Thermodynamical Properties of Hall Systems

We study quantum Hall effect within the framework of a newly proposed approach, which captures the principal results of some proposals. This can be established by considering a system of particles living on the non-commutative plane in the presence of an electromagnetic field and quantum statistical mechanically investigate its basic features. Solving the eigenvalue equation, we analytically derive the energy levels and the corresponding wavefunctions. These will be used, at low temperature and weak electric field, to determine the thermodynamical potential \Omega^{nc} and related physical quantities. Varying \Omega^{nc} with respect to the non-commutativity parameter \theta, we define a new function that can be interpreted as a \Omega^{nc} density. Evaluating the particle number, we show that the Hall conductivity of the system is \theta-dependent. This allows us to make contact with quantum Hall effect by offering different interpretations. We study the high temperature regime and discuss the magnetism of the system. We finally show that at \theta=2l_B^2, the system is sharing some common features with the Laughlin theory