Heat Conduction in Solids with Fractal Cracks

Павел Абрамович Головинский

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Heat Conduction in Solids with Fractal Cracks

П. А. Головинский

2026-05-01

https://doi.org/10.24108/preprints-3115120

We develop an approach to the description of steady-state heat conduction in solids containing a hierarchical (fractal) network of cracks. The temperature field and the crack geometry are treated as dynamical variables, and the full partition function is written as a double functional integral over the temperature field and the crack-surface configurations. Using the exact duality between the temperature field and an ensemble of Brownian streamlines, we show that averaging over fractal boundaries induces an effective two-dimensional quantum gravity of the Polyakov type on the worldsheets of the streamlines. A closed system of mean-field equations is derived: a nonlinear heat-conduction equation and a modified Liouville equation for the crack density. An exact analytical solution is found for the one-dimensional steady-state temperature and crack-density profiles, demonstrating the existence of a critical heat flux~$J_c$ that separates two qualitatively different regimes: for~$J < J_c$ the cracks are suppressed by surface tension; for~$J > J_c$ the heat flux concentrates the cracks into a localized cluster, which can lead to a percolation transition.

Ссылка для цитирования:

Головинский П. А. 2026. Heat Conduction in Solids with Fractal Cracks. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115120

Список литературы

1. [1] A. M. Polyakov, “Quantum geometry of bosonic strings”, Phys. Lett. B, 103(3), 207–

2. (1981).

3. [2] A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, Chur,

4. [3] Ph. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer, New

5. York, 1997.

6. [4] V. G. Knizhnik, A. M. Polyakov, A. B. Zamolodchikov, “Fractal structure of 2D quantum

7. gravity”, Mod. Phys. Lett. A, 3(8), 819–826 (1988).

8. [5] F. David, “Conformal field theories coupled to 2-D gravity in the conformal gauge”,

9. Mod. Phys. Lett. A, 3(17), 1651–1656 (1988).

10. [6] J. Distler, H. Kawai, “Conformal field theory and 2D quantum gravity”, Nucl. Phys.

11. B, 321(2), 509–527 (1989).

12. [7] P. Ginsparg, G. Moore, “Lectures on 2D gravity and 2D string theory”, arXiv:hepth/

13. 9304011, 1993.

14. [8] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and

15. Financial Markets, 5th ed., World Scientific, Singapore, 2009.

16. [9] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford

17. University Press, Oxford, 2021.

18. [10] R.P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill,

19. New York, 1965.

20. [11] B. A. Dubrovin, A. T. Fomenko, S.P. Novikov, Modern Geometry — Methods and

21. Applications, vols. I–III, Springer, New York, 1984–1990.

22. [12] M. Nakahara, Geometry, Topology and Physics, 2nd ed., IOP Publishing, Bristol,

23. [13] J. Polchinski, String Theory, vol. 1, Cambridge University Press, Cambridge, 1998.

24. [14] S. W. Hawking, “Zeta function regularization of path integrals in curved spacetime”,

25. Commun. Math. Phys., 55(2), 133–148 (1977).

26. [15] E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, 2nd ed., Springer,

27. Berlin, 2012.

28. [16] M. Kac, “Can one hear the shape of a drum?”, Amer. Math. Monthly, 73(4P2), 1–23

29. (1966).

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