Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: a review. (English) Zbl 1467.82001
Summary: Fractal calculus generalizes ordinary calculus, offering a way to differentiate otherwise non-differentiable domains and phenomena. This paper discusses the equilibrium and non-equilibrium statistical mechanics involving fractal structure, as well as fractal temperature in the partition function.
MSC:
| 82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |
| 26A33 | Fractional derivatives and integrals |
| 82B05 | Classical equilibrium statistical mechanics (general) |
| 82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |
Keywords:
fractal calculus; fractal temperature; fractal anomalous diffusion; fractal Cantor tartan; fractal Langevin equation; density of states in fractalReferences:
| [1] | Frankel, T., The Geometry of Physics: An Introduction (Cambridge Univ. Press, 2011). · Zbl 1250.58001 |
| [2] | Mandelbrot, B. B., The Fractal Geometry of Nature, Vol. 173 (W.H. Freeman, 1983). |
| [3] | Blackledge, J. M., Evans, A. K. and Turner, M. J., Fractal Geometry: Mathematical Methods, Algorithms, Applications (Elsevier, 2002). · Zbl 0996.00016 |
| [4] | Pesin, Y. B., Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Univ. Chicago Press, 2008). · Zbl 0895.58033 |
| [5] | Peters, E. E., Fractal Market Analysis: Applying Chaos Theory to Investment and Economics, Vol. 24 (John Wiley & Sons, 1994). |
| [6] | Dewey, T. G., Fractals in Molecular Biophysics (Oxford Univ. Press, 1998). · Zbl 0929.92006 |
| [7] | Pietronero, L. and Tosatti, E., Fractals in Physics (Elsevier, 2012). · Zbl 0652.58001 |
| [8] | Bercioux, D. and Iñiguez, A., Nat. Phys.15, 111 (2019). |
| [9] | Deppman, A. and Megías, E., Physics1, 103 (2019). |
| [10] | Calcagni, G., Eur. Phys. J. C76, 181 (2016). |
| [11] | Calcagni, G., Phys. Lett. B697, 251 (2011). |
| [12] | Fernández-Martínez, M. and Sánchez-Granero, M. A., Topol. Appl.163, 93 (2014). · Zbl 1296.37038 |
| [13] | Family, F. and Vicsek, T., Dynamics of Fractal Surfaces (World Scientific, 1991). · Zbl 0866.28009 |
| [14] | Massopust, P. R., Fractal Functions, Fractal Surfaces, and Wavelets (Academic Press, 2016). · Zbl 1343.28003 |
| [15] | Ieva, A. D. (ed.), The Fractal Geometry of the Brain (Springer, 2016). |
| [16] | Losa, G. A., Merlini, D., Nonnenmacher, T. F. and Weibel, E. R., Fractals in Biology and Medicine (Birkhäuser, 2005). · Zbl 0997.00037 |
| [17] | Kaandorp, J. A. and Prusinkiewicz, P., Fractal Modelling: Growth and Form in Biology (Springer, 1994). · Zbl 0796.92001 |
| [18] | Schroeder, M., Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Courier Corporation, 2009). |
| [19] | Bandt, C., Graf, S. and Zähle, M. (eds.), Fractal Geometry and Stochastics (Birkhäuser, 1995). · Zbl 0829.00021 |
| [20] | Edgar, G., Measure, Topology, and Fractal Geometry (Springer Science & Business Media, 2007). |
| [21] | Falconer, K., Fractal Geometry: Mathematical Foundations and Applications (John Wiley & Sons, 2004). |
| [22] | Falconer, K., Techniques in Fractal Geometry (John Wiley and Sons, 1997). · Zbl 0869.28003 |
| [23] | Samayoa, D., Ochoa Ontiveros, L. A., Damián Adame, L., Reyes de Luna, E., Álvarez Romero, L. and Romero-Paredes, G., Rev. Mex. Fís.66, 283 (2020). |
| [24] | Lapidus, M. L. and Van Frankenhuijsen, M., Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer, 2012). · Zbl 1261.28011 |
| [25] | Strichartz, R. S., Differential Equations on Fractals: A Tutorial (Princeton Univ. Press, 2006). · Zbl 1190.35001 |
| [26] | Czachor, M., Acta Phys. Pol. B50, 813 (2019). · Zbl 07913270 |
| [27] | Uchaikin, V. V. and Sibatov, R. T., Fractional Kinetics in Space: Anomalous Transport Models (World Scientific, 2017). · Zbl 1387.85001 |
