Advanced Search
Article Contents
Article Contents

Einstein relation on fractal objects

Abstract Related Papers Cited by
  • Many physical phenomena proceed in or on irregular objects which are often modeled by fractal sets. Using the model case of the Sierpinski gasket, the notions of Hausdorff, spectral and walk dimension are introduced in a survey style. These characteristic numbers of the fractal are essential for the Einstein relation, expressing the interaction of geometric, analytic and stochastic aspects of a set.
    Mathematics Subject Classification: Primary: 28A80; Secondary: 31C25, 34B45, 35P20, 60J45.


    \begin{equation} \\ \end{equation}
  • [1]

    N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev, Vibration modes of $3n$-gaskets and other fractals, J. Phys. A, 41 (2008), Article ID 015101, 21 pp.


    M. Barlow, Diffusions on fractals, in "Lectures on Probability Theory and Statistics" (Saint-Flour, 1995), Lecture Notes in Math., 1690, Springer, Berlin, 1998.


    M. Barlow and R. Bass, Transition densities for Brownian motion on the Sierpinski carpet, Probab. Theory Relat. Fields, 91 (1992), 307-330.doi: 10.1007/BF01192060.


    M. Barlow and R. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 51 (1999), 673-744.


    M. Barlow and B. Hambly, Transition density estimates for Brownian motion for scale irregular Sierpinski gaskets, Ann. Inst. Henri Poincaré, 33 (1997), 531-557.


    M. Barnsley and U. Freiberg, Fractal transformations of harmonic functions, in "Complexity and Nonlinear Dynamics" (ed. Axel Bender), Proc. SPIE, 64170C, (2007).


    M. Barnsley, J. Hutchinson and Ö. Stenflo, A fractals valued random iteration algorithm and fractal hierarchy, Fractals, 13 (2005), 111-146.doi: 10.1142/S0218348X05002799.


    M. Barnsley, J. Hutchinson and Ö. Stenflo, $V$-variable fractals: Fractals with partial self similarity, Adv. Math., 218 (2008), 2051-2088.doi: 10.1016/j.aim.2008.04.011.


    M. Barnsley, J. Hutchinson and Ö. Stenflo$V$-variable fractals: Dimension results, to appear in Forum Mathematicum.


    M. Berry, Some geometric aspects of wave motion: Wavefront dislocations, diffraction catastrophes, diffractals, in "Geometry of the Laplace Operator" (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Symp. Pure Math., XXXVI, Am. Math. Soc., Providence, R.I., (1980), 13-28.


    B. Boyle, K. Cekala, D. Ferrone, N. Rifkin and A. Teplyaev, Electrical resistance of $n$-gasket fractal networks, Pac. J. Math., 233 (2007), 15-40.doi: 10.2140/pjm.2007.233.15.


    K. Falconer, "The Geometry of Fractal Sets," Cambridge Tracts in Mathematics, 85, Cambridge Univ. Press., Cambridge, 1986.


    K. Falconer, Random fractals, Math. Proc. Cambridge Philos. Soc., 100 (1986), 559-582.doi: 10.1017/S0305004100066299.


    U. Freiberg, Analytic properties of measure geometric Krein-Feller-operators on the real line, Math. Nach., 260 (2003), 34-47.doi: 10.1002/mana.200310102.


    U. FreibergDirichlet forms on fractal subsets of the real line, Real Analysis Exchange, 30 (2004/05), 589-603.


    U. Freiberg, Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets, Forum Math., 17 (2005), 87-104.doi: 10.1515/form.2005.17.1.87.


    U. Freiberg, Analysis on fractal objects, Meccanica, 40 (2005), 419-436.doi: 10.1007/s11012-005-2107-0.


    U. Freiberg, Some remarks on the Hausdorff and spectral dimension of $V$-variable nested fractals, in "Recent Developments in Fractals and Related Fields" (eds. Julien Barral, et al.), Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, (2010), 267-282.doi: 10.1007/978-0-8176-4888-6_17.


    U. Freiberg, Tailored $V$-variable models, European Congress of Stereology and Image Analysis [Online], 2008.


    U. Freiberg, B. Hambly and J. E. HutchinsonSpectral asymptotics for $V$-variable Sierpinski gaskets, preprint.


    U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen, 23 (2004), 115-137.


