“FRACTALIZATION” OF THE SOLID-STATE PHYSICS
Keywords:
nonlinear paradigm, fractal paradigm, fractal approach, Hurst fractal dimension, fractal image, two-dimensional Weierstrass function
Abstract
It has been demonstrated that in modern solid-state physics, in accordance with the nonlinear and systems paradigms formulated by L. F. Chernogor in the late 1980s, many processes in open, nonlinear, dynamical systems are occurred to be very complex, nonlinear, short-time, ultra-wideband, or fractal.
Moreover, from the point of view of the fractal paradigm put forward in the early 2000s by V. V. Yanovsky, fractality is generally considered as one of the fundamental properties of the surrounding world. Therefore, the study of fractal characteristics, in particular, of natural physical processes and objects in the field of solid-state physics, is appeared to be relevant, interesting, useful and promising.
The main stages of the development of the fractal approach in general are briefly discussed. It is pointed that it was occurred to be too hard to formulate a strict and clear mathematical definition of a fractal. The definition of a fractal been formulated by. K. Falconer and been agreed by the most speciallists as the best for practical usage is considered in detail. Main numerical characteristics of a fractal as well as the modern classification of fractals are considered. The general principal differences existing between the mathematical fractals and physical (or natural) ones as well as between the mono-fractals and the multi-fractals are clearly explained. A review of the main methods for estimating the Hurst fractal dimension is provided.
The main existing directions of "fractalization" of modern solid-state physics are highlighted. Relevant examples are given.
It is noted that images with certain fractal properties play an important role in the "fractalization" of solid-state physics. The use of the two-dimensional Weierstrass function is proposed for modeling images with fractal properties. As an example, the modelling of the unidirectional twin structure observed in the YBa2Cu3O7-δ crystal is considered. A comparison between the model image based on one-dimensional Weierstrass function with defined value of the Hurst fractal dimension and real experimental one is demonstrated.
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References
1. C. Kittel. Introduction to Solid State Physics, Wiley (2004), 704 p.
2. M. A. Wahab. Solid State Physics: Structure and Properties of Materials, Alpha Science (2017), 620 p.
3. L. F. Chernogor. On the Nonlinearity in Nature and Science, Kharkiv, V. N. Karazin Kharkiv National University (2008), 528 p. (In Ukrainian).
4. V. V. Yanovsky. Universitates, 3, 32 (2003) (In Ukrainian).
5. V. V. Yanovsky. Lectures on Nonlinear Phenomena. Volume 1, Kharkiv, Institut monokristallov Publ. (2006), 456 p. (in Ukrainian).
6. B. B. Mandelbrot. Fractal Objects: Shape, Randomness and Dimension, Paris: Flammarion (1975), 190 p. (in French).
7. B. B. Mandelbrot Fractals: Form, Chance and Dimension. San Francisco: W. H. Freeman and Company (1977), 468 p. https://doi.org/10.1002/zamm.19790590830
8. B. B. Mandelbrot. The Fractal Geometry of Nature, San Francisco, CA-Freeman (1982), 460 p. https://doi.org/10.1119/1.13295
9. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 25 (1), 3 (2020) (in Ukrainian). https://doi.org/10.15407/rpra25.01.003
10. G. H. Hardy Weierstrass’s non-differentiable function. Trans. AMS, 17 (1916). https://doi.org/10.2307/1989005
11. C. Bandt, M. Barnsley, R. Devaney, K. J. Falconer, V. Kannan, P. B. Vinod Kumar, eds. Fractals, Wavelets, and their Applications: Contributions from the Int. Conference and Workshop on Fractals and Wavelets (Springer Proceedings in Mathematics & Statistics), Switzerland, Springer Int. Publ. (2014), 508 p.
