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The conductance of finite systems plays a central role in the scaling theoryof localization (Abrahams et al, 1979). Usually it is defined by theLandauer-type formulas, which remain open the following questions: (a)exclusion of the contact resistance in the many-channel case; (b)correspondence of the Landauer conductance with internal properties of thesystem; (c) relation with the diffusion coefficient D(\omega,q) of an infinitesystem. The answers to these questions are obtained below in the framework oftwo approaches: (1) self-consistent theory of localization by Vollhardt andWoelfle, and (2) quantum mechanical analysis based on the shell model. Bothapproaches lead to the same definition for the conductance of a finite system,closely related to the Thouless definition. In the framework of theself-consistent theory, the relations of finite-size scaling are derived andthe Gell-Mann - Low functions \beta(g) for space dimensions d=1,2,3 arecalculated. In contrast to the previous attempt by Vollhardt and Woelfle(1982), the metallic and localized phase are considered from the samestandpoint, and the conductance of a finite system has no singularity at thecritical point. In the 2D case, the expansion of \beta(g) in 1/g coincides withresults of the \sigma-model approach on the two-loop level and depends on therenormalization scheme in higher loops; the use of dimensional regularizationfor transition to dimension d=2+\epsilon looks incompatible with the physicalessence of the problem. The obtained results are compared with numerical andphysical experiments. A situation in higher dimensions and the conditions forobservation of the localization law \sigma\propto -i\omega for conductivity arediscussed.

Conductance of Finite Systems and Scaling in Localization Theory