition law of κ-entropy S_{κ}(f)=1/2κ∑{i}(f{i}^{1-κ}-f_{i}^{1+κ}) with 0<κ<1 and ∑{i}f{i}=1 is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution f={f_{ij}}, made up of two statistically independent subsystems, described through the probability distributions p={p_{i}} and q={q_{j}}, respectively, with f_{ij}=p_{i}q_{j}, the joint entropy S_{κ}(pq) can be obtained starting from the S_{κ}(p) and S_{κ}(q) entropies, and additionally from the entropic functionals S_{κ}(p/e_{κ}) and S_{κ}(q/e_{κ}),e_{κ} being the κ-Napier number. The composition law of the κ-entropy is given in closed form and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the κ→0 limit.
这里重新考虑并解决了关于 κ 熵 S_{κ}(f)=1/2κ∑{i}(f{i}^{1-κ}-f_{i}^{1+κ})，0<κ<1，∑{i}f{i}=1 的组成定律的这个有趣且仍未解决的问题。结果表明，对于由概率分布 f={f_{ij}}描述的统计系统，该系统由两个统计上独立的子系统组成，分别通过概率分布 p={p_{i}}和 q={q_{j}}描述，其中 f_{ij}=p_{i}q_{j}，从 S_{κ}(p)和 S_{κ}(q)熵以及从熵泛函 S_{κ}(p/e_{κ})和 S_{κ}(q/e_{κ})可以得到联合 κ 熵 S_{κ}(pq)，e_{κ}是 κ-Napier 数。κ 熵的组成定律以封闭形式给出，并且作为在 κ→0 极限下恢复的 Boltzmann-Shannon 熵的普通可加性定律的一个参数推广出现。
