Application of fractals in condensed-matter physics

Basic information about the theory of mono- and multifractal sets is presented. Geometric and thermodynamic descriptions are developed. The geometric picture is presented on the basis of the simplest examples of the Koch and Cantor fractal sets. An ultrametric space, representing the metric of a fractal set, is introduced on the basis of Cayley's hierarchical tree. The spectral characteristics of a multifractal formation are described. Attention is focused mainly on the application of the fractal concept for a thermodynamic system with partial memory loss, turbulent fluid flow, hierarchically coordinated set of statistical ensembles, Anderson's transition, and incommensurable and quasicrystalline structures.

PACS: 64.60.Ak, 47.53.+n, 61.43.Hv, 61.44.−n (all)
DOI: 10.1070/PU1993v036n12ABEH002208
URL: https://ufn.ru/en/articles/1993/12/a/
Citation: Olemskoi A I, Flat A Ya "Application of fractals in condensed-matter physics" Phys. Usp. 36 (12) 1087–1128 (1993)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Олемской А И, Флат А Я «Использование концепции фрактала в физике конденсированной среды» УФН 163 (12) 1–50 (1993); DOI: 10.3367/UFNr.0163.199312a.0001

References (60) Cited by (108) Similar articles (20) ↓

V.V. Zosimov, L.M. Lyamshev “Fractals in wave processes” 38 347–384 (1995)
V.V. Uchaikin “Self-similar anomalous diffusion and Levy-stable laws” 46 821–849 (2003)
A.I. Olemskoi, I.A. Sklyar “Evolution of the defect structure of a solid during plastic deformation” 35 (6) 455–480 (1992)
L.M. Zelenyi, A.V. Milovanov “Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics” 47 749–788 (2004)
M.I. Rabinovich, M.M. Sushchik “The regular and chaotic dynamics of structures in fluid flows” 33 (1) 1–35 (1990)
V.P. Budaev, S.P. Savin, L.M. Zelenyi “Investigation of intermittency and generalized self-similarity of turbulent boundary layers in laboratory and magnetospheric plasmas: towards a quantitative definition of plasma transport features” 54 875–918 (2011)
A.I. Olemskoi “Theory of stochastic systems with singular multiplicative noise” 41 269–301 (1998)
A.I. Olemskoi, I.V. Koplyk “The theory of spatiotemporal evolution of nonequilibrium thermodynamic systems” 38 1061–1097 (1995)
I.M. Sokolov “Dimensionalities and other geometric critical exponents in percolation theory” 29 924–945 (1986)
A.V. Batunin “Fractal analysis and Feigenbaum universality in hadron physics” 38 609–622 (1995)
E.P. Emets, A.E. Novoselova, P.P. Poluektov “In situ determination of the fractal dimensions of aerosol particles” 37 881–887 (1994)
N.M. Astaf’eva “Wavelet analysis: basic theory and some applications” 39 1085–1108 (1996)
E.F. Mikhailov, S.S. Vlasenko “The generation of fractal structures in gaseous phase” 38 253–271 (1995)
V.I. Alkhimov “Excluded volume effect in statistics of self-avoiding walks” 37 527–561 (1994)
G.V. Kozlov “Structure and properties of particulate-filled polymer nanocomposites” 58 33–60 (2015)
B.M. Smirnov “Fractal clusters” 29 481–505 (1986)
V.S. Dotsenko “Physics of the spin-glass state” 36 (6) 455–485 (1993)
V.S. Dotsenko “Critical phenomena and quenched disorder” 38 457–496 (1995)
I.R. Stakhovsky “Scale invariance of shallow seismicity and the prognostic signs of earthquakes” 60 472–489 (2017)
V.I. Alkhimov “The self-avoiding walk problem” 34 (9) 804–816 (1991)

The list is formed automatically.