By Alfred J.Lotka

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Right: The unambiguous dependence var g vs g in the critical region of the metal-insulator transition. This agrees with the single parameter scaling theory percentile gα is defined as gα P (g)dg. α= (4) 0 Owing to (4), the probability to find g < gα equals to α. Of course, the percentile gα is a function of disorder and system size: gα = gα (L, W ). Single parameter scaling of percentiles has been proved for several values of α [34]. Suppose that gα and gβ (α < β) obey the single parameter scaling.

These results can not be applied to three dimensional (3D) systems ( = 1) where g ∼ 1 [27]. Numerical simulations are therefore crucial in 3D systems. The first systematic numerical analysis of the conductance statistics in 3D was done in [27] and was followed by a series of papers [28–34]. In 2D systems, the critical conductance distribution was numerically studied in the regime of the quantum Hall effect [35] and in systems with the spin-orbit interaction [28,36,37]. This paper reviews our recent numerical data for the conductance distribution.

Phys. (Leipzig) 8, 655 (1999) 7. K. Slevin, P. MarKo˘s, and T. Ohtsuki: Phys. Rev. Lett. 86, 3597 (2001) 8. A. Kawabata: Prog. Theor. Phys. Suppl. No. 84, 16 (1985) 9. S. Neal: Phys. Rev. Lett. 16, 984 (1966) 10. G. Bergmann: Phys. Rep. 107, 1 (1984) 11. Y. Nagaoka: Prog. Theor. Phys. Suppl. No. 84, 1 (1985) 12. J. Wegner: Nucl. Phys. B 316, 663 (1989) On the Critical Exponent of the Anderson Transition 51 13. S. Hikami: Proc. Low Dimensional Field Theories and Condensed Matter Physics, 4th Yukawa International Seminar, 1991 Kyoto, Prog.

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Elements of Physical Biology by Alfred J.Lotka


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