Dobreci Lucian, Rusu Oana, Vasincu Decebal, Jarcău Mihaela, Rusu Cristina Marcela, Gurlui Silviu, Ghizdovat Vlad, Gavrilut Alina, Agop Maricel
Department of Physical and Occupational Therapy, "Vasile Alecsandri" University of Bacau, 600115 Bacau, Romania.
Faculty of Material Science and Engineering, "Gheorghe Asachi" Technical University, 700050 Iasi, Romania.
Entropy (Basel). 2022 Mar 30;24(4):484. doi: 10.3390/e24040484.
Possible implications and consequences of using SL(2R) as invariance groups in the description at any scale resolution of the dynamics of any complex system are analyzed. From this perspective and based on Jaynes' remark (any circumstance left unspecified in the description of any complex system dynamics has the concrete expression in the existence of an invariance group), in the present paper one specifies such unspecified circumstances that result directly from the consideration of the canonical formalism induced by the SL(2R) as invariance group. It follows that both the Hamiltonian function and the Guassian distribution acquire the status of invariant group functions, the parameters that define the Hamiltonian acquire statistical significances based on a principle of maximizing informational energy, the class of statistical hypotheses specific to Gaussians of the same average acts as transitivity manifolds of the group (transitivity manifolds which can be correlated with the multifractal-non-multifractal scale transitions), joint invariant functions induced through SL(2R) groups isomorphism (the SL(2R) variables group, and the SL(2R) parameters group, etc.). For an ensemble of oscillators of the same frequency, the unspecified circumstances return to the ignorance of the amplitude and phase of each of the oscillators, which forces the recourse to a statistical ensemble traversed by the transformations of the Barbilian-type group. Finally, the model is validated based on numerical simulations and experimental results that refer to transient phenomena in ablation plasmas. The novelty of our model resides in the fact that fractalization through stochasticization is imposed through group invariance, situation in which the group's transitivity manifolds can be correlated with the scale resolution.
分析了在任何复杂系统动力学的任意尺度分辨率描述中使用 SL(2R) 作为不变性群的可能影响和后果。从这个角度出发，并基于杰恩斯的评论（在任何复杂系统动力学描述中未明确指定的任何情况，在不变性群的存在中都有具体表现），在本文中，我们明确了由于将 SL(2R) 视为不变性群而直接导致的这种未明确指定的情况。结果表明，哈密顿函数和高斯分布都获得了不变性群函数的地位，定义哈密顿量的参数基于最大化信息能量的原则获得了统计意义，同一平均值的高斯特定的统计假设类别充当该群的可迁流形（可与多分形 - 非多分形尺度转变相关的可迁流形），通过 SL(2R) 群同构诱导的联合不变函数（SL(2R) 变量群和 SL(2R) 参数群等）。对于相同频率的振子系综，未明确指定的情况归结为对每个振子的振幅和相位的无知，这迫使我们求助于由巴尔比里安型群的变换遍历的统计系综。最后，基于涉及烧蚀等离子体瞬态现象的数值模拟和实验结果对该模型进行了验证。我们模型的新颖之处在于，通过随机化进行分形化是通过群不变性实现的，在这种情况下，群的可迁流形可以与尺度分辨率相关联。