12. Y. Baryshev and P. Teerikorpi. Discovery of Cosmic Fractals. New Jersey: World Scientific Publ. (2002), 373 p. https://doi.org/10.1142/4896
13. R. M. Crownover. Introduction to Fractals and Chaos, Boston, Jones and Barlett Publishers (1995), 320 p.
14. O. R. Shenker. Studies in History and Philosophy of Science, Part A. 25 (6), 967 (1994). https://doi.org/10.1016/0039-3681(94)90072-8
15. H.-O. Peitgen, P. H. Richter. The Beauty of Fractals: Images of Complex Dynamical Systems, Springer (1986), 214 p. https://doi.org/10.1007/978-3-642-61717-1
16. B. B. Mandelbrot. Fractals and Chaos: The Mandelbrot Set and Beyond, Springer (2005), 400 p. https://doi.org/10.1186/1475-925X-4-30
17. B. B. Mandelbrot. Multifractals and 1/f Noise, Springer (1999), 442 p. https://doi.org/10.1007/978-1-4612-2150-0
18. B. B. Mandelbrot Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E. New York, Springer-Verlag (1997), 541 p.
19. C. H. Scholz and B. B. Mandelbrot, eds. Fractals in Geophysics, Basel: Birkhäuser (1989), 314 p.
20. B. B. Mandelbrot and R. L. Hudson. The Misbehavior of Markets: A fractal View of Risk, Ruin and Reward, New York, Basic Books (2005), 288 p.
21. B. Mandelbrot. Phys. Scr. Vol. 32 (4), 257 (1985). https://doi.org/10.1088/0031-8949/32/4/001
22. А. F. Bulat, А. S. Kobets, V. І. Dyrda, B. О. Bluess, А. М. Pugach. Fractals in Geomechanics, Dnipro, Zhurfond (2022), 367 p. (In Ukrainian).
23. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 28 (1), 5 (2023) (in Ukrainian). https://doi.org/10.15407/rpra28.01.005
24. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 29 (1), 46 (2024) (in Ukrainian). https://doi.org/10.15407/rpra29.01.046
25. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 29 (3), 180 (2024) (in Ukrainian). https://doi.org/10.26565/2222-5617-2023-40-02
26. B. Ghosh, S. Sinha, and M. V. Kartikeyan. Fractal Apertures in Waveguides, Conducting Screens and Cavities: Analysis and Design. Switzerland, Springer Int. Publ. (2014), 201 p. https://doi.org/10.1007/978-3-319-06535-9
27. K. Falconer Fractal Geometry. Mathematical Foundations and Applications, Chichester, John Wiley & Sons Ltd. (2014), 400 p. http://dx.doi.org/10.1002/0470013850
28. K. Falconer. Techniques in Fractal Geometry. Chichester, John Wiley & Sons Ltd. (1997), 274 p. https://doi.org/10.1017/s0013091500020137
29. L. Seuront. Fractals and Multifractals in Ecology and Aquatic Science, Boca Raton, London, New York, CRC Press (2010), 344 p. https://doi.org/10.1201/9781420004243
30. T. Nakayama and K. Yakubo. Fractal Concepts in Condensed Matter Physics. Berlin, Heidelberg, Springer-Verlag (2010), 203 p. https://doi.org/10.1007/978-3-662-05193-1
31. D. P. Feldman. Chaos and Fractals. An Elementary Introduction, Oxford, University Press (2012), 408 p. https://doi.org/10.1093/acprof:oso/9780199566433.001.0001
32. M. Schroeder. Fractals, Chaos, Power Laws. Minutes from Infinite Paradise, New York, W. H. Freeman and Company (1991), 528 p.
33. F. C. Moon. Chaotic Vibrations. An Introduction for Applied Scientists and Engineers, New York, Wiley and Sons (2004), 309 p. https://doi.org/10.1002/3527602844
34. J. M. Li., L. Lu, M. O. Lai, and B. Ralph. Image-Based Fractal Description of Microstructures. New York, Springer (2003), 272 p. https://doi.org/10.1007/978-1-4757-3773-8
35. H.-O. Peintgen and D. Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag (1988), 312 p. https://link.springer.com/book/10.1007/978-1-4612-3784-6
36. R. C. Hilborn. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. New York, Oxford University Press (2000), 650 p.
