Abstract
We describe phase transitions using a one-dimensional fractional differential kinetic equation of the Fokker-Planck type. We find a general solution describing the growth of nuclei during phase transitions in a fractal medium.
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References
Ya. B. Zel’dovich, Sov. Phys. JETP, 12, 525–538 (1942).
L. P. Pitaevskii and E. M. Lifshitz, Course of Theoretical Physics [in Russian], Vol. 10, Physical Kinetics, Fizmatlit, Moscow (2002); English transl. prev. ed., Pergamon, London (1981).
V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields, and Media, Springer, Heidelberg (2010).
G. I. Kapel’, S. V. Razorenov, A. V. Utkin, and V. E. Fortov, Shockwave Phenomena in Condensed Media [in Russian], Yanus-K, Moscow (1996).
H. von L. Pietronero and E. Tosatti, eds., Fractals in Physics (Proc. 6th Trieste Intl. Symp. Fractals in Physics, ICTP, Trieste, Italy, 9–12 July 1985), North-Holland, Amsterdam (1986).
A. V. Milovanov, L. M. Zelenyi, G. Zimbardo, and P. Veltri, J. Geophys. Res., 106, 6291–6307 (2001).
K. Rypdal, J.-V. Paulsen, O. E. Garcia, S. V. Ratynskaia, and V. I. Demidov, Nonlin. Processes Geophys., 10, 139–149 (2003).
A. V. Milovanov and J. J. Rasmussen, Phys. Rev. B, 66, 134505 (2002).
A. I. Olemskoi, Synergy of a Complex System: Phenomonology and Statistical Theory [in Russian], Editorial URSS, Moscow (2009).
G. M. Zaslavsky, “Fractional kinetics of Hamiltonian chaotic systems,” in: Application of Fractional Calculus in Physics (R. Hilfer, ed.), World Scientific, River Edge, N. J. (2000), pp. 203–239.
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, and Some of their Applications (Math. Sci. Engin., Vol. 198), Acad. Press, San Diego, Calif. (1999).
K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Math. Sci. Engin., Vol. 111), Acad. Press, New York (1974).
A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Application, Springer, New York (2010).
K. Tsallis, Introduction in Nonextensive Statistical Mechanics: Approaching a Complex World, Springer, Berlin (2009).
B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco (1982).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series [in Russian], Vol. 3, Elementary Functions: Complementary Chapters, Fizmatlit, Moscow (2003); English transl. prev. ed. Integrals and Series, Vol. 1, Elementary Functions, Gordon and Breach, New York (1986).
H. Bateman, Higher Transcedental Functions, Vol. 1, McGraw-Hill, New York (1953).
L. D. Landau and E. M. Lifshitz, Theoretical Physics [in Russian], Vol. 5, Statistical Physics: Part 1, Nauka, Moscow (1976); English transl., Pergamon, Oxford (1980).
L. M. Zelenyi and A. V. Milovanov, Phys. Usp., 47, 749–788 (2004).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 1, pp. 126–136, October, 2013.
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Borodikhin, V.N. Phase transitions and peculiarities of the growth of nuclei of the new phase of a substance. Theor Math Phys 177, 1390–1399 (2013). https://doi.org/10.1007/s11232-013-0111-4
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DOI: https://doi.org/10.1007/s11232-013-0111-4