    U. Freiberg and M. R. Lancia, Energy forms on conformal $\mathcalC^1$-diffeomorphic images of the Sierpinski gasket, Math. Nachr., 281 (2008), 337-349.doi: 10.1002/mana.200510606.


    U. Freiberg and J.-U. Löbus, Zeros of eigenfunctions of a class of generalized second order differential operators on the Cantor set, Math. Nach., 265 (2004), 3-14.doi: 10.1002/mana.200310133.


    U. Freiberg and C. Thäle, A Markov chain algorithm in determining crossing times through nested graphs, in "Fifth Colloquium on Mathematics and Computer Science," Discrete Mathematics and Theoretical Computer Science Proc., Al, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, (2008), 501-517.


    U. Freiberg and C. ThäleExact computation and approximation of stochastic and analytic parameters of generalized Sierpinski gaskets, preprint.


    M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," de Gruyter Studies in Mathematics, 19, Walter de Gruyter & Co., Berlin, 1994.


    S. Graf, Statistically self-similar fractals, Probab. Theory Relat. Fields, 74 (1987), 357-392.doi: 10.1007/BF00699096.


    J. A. Given and B. B. Mandelbrot, Diffusion on fractal lattices and the fractal Einstein relation, J. Phys. A, 16 (1983), L565-L569.doi: 10.1088/0305-4470/16/15/003.


    B. Hambly, Brownian motion on a homogenuous random fractal, Probab. Theory Rel. Fields, 94 (1992), 1-38.doi: 10.1007/BF01222507.


    B. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Prob., 25 (1997), 1059-1102.doi: 10.1214/aop/1024404506.


    B. Hambly, On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets, Probab. Theory Rel. Fields, 117 (2000), 221-247.doi: 10.1007/s004400050005.


    J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.doi: 10.1512/iumj.1981.30.30055.


    J. E. Hutchinson and L. Rüschendorf, Random fractal measures via the contraction method, Indiana Univ. Math. J., 47 (1998), 471-487.doi: 10.1512/iumj.1998.47.1461.


    J. E. Hutchinson, Deterministic and random fractals, in "Complex Systems," Cambridge Univ. Press, Cambridge, (2000), 127-166.doi: 10.1017/CBO9780511758744.005.


    Y. Kifer, Fractals via random iterated function systems and random geometric constructions, in "Fractal Geometry and Stochastics" (Finsterbergen, 1994), Progr. Probab., 37, Birkhäuser, Basel, (1995), 145-164.


    J. Kigami, "Analysis on Fractals," Cambridge Tracts in Mathematics, 143, Cambridge Univ. Press, Cambridge, 2001.


    J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Commun. Math. Phys., 158 (1993), 93-125.doi: 10.1007/BF02097233.


    S. Kusuoka, "Diffusion Processes on Nested Fractals," Lecture Notes in Math., 1567, Springer, 1993.


    T. Lindstrøm, Brownian motion on nested fractals, Memoirs Amer. Math. Soc., 83 (1990), iv+128 pp.


    R. D. Mauldin and S. C. Williams, Random recursive constructions: Asymptotic geometric and topological properties, Trans. Amer. Math. Soc., 295 (1986), 325-346.doi: 10.1090/S0002-9947-1986-0831202-5.


    S. O. Nyberg, The discrete Einstein relation, Circuits Systems Signal Process, 16 (1997), 547-557.doi: 10.1007/BF01185004.


    R. O. Schonmann, Einstein relation for a class of interface models, Comm. Math. Phys., 232 (2003), 279-302.doi: 10.1007/s00220-002-0749-5.


    R. Strichartz, "Differential Equations on Fractals. A Tutorial," Princeton University Press, Princeton, NJ, 2006.


    A. Telcs, The Einstein relation for random walks on graphs, J. Stat. Phys., 122 (2006), 617-645.doi: 10.1007/s10955-005-8002-1.


    P. Tetali, Random walks and the effective resistance of networks, J. Theo. Prob., 4 (1991), 101-109.doi: 10.1007/BF01046996.


    H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers, Rend. Cir. Mat. Palermo, 39 (1915), 1-50.doi: 10.1007/BF03015971.


    X. Y. Zhou, The resistance dimension, random walk dimension and fractal dimension, J. Theo. Prob., 6 (1993), 635-652.doi: 10.1007/BF01049168.

  • 加载中

Article Metrics

HTML views() PDF downloads(139) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint