Interpretation of percolation in terms of infinity computations
Interpretation of percolation in terms of infinity computations
复制标题
用无穷大计算解释渗滤
DOI：
10.1016/j.amc.2011.11.044
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
M. Hayakawa
中科院分区：
文献类型：
--
作者：
D. Iudin;Y. Sergeyev;Y. Sergeyev;M. Hayakawa
本文采用一种新的计算方法来考虑与渗流理论相关的一些传统模型，这种方法不使用康托尔的思想，而是按照“部分小于整体”的原则来描述无限和无穷小的数。它提供了一种可能，通过使用一种新的计算机——无限计算机——在数值上处理有限、无限和无穷小的量。这种新方法与康托并不矛盾。相反，它可以被看作是他关于不同无限数存在的深刻思想在更实用的方式上的演变。利用新的计算工具研究了现场渗流和梯度渗流。已经证明，在无限系统中，相变点并不像传统方法那样是一个点。根据新的算法，它表现为一个临界区间，而不是一个临界点。根据我们使用的“显微镜”，这个区间可以被看作是有限的、无限的和无穷小的短区间。利用新的方法，我们观察到在渗透阈值附近有许多不同的无限簇，而不是传统考虑中出现的一个无限簇。
In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The part is less than the whole’. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a computer – the Infinity Computer – introduced recently in [18]. The new approach does not contradict Cantor. In contrast, it can be viewed as an evolution of his deep ideas regarding the existence of different infinite numbers in a more applied way. Site percolation and gradient percolation have been studied by applying the new computational tools. It has been established that in an infinite system the phase transition point is not really a point as with respect of traditional approach. In light of new arithmetic it appears as a critical interval, rather than a critical point. Depending on “microscope” we use this interval could be regarded as finite, infinite and infinitesimal short interval. Using new approach we observed that in vicinity of percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional consideration.