论文标题
非加工熵的时间演变：逻辑图
Time evolution of nonadditive entropies: The logistic map
论文作者
论文摘要
由于热力学的第二个原理，各种物理环境下熵对各种系统的时间依赖总是会产生兴趣。 logistic映射$ x_ {t+1} = 1-a x_t^2 \ in [-1,1] \;（[in [0,2]）$既不大，因为它只有一个自由度，也没有封闭，因为它是散落的。尽管如此，它还是展示了其自然熵的特殊时间演变，这是添加剂的boltzmann-gibbs-shannon One，$ s_ {bg} = - \ sum_ {i = 1}^w p_i \ ln p_i \ ln p_i $ a $ a $ a $ a $ a $ a $ a $ a $ a lyapunov exportion and nonondive and s $ s $ \ frac {1- \ sum_ {i = 1}^w p_i^q} {q-1} $，$ q = 0.2445 \ dots $在混乱的边缘，lyapunov指数消失，$ w $是相位空间分区的窗口的数量。我们从数字上表明，在增加时间空间平均熵的时间内，在所有情况下，均高于其固定态值。但是，当$ w \ to \ infty $时，对于最混乱的案例（$ a = 2 $）而逐渐消失，而在显着的对比中，它似乎在Feigenbaum Point（$ a = 1.4011 \ dots $）单调差异。因此，固定状态熵值是从{\ it上方}中实现的，而不是从{\ it下方}中实现的，因为它可能是先验的预期。这些结果提出了一个问题，即对于第二个主要有效性可能是必要但不够的，通常的要求 - 大，封闭和对通用初始条件 - 对于通用的初始条件而言是否足够。
Due to the second principle of thermodynamics, the time dependence of entropy for all kinds of systems under all kinds of physical circumstances always thrives interest. The logistic map $x_{t+1}=1-a x_t^2 \in [-1,1]\;(a\in [0,2])$ is neither large, since it has only one degree of freedom, nor closed, since it is dissipative. It exhibits, nevertheless, a peculiar time evolution of its natural entropy, which is the additive Boltzmann-Gibbs-Shannon one, $S_{BG}=-\sum_{i=1}^W p_i \ln p_i$, for all values of $a$ for which the Lyapunov exponent is positive, and the nonadditive one $S_q= \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ with $q=0.2445\dots$ at the edge of chaos, where the Lyapunov exponent vanishes, $W$ being the number of windows of the phase space partition. We numerically show that, for increasing time, the phase-space-averaged entropy overshoots above its stationary-state value in all cases. However, when $W\to\infty$, the overshooting gradually disappears for the most chaotic case ($a=2$), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point ($a=1.4011\dots$). Consequently, the stationary-state entropy value is achieved from {\it above}, instead of from {\it below}, as it could have been a priori expected. These results raise the question whether the usual requirements -- large, closed, and for generic initial conditions -- for the second principle validity might be necessary but not sufficient.
