Abstract
In discussing transport quantities in one-dimensional disordered systems, care must be taken to assure that averaged quantities are statistically meaningful. ANDERSON, THOULESS, ABRAHAMS, and FISHER [1] have recently pointed to a, the inverse localization length, as a proper quantity to average. The mathematical properties of a have been discussed in the past by FURSTENBURG [2] and O’CONNOR [3], for example. The connection of a to resistance and conductance was made by LANDAUER [4]. He considered a collection of random barriers of total length L, and described by transmission coefficient T and reflection coefficient R = 1-T. By using the Einstein relation between conductivity and the diffusion constant, he showed that the conductance, G, is given by
where e is the electronic charge and h is Planck’s constant. LANDAUER [4] and ANDERSON et al. [1] agree that averaging the resistances of a collection of random systems gives a quantity proportional to exp(2αL)-1, where a is a parameter independent of length. While this accurately describes average resistance its usefulness is questionable, for if we instead average G, we will find that 1/<G> ≠ <1/G>.
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References
P.W.Anderson, D.J.Thouless, E.Abrahams, D.S.Fisher: to be published
H. Furstenberg: Trans. Amer. Math. Soc. 108, 377 (1963)
A. J. O’Connor: Commun. Math. Phys. 45, 63 (1975)
R. Landauer: Phil. Mag. 21, 863 (1970)
α, the transmission coefficient, has been shown to be equivalent to the inverse localization length by K. Ishii: Sup. Prog. Theor. Phys. 53, 77 (1973)
E. Abrahams, M. J. Stephen: J. Phys. C 13, L377 (1980)
For L ≳ 1/<α>, a should be described by a Γ-distribution (T.Lukes, preprint) which is indistinguishable from a Gaussian in the large L limit.
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© 1981 Springer-Verlag Berlin Heidelberg
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Andereck, B.S., Abrahams, E. (1981). Transport Quantities in One-Dimensional Disordered Systems. In: Bernasconi, J., Schneider, T. (eds) Physics in One Dimension. Springer Series in Solid-State Sciences, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81592-8_37
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DOI: https://doi.org/10.1007/978-3-642-81592-8_37
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