37. D. Harte. Multifractals. Theory and Applications, Boca Raton, Chapman and Hall/CRC Press (2001), 264 p. https://doi.org/10.1201/9781420036008
38. L. Pietronero, ed. Fractals’ Physical Origin and Properties. New York, Springer Science (1989), 370 p.
39. B. B. Mandelbrot. Fractals, Hazard and Finance (1959 – 1997). Paris, Flammarion (1998), 255 p. (in French).
40. H. Hurst. Trans. Am. Soc. Civ. Eng., 116, 770 (1951).
41. H. E. Hurst, R. P. Black, Y. M. Simaika. Long-term storage: an experimental study. London, Constable (1965).
42. N. Scafetta, P. Grigolini Phys. Rev. E. 66(3) (2002). https://doi.org/10.1103/physreve.66.036130
43. C. J. Bishop and Y. Peres. Fractals in Probability and Analysis. Cambridge, Cambridge University Press (2016), 402 p. https://doi.org/10.1017/9781316460238
44. F. Brambila, ed. Fractal Analysis. Applications in Health Sciences and Social Sciences. Rijeka, Croatia, InTech (2017), 216 p. https://doi.org/10.5772/intechopen.68898
45. D. Kumar, S. P. Arjunan, and B. Aliahmad. Fractals: Application in Biological Signalling and Image Processing. Boca Raton, CRC Press (2017), 174 p. https://doi.org/10.1201/9781315165868
46. Y. Yamamoto, R. L. Hughson. J. Appl. Physiol., 71(3) 1143 (1991). https://doi.org/10.1152/jappl.1991.71.3.1143
47. Y. Yamamoto, R. L. Hughson. Physica D, 68(2), 250 (1993). https://doi.org/10.1016/0167-2789(93)90083-d
48. B. Mandelbrot, and J. R. Wallis. Water Resources Res., 5(1), 228 (1969). https://doi.org/10.1029/WR005i001p00228
49. J. H. Van Beek, S. A. Roger, J. B. Bassingthwaighte. Am. J. Physiol. 257(26), H1670 (1989). https://doi.org/10.1152/ajpheart.1989.257.5.H1670
50. N. Vandewalle, M. Ausloos. Phys. Rev. E., 58(5), 6832 (1998). https://doi.org/10.1103/physreve.58.6832
51. H. M. Hastings, G. Sugihara. Fractals: A User’s Guide for the Natural Science. Oxford, Oxford University Press (1993), 248 p. https://doi.org/10.1093/oso/9780198545989.001.0001
52. R. F. Peltier and J. Lévy-Véhel. Research report, INRIA Rocqencourt (1994). https://hal.inria.fr/inria-00074279
53. J. Beran. Statistics for Long-Memory Processes, Chapman and Hall (1994), 328 p. https://doi.org/10.2307/2983481
54. G. Matheron. Economic Geology, 58(8), 1246 (1963) https://doi.org/10.2113/gsecongeo.58.8.1246
55. M. A. Riley, S. Bonnette, N. Kuznetsov, S. Wallot, J. Gao. Frontiers in Physiology, 3 (2012). https://doi.org/10.3389/fphys.2012.00371
56. T. Gneiting, H. Sevcikova and D. B. Percival. Statist. Sci., 27, 247 (2012). https://doi.org/10.1214/11-STS370
57. M. S. Taqqu, V. Teverovsky, W. Willinger. Fractals, 3(4), 785 (2012). https://doi.org/10.1142/s0218348x95000692
58. J. Geweke and S. Porter-Hudak. J. Time Ser. Anal., 4(4), 221 (2008). https://doi.org/10.1111/j.1467-9892.1983.tb00371.x
59. T. Alieva. Journal of the Optical Society of America A, 13(6), 1189 (1996). https://doi.org/10.1364/josaa.13.001189
60. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng. Proc. of the Royal Society A: Mathematical, Physical and Engineering Sciences, 454(1971), 903 (1998). https://doi.org/10.1098/rspa.1998.0193
61. J. B. Bassingthwaighte. News Physiol. Sci., 3, 5 (1988). https://doi.org/10.1152/physiologyonline
62. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, J. B. Bassingthwaighte. Physica A: Statistical Mechanics and Its Applications, 241(3–4), 606 (1997). https://doi.org/10.1016/s0378-4371(97)00252-5
63. A. Eke, P. Hermán, J. Bassingthwaighte, G. Raymond, D. Percival, M. Cannon, C. Ikrényi. Pflügers Archiv - European Journal of Physiology, 439(4), 403 (2000). https://doi.org/10.1007/s004249900135
64. Y. Nyvlt, J. Ulrich. Admixtures in Crystallization. Weinheim, New York, VCH Verlagsgesellschaft (1995), 397 p. https://doi.org/10.1002/9783527615315
