Quantum conductance fluctuations and maximum entropy ensembles for the transfer matrix

As an introduction, we present conductance fluctuations which have been observed in a mesoscopic wire at very low temperature. Those experiments show either time independent quantum fluctuations, observed by varying an applied magnetic field of the Fermi energy, or quantum noises induced by short applied voltage pulses. We review and extend a new approach to the theory of those quantum fluctuations, based on the study of multiplicative random matrix ensembles defined using a hypothesis of maximum information entropy. The distribution of the radial parameters {λa} of the transfer matrix M is given both in the metallic and localized regime. For disordered conductors, it is shown that this approach gives a distribution of the {λa} of a type similar to those describing the spectral fluctuations of the Gaussian ensembles in random matrix theory. For disordered insulators, the logarithms of the {λa} have quasi-independent Gaussian fluctuations. The conductance g being a linear statistic of the {λa}, the universal conductance fluctuations in the metallic regime and lognormal fluctuations in the localized regime are then a consequence of those macroscopic maximum entropy models, which can be understood by simple Coulomb gas analogies. Universal results relative to localization lengths and conductance fluctuations of Anderson insulators are given. The importance of the multiplicative composition law of M in explaining the success of this approach is underlined.