On Mott’s formula for the ac-conductivity in the Anderson model. (English) Zbl 1141.82006

Klein, Abel; Lenoble, Olivier; Müller, Peter

Ann. Math. (2) 166, No. 2, 549-577 (2007).

The so-called Mott’s formula for the ac-conductivity at low temperature, deals with Anderson’s model and makes the assumption that ‘at sufficiently low density, transport does not take place; and the exact wave functions are localized in a small region of space’. The present article comeback on the topic, and provides a rigorous mathematical approach to this formula, via a new concept referred to as ‘conductivity measure’. Anderson’s model is defined and the main result is stated, electrical conductivity is defined and calculated in the framework of linear response theory, and the proof of the main result is postponed to the end of the article. Whilst Mott’s formula provides an approximation for the real-value of the ac-conductivity, here one yields an upper bound for the mean value of this variable.

Reviewer: Guy Jumarie (Montréal)

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80	Random linear operators

Keywords:

Mott’s formula; ac-conductivity; Anderson’s model