Section for Science of Complex Systems, CeMDS, Medical University of Vienna, Vienna, Austria.
Complexity Science Hub Vienna, Vienna, Austria.
PLoS One. 2023 Sep 6;18(9):e0290695. doi: 10.1371/journal.pone.0290695. eCollection 2023.
Complex systems with strong correlations and fat-tailed distribution functions have been argued to be incompatible with the Boltzmann-Gibbs entropy framework and alternatives, so-called generalised entropies, were proposed and studied. Here we show, that this perceived incompatibility is actually a misconception. For a broad class of processes, Boltzmann entropy -the log multiplicity- remains the valid entropy concept. However, for non-i.i.d. processes, Boltzmann entropy is not of Shannon form, -k∑ipi log pi, but takes the shape of generalised entropies. We derive this result for all processes that can be asymptotically mapped to adjoint representations reversibly where processes are i.i.d. In these representations the information production is given by the Shannon entropy. Over the original sampling space this yields functionals identical to generalised entropies. The problem of constructing adequate context-sensitive entropy functionals therefore can be translated into the much simpler problem of finding adjoint representations. The method provides a comprehensive framework for a statistical physics of strongly correlated systems and complex processes.
具有强相关性和长尾分布函数的复杂系统被认为与玻尔兹曼-吉布斯熵框架和替代的所谓广义熵不兼容，因此提出并研究了这些替代的广义熵。在这里，我们表明，这种被认为的不兼容性实际上是一种误解。对于广泛的一类过程，玻尔兹曼熵——对数多重性——仍然是有效的熵概念。然而，对于非独立同分布过程，玻尔兹曼熵不是香农形式的-k∑ipi log pi，而是广义熵的形式。我们推导出所有可以渐近地可逆映射到伴随表示的过程的这个结果，其中过程是独立同分布的。在这些表示中，信息产生由香农熵给出。在原始采样空间上，这产生与广义熵相同的泛函。因此，构建适当的上下文敏感熵泛函的问题可以转化为寻找伴随表示的简单得多的问题。该方法为强相关性系统和复杂过程的统计物理提供了一个全面的框架。
