Skip to main content
Log in

Phase transitions and peculiarities of the growth of nuclei of the new phase of a substance

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ya. B. Zel’dovich, Sov. Phys. JETP, 12, 525–538 (1942).

    Google Scholar 

  2. 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).

    Google Scholar 

  3. V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields, and Media, Springer, Heidelberg (2010).

    Google Scholar 

  4. G. I. Kapel’, S. V. Razorenov, A. V. Utkin, and V. E. Fortov, Shockwave Phenomena in Condensed Media [in Russian], Yanus-K, Moscow (1996).

    Google Scholar 

  5. 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).

    Google Scholar 

  6. A. V. Milovanov, L. M. Zelenyi, G. Zimbardo, and P. Veltri, J. Geophys. Res., 106, 6291–6307 (2001).

    Article  ADS  Google Scholar 

  7. K. Rypdal, J.-V. Paulsen, O. E. Garcia, S. V. Ratynskaia, and V. I. Demidov, Nonlin. Processes Geophys., 10, 139–149 (2003).

    Article  ADS  Google Scholar 

  8. A. V. Milovanov and J. J. Rasmussen, Phys. Rev. B, 66, 134505 (2002).

    Article  ADS  Google Scholar 

  9. A. I. Olemskoi, Synergy of a Complex System: Phenomonology and Statistical Theory [in Russian], Editorial URSS, Moscow (2009).

    Google Scholar 

  10. 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.

    Chapter  Google Scholar 

  11. 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).

    Google Scholar 

  12. 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).

    Google Scholar 

  13. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Application, Springer, New York (2010).

    Book  Google Scholar 

  14. K. Tsallis, Introduction in Nonextensive Statistical Mechanics: Approaching a Complex World, Springer, Berlin (2009).

    Google Scholar 

  15. B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco (1982).

    MATH  Google Scholar 

  16. 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).

    MATH  Google Scholar 

  17. 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).

    Google Scholar 

  18. H. Bateman, Higher Transcedental Functions, Vol. 1, McGraw-Hill, New York (1953).

    Google Scholar 

  19. 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).

    Google Scholar 

  20. L. M. Zelenyi and A. V. Milovanov, Phys. Usp., 47, 749–788 (2004).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. N. Borodikhin.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 1, pp. 126–136, October, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-013-0111-4

Keywords

Navigation