Advantages of q-logarithm representation over q-exponential representation from the sense of scale and shift on nonlinear systems
Advantages of q-logarithm representation over q-exponential representation from the sense of scale and shift on nonlinear systems
复制标题
DOI：
10.1140/epjst/e2020-900196-x
复制
发表时间：
2019-09
期刊：
影响因子：
--
通讯作者：
H. Suyari;H. Matsuzoe;A. M. Scarfone
中科院分区：
文献类型：
--
作者：
H. Suyari;H. Matsuzoe;A. M. Scarfone
观测值的加减运算可以在所有参数的尺度单位相同这一明显和隐含的假设下进行，这一假设甚至对任何非线性系统都有效。本文首先从尺度度量单位的意义上区分指数族和非指数族。在最简单的非线性模型dy / dx=yq中，显示了非线性系统中出现的典型效应，如重标度和移位，并影响观测数据。在此基础上，提出了q-指数和q-对数两种表示形式。前者用于重新缩放，后者用于统一理解，具有固定的缩放单位。作为这些表示的应用，给出了相应的熵和具有固定尺度单位的统一理解的一般概率表达式。对于非线性系统的理论研究，q-对数表示比q-指数表示有显著的优势。
Addition and subtraction of observed values can be computed under the obvious and implicit assumption that the scale unit of measurement should be the same for all arguments, which is valid even for any nonlinear systems. This paper starts with the distinction between exponential and non-exponential family in the sense of the scale unit of measurement. In the simplest nonlinear modeldy∕dx=yq, it is shown how typical effects such as rescaling and shift emerge in the nonlinear systems and affect observed data. Based on the present results, the two representations, namely theq-exponential and theq-logarithm ones, are proposed. The former is for rescaling, the latter for unified understanding with a fixed scale unit. As applications of these representations, the corresponding entropy and the general probability expression for unified understanding with a fixed scale unit are presented. For the theoretical study of nonlinear systems,q-logarithm representation is shown to have significant advantages overq-exponential representation.