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Lattice statistical theory of random walks on a fractal-like geometry

Kozak, John J. and Garza-López, Roberto A. and Abad, Enrique (2014) Lattice statistical theory of random walks on a fractal-like geometry. Physical Review E, 89 (3). Art. No. 032147. ISSN 1539-3755.

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We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.

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Additional Information:© 2014 American Physical Society. Received 5 December 2013; published 31 March 2014; corrected 3 April 2014. The authors would like to thank S. B. Yuste, J. Wm. Turner, and S. Lelić for valuable insights on aspects of the problem discussed here. Technical assistance has been provided by P. Panagiotopolous. This study was initiated while J.J.K. was in residence at the Beckman Institute, and he is grateful to H. B. Gray for his kind hospitality. Financial support for R.A.G.-L. was provided by the Hirsh Research Initiation Grant, the Howard Hughes Medical Institute Research Program, and the Summer Undergraduate Research Program from Pomona College. E.A. gratefully acknowledges financial support from Ministerio de Ciencia y Tecnología (Spain) through Grant No. FIS2010-16587 (partially funded by FEDER funds) as well as additional financial support from Junta de Extremadura through Grant No. GRU10158.
Funding AgencyGrant Number
Hirsh Research Initiation GrantUNSPECIFIED
Howard Hughes Medical Institute (HHMI)UNSPECIFIED
Pomona College Summer Undergraduate Research ProgramUNSPECIFIED
Ministerio de Ciencia y Tecnología (Spain)FIS2010-16587
Junta de ExtremaduraGRU10158
Issue or Number:3
Classification Code:PACS: 05.40.Fb, 61.43.Hv, 05.50.+q
Record Number:CaltechAUTHORS:20140516-090326993
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:45789
Deposited By: Tony Diaz
Deposited On:16 May 2014 19:50
Last Modified:03 Oct 2019 06:36

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