  This paper is devoted to the Nernst-Planck system of equations with an
external potential of confinement. The main result is concerned with the
asymptotic behaviour of the solution of the Cauchy problem. We will prove that
the optimal exponential rate of convergence of the solution to the unique
stationary solution is determined by the spectral gap of the linearized problem
around the minimizer of the free energy. The key issue is to consider an
adapted notion of scalar product.
