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Electron transport and electron density inside quasi-one-dimensional disordered conductors

We consider the problem of electron transport across a quasi-one-dimensional disordered multiply-scattering medium, and study the statistical properties of the electron density inside the system. In the physical setup that we contemplate, electrons of a given energy feed the disordered conductor from one end. The physical quantity that is mainly considered is the logarithm of the electron density, $\ln {\cal W}(x)$, since its statistical properties exhibit a self-averaging behavior. We also describe a {\em gedanken} experiment, as a possible setup to measure the electron density. We study analytically and through computer simulations the ballistic, diffusive and localized regimes. We generally find a good agreement between the two approaches. The extension of the techniques that were used in the past to find information outside the sample is done in terms of the scattering properties of the two segments that form the entire conductor on each side of the observation point. The problem is of interest in various other branches of physics, as electrodynamics and elasticity.