March  2012, 17(2): 509-525. doi: 10.3934/dcdsb.2012.17.509

Einstein relation on fractal objects

1. 

Universität Siegen, Département für Mathematik, Walter–Flex–Straße 3, D–57068 Siegen, Germany

Received  August 2010 Revised  July 2011 Published  December 2011

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.
Citation: Uta Renata Freiberg. Einstein relation on fractal objects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509
References:
[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).   Google Scholar

[2]

M. Barlow, Diffusions on fractals,, in, 1690 (1995).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

M. Barnsley and U. Freiberg, Fractal transformations of harmonic functions,, in, 64170C (2007).   Google Scholar

[7]

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

[8]

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

[9]

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

[10]

M. Berry, Some geometric aspects of wave motion: Wavefront dislocations, diffraction catastrophes, diffractals,, in, (1980), 13.   Google Scholar

[11]

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

[12]

K. Falconer, "The Geometry of Fractal Sets,", Cambridge Tracts in Mathematics, 85 (1986).   Google Scholar

[13]

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

[14]

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

[15]

U. Freiberg, Dirichlet forms on fractal subsets of the real line,, Real Analysis Exchange, 30 (): 589.   Google Scholar

[16]

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

[17]

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

[18]

U. Freiberg, Some remarks on the Hausdorff and spectral dimension of $V$-variable nested fractals,, in, (2010), 267.  doi: 10.1007/978-0-8176-4888-6_17.  Google Scholar

[19]

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

[20]

U. Freiberg, B. Hambly and J. E. Hutchinson, Spectral asymptotics for $V$-variable Sierpinski gaskets,, preprint., ().   Google Scholar

[21]

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

[22]

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

[23]

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.  doi: 10.1002/mana.200310133.  Google Scholar

[24]

U. Freiberg and C. Thäle, A Markov chain algorithm in determining crossing times through nested graphs,, in, (2008), 501.   Google Scholar

[25]

U. Freiberg and C. Thäle, Exact computation and approximation of stochastic and analytic parameters of generalized Sierpinski gaskets,, preprint., ().   Google Scholar

[26]

M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", de Gruyter Studies in Mathematics, 19 (1994).   Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

J. E. Hutchinson, Deterministic and random fractals,, in, (2000), 127.  doi: 10.1017/CBO9780511758744.005.  Google Scholar

[35]

Y. Kifer, Fractals via random iterated function systems and random geometric constructions,, in, 37 (1995), 145.   Google Scholar

[36]

J. Kigami, "Analysis on Fractals,", Cambridge Tracts in Mathematics, 143 (2001).   Google Scholar

[37]

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.  doi: 10.1007/BF02097233.  Google Scholar

[38]

S. Kusuoka, "Diffusion Processes on Nested Fractals,", Lecture Notes in Math., 1567 (1567).   Google Scholar

[39]

T. Lindstrøm, Brownian motion on nested fractals,, Memoirs Amer. Math. Soc., 83 (1990).   Google Scholar

[40]

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

[41]

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

[42]

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

[43]

R. Strichartz, "Differential Equations on Fractals. A Tutorial,", Princeton University Press, (2006).   Google Scholar

[44]

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

[45]

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

[46]

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

[47]

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

show all references

References:
[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).   Google Scholar

[2]

M. Barlow, Diffusions on fractals,, in, 1690 (1995).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

M. Barnsley and U. Freiberg, Fractal transformations of harmonic functions,, in, 64170C (2007).   Google Scholar

[7]

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

[8]

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

[9]

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

[10]

M. Berry, Some geometric aspects of wave motion: Wavefront dislocations, diffraction catastrophes, diffractals,, in, (1980), 13.   Google Scholar

[11]

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

[12]

K. Falconer, "The Geometry of Fractal Sets,", Cambridge Tracts in Mathematics, 85 (1986).   Google Scholar

[13]

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

[14]

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

[15]

U. Freiberg, Dirichlet forms on fractal subsets of the real line,, Real Analysis Exchange, 30 (): 589.   Google Scholar

[16]

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

[17]

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

[18]

U. Freiberg, Some remarks on the Hausdorff and spectral dimension of $V$-variable nested fractals,, in, (2010), 267.  doi: 10.1007/978-0-8176-4888-6_17.  Google Scholar

[19]

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

[20]

U. Freiberg, B. Hambly and J. E. Hutchinson, Spectral asymptotics for $V$-variable Sierpinski gaskets,, preprint., ().   Google Scholar

[21]

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

[22]

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

[23]

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.  doi: 10.1002/mana.200310133.  Google Scholar

[24]

U. Freiberg and C. Thäle, A Markov chain algorithm in determining crossing times through nested graphs,, in, (2008), 501.   Google Scholar

[25]

U. Freiberg and C. Thäle, Exact computation and approximation of stochastic and analytic parameters of generalized Sierpinski gaskets,, preprint., ().   Google Scholar

[26]

M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", de Gruyter Studies in Mathematics, 19 (1994).   Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

J. E. Hutchinson, Deterministic and random fractals,, in, (2000), 127.  doi: 10.1017/CBO9780511758744.005.  Google Scholar

[35]

Y. Kifer, Fractals via random iterated function systems and random geometric constructions,, in, 37 (1995), 145.   Google Scholar

[36]

J. Kigami, "Analysis on Fractals,", Cambridge Tracts in Mathematics, 143 (2001).   Google Scholar

[37]

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.  doi: 10.1007/BF02097233.  Google Scholar

[38]

S. Kusuoka, "Diffusion Processes on Nested Fractals,", Lecture Notes in Math., 1567 (1567).   Google Scholar

[39]

T. Lindstrøm, Brownian motion on nested fractals,, Memoirs Amer. Math. Soc., 83 (1990).   Google Scholar

[40]

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

[41]

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

[42]

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

[43]

R. Strichartz, "Differential Equations on Fractals. A Tutorial,", Princeton University Press, (2006).   Google Scholar

[44]

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

[45]

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

[46]

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

[47]

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

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