65. S. M. Liu. Solid State Physics, 39, 207 (1986). https://doi.org/10.1016/S0081-1947(08)60370-7
66. R. Jullien. R. Botet. Aggregation and Fractal Aggregates. Singapore, World Scientific Publ. (1987), 120 p.
67. A. Bunde, S. Havlin, eds. Fractals and Disordered Systems. Berlin, Springer (2012), 408 p. https://doi.org/10.1007/978-3-642-51435-7
68. H. Takayasu. Fractals in the Physical Sciences. Manchester, Manchester University Press (1990), 170 p.
69. A. Amann, L. Cederbaum, W. Gans, eds. Fractals, Quasicrystals, Chaos, Knots and Algebraic Quantum Mechanics. Dordrecht, Kluwer Academic Publishers (1988), 331 p. https://doi.org/10.1007/978-94-009-3005-6
70. T. J. Cui, W. X. Tang, X. M. Yang, Z. L. Mei, W. X. Jiang. Metamaterials. Beyond Crystals, Noncrystals, and Quasicrystals. CRC Press (2016), 311 p. https://doi.org/10.1201/9781315373614
71. P. Papon, J. Leblond, P. H. E. Meijer, eds. The Physics of Phase Transitions. Concepts and Applications, Springer, Berlin (2007), 409 p. https://doi.org/10.1007/3-540-33390-8
72. H.-O. Peitgen, P. H. Richter. The Beauty of Fractals. Springer, Berlin (1986), 199 p. https://doi.org/10.1007/978-3-642-61717-1
73. A. Samoilenka. E. Babaev. Physical Review B, 107, 184104 (2023). https://doi.org/10.1103/PhysRevB.107.184104
74. R. C. Gonzalez, R. E. Woods, S. L. Eddins. Digital Image Processing Using MATLAB. Pearson Education, Prentice Hall (2004), 609 p.
75. A. T. Fiory, M. Gurvitch, R. J. Cava, C. P. Espinosa. Phys. Rev. B, 36(13), 7262 (1987). https://doi.org/10.1103/PhysRevB.36.7262
2. M. A. Wahab. Solid State Physics: Structure and Properties of Materials, Alpha Science (2017), 620 p.
3. L. F. Chernogor. On the Nonlinearity in Nature and Science, Kharkiv, V. N. Karazin Kharkiv National University (2008), 528 p. (In Ukrainian).
4. V. V. Yanovsky. Universitates, 3, 32 (2003) (In Ukrainian).
5. V. V. Yanovsky. Lectures on Nonlinear Phenomena. Volume 1, Kharkiv, Institut monokristallov Publ. (2006), 456 p. (in Ukrainian).
6. B. B. Mandelbrot. Fractal Objects: Shape, Randomness and Dimension, Paris: Flammarion (1975), 190 p. (in French).
7. B. B. Mandelbrot Fractals: Form, Chance and Dimension. San Francisco: W. H. Freeman and Company (1977), 468 p. https://doi.org/10.1002/zamm.19790590830
8. B. B. Mandelbrot. The Fractal Geometry of Nature, San Francisco, CA-Freeman (1982), 460 p. https://doi.org/10.1119/1.13295
9. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 25 (1), 3 (2020) (in Ukrainian). https://doi.org/10.15407/rpra25.01.003
10. G. H. Hardy Weierstrass’s non-differentiable function. Trans. AMS, 17 (1916). https://doi.org/10.2307/1989005
11. C. Bandt, M. Barnsley, R. Devaney, K. J. Falconer, V. Kannan, P. B. Vinod Kumar, eds. Fractals, Wavelets, and their Applications: Contributions from the Int. Conference and Workshop on Fractals and Wavelets (Springer Proceedings in Mathematics & Statistics), Switzerland, Springer Int. Publ. (2014), 508 p.
12. Y. Baryshev and P. Teerikorpi. Discovery of Cosmic Fractals. New Jersey: World Scientific Publ. (2002), 373 p. https://doi.org/10.1142/4896
13. R. M. Crownover. Introduction to Fractals and Chaos, Boston, Jones and Barlett Publishers (1995), 320 p.
14. O. R. Shenker. Studies in History and Philosophy of Science, Part A. 25 (6), 967 (1994). https://doi.org/10.1016/0039-3681(94)90072-8
15. H.-O. Peitgen, P. H. Richter. The Beauty of Fractals: Images of Complex Dynamical Systems, Springer (1986), 214 p. https://doi.org/10.1007/978-3-642-61717-1
16. B. B. Mandelbrot. Fractals and Chaos: The Mandelbrot Set and Beyond, Springer (2005), 400 p. https://doi.org/10.1186/1475-925X-4-30
17. B. B. Mandelbrot. Multifractals and 1/f Noise, Springer (1999), 442 p. https://doi.org/10.1007/978-1-4612-2150-0
18. B. B. Mandelbrot Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E. New York, Springer-Verlag (1997), 541 p.
19. C. H. Scholz and B. B. Mandelbrot, eds. Fractals in Geophysics, Basel: Birkhäuser (1989), 314 p.
20. B. B. Mandelbrot and R. L. Hudson. The Misbehavior of Markets: A fractal View of Risk, Ruin and Reward, New York, Basic Books (2005), 288 p.
21. B. Mandelbrot. Phys. Scr. Vol. 32 (4), 257 (1985). https://doi.org/10.1088/0031-8949/32/4/001
22. А. F. Bulat, А. S. Kobets, V. І. Dyrda, B. О. Bluess, А. М. Pugach. Fractals in Geomechanics, Dnipro, Zhurfond (2022), 367 p. (In Ukrainian).
23. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 28 (1), 5 (2023) (in Ukrainian). https://doi.org/10.15407/rpra28.01.005
24. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 29 (1), 46 (2024) (in Ukrainian). https://doi.org/10.15407/rpra29.01.046
25. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 29 (3), 180 (2024) (in Ukrainian). https://doi.org/10.26565/2222-5617-2023-40-02
26. B. Ghosh, S. Sinha, and M. V. Kartikeyan. Fractal Apertures in Waveguides, Conducting Screens and Cavities: Analysis and Design. Switzerland, Springer Int. Publ. (2014), 201 p. https://doi.org/10.1007/978-3-319-06535-9
27. K. Falconer Fractal Geometry. Mathematical Foundations and Applications, Chichester, John Wiley & Sons Ltd. (2014), 400 p. http://dx.doi.org/10.1002/0470013850
28. K. Falconer. Techniques in Fractal Geometry. Chichester, John Wiley & Sons Ltd. (1997), 274 p. https://doi.org/10.1017/s0013091500020137
29. L. Seuront. Fractals and Multifractals in Ecology and Aquatic Science, Boca Raton, London, New York, CRC Press (2010), 344 p. https://doi.org/10.1201/9781420004243
30. T. Nakayama and K. Yakubo. Fractal Concepts in Condensed Matter Physics. Berlin, Heidelberg, Springer-Verlag (2010), 203 p. https://doi.org/10.1007/978-3-662-05193-1
31. D. P. Feldman. Chaos and Fractals. An Elementary Introduction, Oxford, University Press (2012), 408 p. https://doi.org/10.1093/acprof:oso/9780199566433.001.0001
32. M. Schroeder. Fractals, Chaos, Power Laws. Minutes from Infinite Paradise, New York, W. H. Freeman and Company (1991), 528 p.
