自避行走：修订间差异

删除的内容添加的内容

行内

2020年2月10日 (一) 04:29的版本

在数学中，自避行走（简称：SAW，Self-Avoiding Walk）是一种重要的模型，在数学上与简单随机行走有极大不同。由于已行走的路径不能再次被占据，SAW不再是一种马尔可夫链。因此，一般来说,涉及到SAW的问题往往很难得出严格的解析表达式。然而，SAW模型在物理学、化学、生物学中有广泛的应用。例如在材料科学和生命科学中十分重要的高分子就可以用SAW作为一种基本模型。当一条高分子链的一个端基吸附在某一固体壁上，就可以用端点附壁的随机行走来表示。若高分子链采用SAW作为模型，其行为的预示目前仅限于采用计算机“实验”的方法。本工作在新的计算机“实验”数据基础上，提出了消除奇偶效应的方法，得到了一些新结果。链接=https://fa.wikipedia.org/wiki/%D9%BE%D8%B1%D9%88%D9%86%D8%AF%D9%87:SAWwiki.jpg|缩略图|这是自避行走链接=https://fa.wikipedia.org/wiki/%D9%BE%D8%B1%D9%88%D9%86%D8%AF%D9%87:NSAW.jpg|缩略图|这不是自避行走

应用

溶剂和聚合物
蛋白质
纽结理论
随机漫步
保罗*弗洛里学了化学中的自避行走。^[1]
网络理论^[2]
Gompertz distribution^[3]
Erdos-Renyi graph
有数学家认为自避行走的scaling limit是一个 $κ = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}8/3$ 的Schramm-Loewner演变。

介绍

自避行走是一个分形。^[4]^[5] 例如，在 $d = 2$ ，其分形维数是4/3；在 $d = 3$ ，分形维数是5/3；在 $d \geq 4$ ，分形维数是 $2$ （upper critical dimension）。^[6]

没有已知的公式来计算给格子的SAW数。^[7]^[8]

$m \times n$ 矩形点阵只有

{m+n \choose m,n}

个SAW。

也参看

Critical Phenomena
Universality (dynamical systems)
随机走

参考文献

^ P. Flory. Principles of Polymer Chemistry. Cornell University Press. 1953: 672. ISBN 9780801401343.
^ Carlos P. Herrero. Self-avoiding walks on scale-free networks. Phys. Rev. E. 2005, 71 (3): 1728. Bibcode:2005PhRvE..71a6103H. PMID 15697654. arXiv:cond-mat/0412658 . doi:10.1103/PhysRevE.71.016103.
^ Tishby, I.; Biham, O.; Katzav, E. The distribution of path lengths of self avoiding walks on Erdős–Rényi networks. Journal of Physics A: Mathematical and Theoretical. 2016, 49 (28): 285002. Bibcode:2016JPhA...49B5002T. arXiv:1603.06613 . doi:10.1088/1751-8113/49/28/285002.
^ S. Havlin, D. Ben-Avraham. New approach to self-avoiding walks as a critical phenomenon. J. Phys. A. 1982, 15 (6): L321–L328. Bibcode:1982JPhA...15L.321H. doi:10.1088/0305-4470/15/6/013.
^ S. Havlin, D. Ben-Avraham. Theoretical and numerical study of fractal dimensionality in self-avoiding walks. Phys. Rev. A. 1982, 26 (3): 1728–1734. Bibcode:1982PhRvA..26.1728H. doi:10.1103/PhysRevA.26.1728.
^ A. Bucksch, G. Turk, J.S. Weitz. The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion. PLOS ONE. 2014, 9 (1): e85585. Bibcode:2014PLoSO...985585B. PMC 3899046 . PMID 24465607. arXiv:1304.3521 . doi:10.1371/journal.pone.0085585.
^ Hayes B. How to Avoid Yourself (PDF). American Scientist. Jul–Aug 1998, 86 (4): 314. doi:10.1511/1998.31.3301.
^ Liśkiewicz M; Ogihara M; Toda S. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science. July 2003, 304 (1–3): 129–56. doi:10.1016/S0304-3975(03)00080-X.

阅读

Madras, N.; Slade, G. The Self-Avoiding Walk. Birkhäuser. 1996. ISBN 978-0-8176-3891-7.
Lawler, G. F. Intersections of Random Walks. Birkhäuser. 1991. ISBN 978-0-8176-3892-4.
Madras, N.; Sokal, A. D. The pivot algorithm – A highly efficient Monte-Carlo method for the self-avoiding walk. Journal of Statistical Physics. 1988, 50 (1–2): 109–186. Bibcode:1988JSP....50..109M. doi:10.1007/bf01022990.
Fisher, M. E. Shape of a self-avoiding walk or polymer chain. Journal of Chemical Physics. 1966, 44 (2): 616–622. Bibcode:1966JChPh..44..616F. doi:10.1063/1.1726734.

[1] P. Flory. Principles of Polymer Chemistry. Cornell University Press. 1953: 672. ISBN 9780801401343.

[2] Carlos P. Herrero. Self-avoiding walks on scale-free networks. Phys. Rev. E. 2005, 71 (3): 1728. Bibcode:2005PhRvE..71a6103H. PMID 15697654. arXiv:cond-mat/0412658 . doi:10.1103/PhysRevE.71.016103.

[3] Tishby, I.; Biham, O.; Katzav, E. The distribution of path lengths of self avoiding walks on Erdős–Rényi networks. Journal of Physics A: Mathematical and Theoretical. 2016, 49 (28): 285002. Bibcode:2016JPhA...49B5002T. arXiv:1603.06613 . doi:10.1088/1751-8113/49/28/285002.

[4] S. Havlin, D. Ben-Avraham. New approach to self-avoiding walks as a critical phenomenon. J. Phys. A. 1982, 15 (6): L321–L328. Bibcode:1982JPhA...15L.321H. doi:10.1088/0305-4470/15/6/013.

[5] S. Havlin, D. Ben-Avraham. Theoretical and numerical study of fractal dimensionality in self-avoiding walks. Phys. Rev. A. 1982, 26 (3): 1728–1734. Bibcode:1982PhRvA..26.1728H. doi:10.1103/PhysRevA.26.1728.

[6] A. Bucksch, G. Turk, J.S. Weitz. The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion. PLOS ONE. 2014, 9 (1): e85585. Bibcode:2014PLoSO...985585B. PMC 3899046 . PMID 24465607. arXiv:1304.3521 . doi:10.1371/journal.pone.0085585.

[7] Hayes B. How to Avoid Yourself (PDF). American Scientist. Jul–Aug 1998, 86 (4): 314. doi:10.1511/1998.31.3301.

[8] Liśkiewicz M; Ogihara M; Toda S. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science. July 2003, 304 (1–3): 129–56. doi:10.1016/S0304-3975(03)00080-X.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]