Abstract
This chapter explores the notion of “dimension” of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the generalization to multifractals, and discrete and continuous scale invariance with the emergence of complex dimensions. The idea of renormalization group flow equations can be introduced in this framework, to show how the power laws determined by dimensional analysis (engineering dimensions) get modified by extra anomalous dimensions. As an example of the RG ow equation, the scaling of conductance by disorder in the context of localization is used. A few technicalities, including the connection between entropy and fractal dimension, can be found in the appendices.
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References
[1] Benoit B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Company, 1982).
[2] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, (Wiley, 3rd Edition, 2014).
[3] M. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions Geometry and Spectra of Fractal Strings, (Springer, New York, 2013).
[4] Michel Hans, Am. J. Phys. 51, 694 (1983).
[5] For a list of fractals with fractal dimensions, see https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
[6] See Sec 9.14 of the online notes, J.Preskill, http://www.theory.caltech.edu/~preskill/ph219/topological.pdf
[7] Robert B. Griffiths and Miron Kaufman, Phys. Rev. B 26, 5022 (1982).
[8] S. M. Bhattacharjee, “Critical Phenomena: An Introduction from a modern perspective” in “Field theoretic methods in condensed matter physics” (TRiPS, Hindusthan Publising Agency, Pune, 2001). (cond-mat/0011011).
[9] J. Cardy, Scaling and Renormalization in Statistical Physics, (Cambridge Univ. Press, London, 1996).
[10] John Milnor, Dynamics in One Complex Variable, 3 ed, Hindusthan Book Agency, Pune, India (2012) (Indian edition). For a shorter version, https://arXiv.org/pdf/math/9201272v1.pdf.
[11] D. Ruelle, Ergod. Th. & Dynam. Sys., 2, 99 (1982) (Appendix B).
[12] H. Kröger, Phys. Rept 323, 81 (2000).
[13] J-P. Bouchaud and A. Georges, Phys. Rept. 195, 127 (1990).
[14] Y. Gefen, B. B. Mandelbrot, and A. Aharony, Phys. Rev. Letts 45, 855 (1980); J. Phys A: Math. Gen. 17, 435 (1984); ibid 17, 1277 (1984).
[15] S. Mukherji and S. M. Bhattacharjee, Phys. Rev. E 63, 051103 (2001).
[16] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)
[17] M. Janssen, Int. J. Mod. Phys. B 8, 943 (1994).
[18] For a discussion on definitions of entropy, see, e.g., S. M. Bhattacharjee, “Entropy and perpetual computers”, Physics Teacher. 45, 12 (2003) (condmat/0310332).
[19] D. Sornette, Phys. Rept. 297, 239 (1998).
[20] R. Rammal and G. Toulose, J. Physique Letts. 44, L-13 (1985).
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Bhattacharjee, S.M. (2017). What is dimension?. In: Bhattacharjee, S., Mj, M., Bandyopadhyay, A. (eds) Topology and Condensed Matter Physics. Texts and Readings in Physical Sciences, vol 19. Springer, Singapore. https://doi.org/10.1007/978-981-10-6841-6_10
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DOI: https://doi.org/10.1007/978-981-10-6841-6_10
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