| [28] | Barlow, M. T. and Perkins, E. A., Probab. Theory Rel. Fields79, 543 (1988). · Zbl 0635.60090 |
| [29] | Kigami, J., Analysis on Fractals (Cambridge Univ. Press, 2001). · Zbl 0998.28004 |
| [30] | Freiberg, U. and Zahle, M., Potential Anal.16, 265 (2002). · Zbl 1055.28002 |
| [31] | Zubair, M., Mughal, M. J. and Naqvi, Q. A., Electromagnetic Fields and Waves in Fractional Dimensional Space (Springer, 2012). · Zbl 1244.78001 |
| [32] | Goldfain, E., Quantum Matter3, 256 (2014). |
| [33] | Nigmatullin, R. R. and Baleanu, D., Relationships between 1D and space fractals and fractional integrals and their applications in physics, in Applications in Physics, Part A, pp. 183-220. |
| [34] | Nigmatullin, R. R., Zhang, W. and Gubaidullin, I., Fract. Calc. Appl. Anal.20, 1263 (2017). · Zbl 1374.28014 |
| [35] | Tarasov, V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, 2011). |
| [36] | Stillinger, F. H., J. Math. Phys.18, 1224 (1977). · Zbl 0359.54023 |
| [37] | Balankin, A. S., Eur. Phys. J. B88, 1 (2015). |
| [38] | Kempfle, S., Schäfer, I. and Beyer, H., Nonlinear Dyn.29, 99 (2002). · Zbl 1026.47010 |
| [39] | Nottale, L., Fractals in the quantum theory of spacetime, in Mathematics of Complexity and Dynamical Systems, ed. Meyers, R. (Springer, 2012). |
| [40] | Bohner, M. and Peterson, A. C. (eds.), Advances in Dynamic Equations on Time Scales (Springer, 2002). |
| [41] | Wu, J. and Wang, C., Fractals28, 2050010 (2020). · Zbl 1434.28009 |
| [42] | Nottale, L., Fractal Space-time and Microphysics: Towards a Theory of Scale Relativity (World Scientific, 1993). · Zbl 0789.58003 |
| [43] | Vrobel, S., Fractal Time: Why a Watched Kettle Never Boils, Vol. 14 (World Scientific, 2011). |
| [44] | Welch, K., A Fractal Topology of Time: Deepening into Timelessness, 2nd edn. (Fox Finding Press, 2020). |
| [45] | Shlesinger, M. F., Ann. Rev. Phys. Chem.39, 269 (1988). |
| [46] | Kröger, H., Phys. Rep.323, 81 (2000). |
| [47] | Schulman, L. S., Techniques and Applications of Path Integration (Courier Corporation, 2012). |
| [48] | Parvate, A. and Gangal, A. D., Fractals17, 53 (2009). · Zbl 1173.28005 |
| [49] | Parvate, A. and Gangal, A. D., Fractals19, 271 (2011). · Zbl 1252.28005 |
| [50] | Satin, S., Parvate, A. and Gangal, A. D., Chaos Solitons Fractals52, 30 (2013). · Zbl 1323.35179 |
| [51] | Parvate, A., Satin, S. and Gangal, A. D., Fractals19, 15 (2011). · Zbl 1217.28014 |
| [52] | Golmankhaneh, A. K., Fernandez, A., Golmankhaneh, A. K. and Baleanu, D., Entropy20, 1 (2018). |
| [53] | Golmankhaneh, A. K., Turk. J. Phys.41, 418 (2017). |
| [54] | Golmankhaneh, A. K. and Cattani, C., Fractal Fract.3, 41 (2019). |
| [55] | Satin, S. and Gangal, A. D., Chaos Solitons Fractals127, 17 (2019). · Zbl 1448.60101 |
| [56] | Satin, S. and Gangal, A. D., Fractals24, 1650028 (2016). · Zbl 1354.28008 |
| [57] | Golmankhaneh, A. K., Ain Shams Eng. J.9, 2499 (2018). |
| [58] | Golmankhaneh, A. K., Golmankhaneh, A. K. and Baleanu, D., Int. J. Theor. Phys.54, 1275 (2015). · Zbl 1328.81119 |
| [59] | Golmankhaneh, A. K. and Fernandez, A., Fractal Fract.2, 30 (2018). |
| [60] | Golmankhaneh, A. K. and Baleanu, D., Open Phys.14, 542 (2016). |
| [61] | Golmankhaneh, A. K. and Baleanu, D., J. Mod. Opt.63, 1364 (2016). |
| [62] | Remón, L., Garcia-Delpech, S., Udaondo, P., Ferrando, V., Monsoriu, J. A. and Furlan, W. D., PLoS One13, 0200197 (2018). |
| [63] | Remón, L., Furlan, W. D. and Monsoriu, J. A., Opt. Pura Apl.48, 1 (2015). |
| [64] | Golmankhaneh, A. K. and Balankin, A. S., Phys. Lett. A382, 960 (2018). · Zbl 1383.76439 |
| [65] | Balankin, A. S., Golmankhaneh, A. K., Patiño-Ortiz, J. and Patiño-Ortiz, M., Phys. Lett. A382, 1534 (2018). · Zbl 1396.28008 |