33. F. C. Moon. Chaotic Vibrations. An Introduction for Applied Scientists and Engineers, New York, Wiley and Sons (2004), 309 p. https://doi.org/10.1002/3527602844
34. J. M. Li., L. Lu, M. O. Lai, and B. Ralph. Image-Based Fractal Description of Microstructures. New York, Springer (2003), 272 p. https://doi.org/10.1007/978-1-4757-3773-8
35. H.-O. Peintgen and D. Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag (1988), 312 p. https://link.springer.com/book/10.1007/978-1-4612-3784-6
36. R. C. Hilborn. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. New York, Oxford University Press (2000), 650 p.
37. D. Harte. Multifractals. Theory and Applications, Boca Raton, Chapman and Hall/CRC Press (2001), 264 p. https://doi.org/10.1201/9781420036008
38. L. Pietronero, ed. Fractals’ Physical Origin and Properties. New York, Springer Science (1989), 370 p.
39. B. B. Mandelbrot. Fractals, Hazard and Finance (1959 – 1997). Paris, Flammarion (1998), 255 p. (in French).
40. H. Hurst. Trans. Am. Soc. Civ. Eng., 116, 770 (1951).
41. H. E. Hurst, R. P. Black, Y. M. Simaika. Long-term storage: an experimental study. London, Constable (1965).
42. N. Scafetta, P. Grigolini Phys. Rev. E. 66(3) (2002). https://doi.org/10.1103/physreve.66.036130
43. C. J. Bishop and Y. Peres. Fractals in Probability and Analysis. Cambridge, Cambridge University Press (2016), 402 p. https://doi.org/10.1017/9781316460238
44. F. Brambila, ed. Fractal Analysis. Applications in Health Sciences and Social Sciences. Rijeka, Croatia, InTech (2017), 216 p. https://doi.org/10.5772/intechopen.68898
45. D. Kumar, S. P. Arjunan, and B. Aliahmad. Fractals: Application in Biological Signalling and Image Processing. Boca Raton, CRC Press (2017), 174 p. https://doi.org/10.1201/9781315165868
46. Y. Yamamoto, R. L. Hughson. J. Appl. Physiol., 71(3) 1143 (1991). https://doi.org/10.1152/jappl.1991.71.3.1143
47. Y. Yamamoto, R. L. Hughson. Physica D, 68(2), 250 (1993). https://doi.org/10.1016/0167-2789(93)90083-d
48. B. Mandelbrot, and J. R. Wallis. Water Resources Res., 5(1), 228 (1969). https://doi.org/10.1029/WR005i001p00228
49. J. H. Van Beek, S. A. Roger, J. B. Bassingthwaighte. Am. J. Physiol. 257(26), H1670 (1989). https://doi.org/10.1152/ajpheart.1989.257.5.H1670
50. N. Vandewalle, M. Ausloos. Phys. Rev. E., 58(5), 6832 (1998). https://doi.org/10.1103/physreve.58.6832
51. H. M. Hastings, G. Sugihara. Fractals: A User’s Guide for the Natural Science. Oxford, Oxford University Press (1993), 248 p. https://doi.org/10.1093/oso/9780198545989.001.0001
52. R. F. Peltier and J. Lévy-Véhel. Research report, INRIA Rocqencourt (1994). https://hal.inria.fr/inria-00074279
53. J. Beran. Statistics for Long-Memory Processes, Chapman and Hall (1994), 328 p. https://doi.org/10.2307/2983481
54. G. Matheron. Economic Geology, 58(8), 1246 (1963) https://doi.org/10.2113/gsecongeo.58.8.1246
55. M. A. Riley, S. Bonnette, N. Kuznetsov, S. Wallot, J. Gao. Frontiers in Physiology, 3 (2012). https://doi.org/10.3389/fphys.2012.00371
56. T. Gneiting, H. Sevcikova and D. B. Percival. Statist. Sci., 27, 247 (2012). https://doi.org/10.1214/11-STS370
