  Up to know all the experimental results concerning the integer and fractional
quantum Hall effect are related to semiconductor heterostructures (and more
recently with graphene). The common characteristic of all these systems is the
presence of a reservoir of electrons, which, in fact, in the initial stage is
the source of the electrons, providing the two-dimensional electron gas (2DES).
Then, any physical realization of a 2DES is necessarily embedded in a 3D
structure, which establishes the Fermi level. Hence, the 2DES appears to be an
open system. In this paper we present an analytical approach to the integer
quantum Hall effect (IQHE) and the Shubnikov-de Haas (SdH) phenomena in the
2DES, basing us in fundamental principles and showing the secondary role of the
localized electron states in both phenomena. In fact, we show that the IQHE is
a consequence of the fluctuations of electrons in the 2DES. Once we obtain the
density of states of the 2DES under the application of a magnetic field we
calculate both magnetoconductivities (diagonal and Hall) deducing them from the
Boltzman semiclassical equation. The model proposed reproduces both phenomena,
the width of the Hall plateaus (with the precision reached in the experimental
measurements, of the order of 10-8-10-9) and the corresponding minima of the
diagonal magnetoresistivity, and also the dependence with temperature of the
IQHE and SdH.