| [66] | Golmankhaneh, A. K. and Tunç, C., Stochastics Int. J. Probab. Stochastic Process.92, 1244 (2020). · Zbl 1490.60158 |
| [67] | Golmankhaneh, A. K. and Fernandez, A., Fractal Fract.3, 31 (2019). |
| [68] | Bodri, L., Theor. Appl. Climatol.49, 53 (1994). |
| [69] | Golmankhaneh, A. K., Fractal Fract.3, 20 (2019). |
| [70] | El-Nabulsi, R. A., Few-Body Syst.61, 25 (2020). |
| [71] | El-Nabulsi, R. A., Few-Body Syst.61, 10 (2020). |
| [72] | Golmankhaneh, A. K., Fractal Fract.3, 11 (2019). |
| [73] | R. DiMartino and W. Urbina, arXiv:1403.6554. |
| [74] | Serpa, C. and Buescu, J., Chaos Solitons Fractals75, 76 (2015). · Zbl 1352.41001 |
| [75] | Iomin, A. and Sandev, T., Fractal Fract.4, 52 (2020). |
| [76] | Pathria, R. K. and Beale, P. D., Statistical Mechanics, 3rd edn. (Academic Press, 2011). · Zbl 1209.82001 |
| [77] | Kittel, C., Introduction to Solid State Physics, 8th edn. (Wiley, 2004). · Zbl 0052.45506 |
| [78] | Bodrova, A. S.et al., Sci. Rep.6, 30520 (2016). |
| [79] | Bodrova, A. S., Chechkin, A. V., Cherstvy, A. G. and Metzler, R., New J. Phys.17, 063038 (2015). |
| [80] | Bodrova, A. S., Dubey, A. K., Puri, S. and Brilliantov, N. V., Phys. Rev. Lett.109, 178001 (2012). |
| [81] | Burov, S. and Barkai, E., Phys. Rev. E78, 031112 (2008). |
| [82] | Godec, A., Chechkin, A. V., Barkai, E., Kantz, H. and Metzler, R., J. Phys. A47, 492002 (2014). · Zbl 1305.60074 |
| [83] | Goychuk, I., Adv. Chem. Phys.150, 187 (2012). |
| [84] | Muniandy, S. V. and Lim, S. C., Phys. Rev. E63, 046104 (2001). |
| [85] | Thiel, F. and Sokolov, I. M., Phys. Rev. E89, 012115 (2014). |
| [86] | Safdari, H., Cherstvy, A. G., Chechkin, A. V., Thiel, F., Sokolov, I. M. and Metzler, R., J. Phys. A48, 375002 (2015). · Zbl 1329.82096 |
| [87] | Lim, S. C. and Muniandy, S. V., Phys. Rev. E66, 021114 (2002). |
| [88] | Kursawe, J., Schulz, J. H. P. and Metzler, R., Phys. Rev. E88, 062124 (2013). |
| [89] | Sen, P. N., Concepts Magn. Reson. A23, 1 (2004). |
| [90] | Molini, A., Talkner, P., Katul, G. G. and Porporato, A., Physica A390, 1841 (2011). · Zbl 1225.60135 |
| [91] | Lanoiselée, Y. and Grebenkov, D. S., J. Phys. A, Math. Theor.51, 145602 (2018). · Zbl 1391.92012 |
| [92] | Xiong, S. J., Xing, D. Y., Evangelou, S. N. and Sheng, D. N., Phys. Lett. A311, 426 (2003). |
| [93] | Liang, Y., Sandev, T. and Lenzi, E. K., Phys. Rev. E101, 042119 (2020). |
| [94] | dos Santos, M. A. F., Fractal Fract.4, 28 (2020). |
| [95] | dos Santos, M. A. F., Indian J. Phys.94, 1123 (2020). |
| [96] | dos Santos, M. A. F., Chaos Soliton. Fract.124, 86 (2019). · Zbl 1448.60193 |
| [97] | Sibatov, R. T. and Sun, H., Fractal Fract.4, 42 (2020). |
| [98] | Uchaikin, V. V. and Sibatov, R. T., Commun. Nonlinear Sci. Numer. Simul.13, 715 (2008). · Zbl 1221.82141 |
| [99] | Uchaikin, V. V. and Sibatov, R. T., Tech. Phys. Lett.30, 316 (2004). |
| [100] | Provata, A., Physica A, Stat. Mech. Appl.381, 148 (2007). |
| [101] | Deppman, A., Phys. Rev. D93, 054001 (2016). |
| [102] | Iliasov, A. A., Katsnelson, M. I. and Yuan, S., Phys. Rev. B101, 045413 (2020). |
| [103] | Yang, Z., Lustig, E., Lumer, Y. and Segev, M., Light, Sci. Appl.9, 1 (2020). |
| [104] | Petreska, I., de Castro, A. S., Sandev, T. and Lenzi, E. K., Phys. Lett. A384, 126866 (2020). · Zbl 1448.81311 |
| [105] | Sen, A. and Perelman, C. C., Eur. Phys. J. B93, 1 (2020). |
| [106] | Tanese, D., Gurevich, E., Baboux, F., Jacqmin, T., Lemaître, A., Galopin, E., Sagnes, I., Amo, A., Bloch, J. and Akkermans, E., Phys. Rev. Lett.112, 146404 (2014). |
| [107] | K. J. Vinoy, Fractal shaped antenna elements for wide-and multi-band wireless applications. Ph.D. Thesis, submitted to The Pennsylvania State University, USA (2002). |
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