57. M. S. Taqqu, V. Teverovsky, W. Willinger. Fractals, 3(4), 785 (2012). https://doi.org/10.1142/s0218348x95000692
58. J. Geweke and S. Porter-Hudak. J. Time Ser. Anal., 4(4), 221 (2008). https://doi.org/10.1111/j.1467-9892.1983.tb00371.x
59. T. Alieva. Journal of the Optical Society of America A, 13(6), 1189 (1996). https://doi.org/10.1364/josaa.13.001189
60. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng. Proc. of the Royal Society A: Mathematical, Physical and Engineering Sciences, 454(1971), 903 (1998). https://doi.org/10.1098/rspa.1998.0193
61. J. B. Bassingthwaighte. News Physiol. Sci., 3, 5 (1988). https://doi.org/10.1152/physiologyonline
62. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, J. B. Bassingthwaighte. Physica A: Statistical Mechanics and Its Applications, 241(3–4), 606 (1997). https://doi.org/10.1016/s0378-4371(97)00252-5
63. A. Eke, P. Hermán, J. Bassingthwaighte, G. Raymond, D. Percival, M. Cannon, C. Ikrényi. Pflügers Archiv - European Journal of Physiology, 439(4), 403 (2000). https://doi.org/10.1007/s004249900135
64. Y. Nyvlt, J. Ulrich. Admixtures in Crystallization. Weinheim, New York, VCH Verlagsgesellschaft (1995), 397 p. https://doi.org/10.1002/9783527615315
65. S. M. Liu. Solid State Physics, 39, 207 (1986). https://doi.org/10.1016/S0081-1947(08)60370-7
66. R. Jullien. R. Botet. Aggregation and Fractal Aggregates. Singapore, World Scientific Publ. (1987), 120 p.
67. A. Bunde, S. Havlin, eds. Fractals and Disordered Systems. Berlin, Springer (2012), 408 p. https://doi.org/10.1007/978-3-642-51435-7
68. H. Takayasu. Fractals in the Physical Sciences. Manchester, Manchester University Press (1990), 170 p.
69. A. Amann, L. Cederbaum, W. Gans, eds. Fractals, Quasicrystals, Chaos, Knots and Algebraic Quantum Mechanics. Dordrecht, Kluwer Academic Publishers (1988), 331 p. https://doi.org/10.1007/978-94-009-3005-6
70. T. J. Cui, W. X. Tang, X. M. Yang, Z. L. Mei, W. X. Jiang. Metamaterials. Beyond Crystals, Noncrystals, and Quasicrystals. CRC Press (2016), 311 p. https://doi.org/10.1201/9781315373614
71. P. Papon, J. Leblond, P. H. E. Meijer, eds. The Physics of Phase Transitions. Concepts and Applications, Springer, Berlin (2007), 409 p. https://doi.org/10.1007/3-540-33390-8
72. H.-O. Peitgen, P. H. Richter. The Beauty of Fractals. Springer, Berlin (1986), 199 p. https://doi.org/10.1007/978-3-642-61717-1
73. A. Samoilenka. E. Babaev. Physical Review B, 107, 184104 (2023). https://doi.org/10.1103/PhysRevB.107.184104
74. R. C. Gonzalez, R. E. Woods, S. L. Eddins. Digital Image Processing Using MATLAB. Pearson Education, Prentice Hall (2004), 609 p.
75. A. T. Fiory, M. Gurvitch, R. J. Cava, C. P. Espinosa. Phys. Rev. B, 36(13), 7262 (1987). https://doi.org/10.1103/PhysRevB.36.7262
Published
2025-05-28
How to Cite
Lazorenko, O. V., Onishchenko, A. A., Taranova, I. A., & Udovenko, M. A. (2025). “FRACTALIZATION” OF THE SOLID-STATE PHYSICS. Journal of V. N. Karazin Kharkiv National University. Series Physics, (42), 43-52. https://doi.org/10.26565/2222-5617-2025-42-05
Section
Articles
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