An analytical solution to the Poisson–Boltzmann equation can be used to describe an electron-electron interaction in a metal-insulator semiconductor (MIS). This can be used to describe both time and position dependence of dissipative systems such as a mesoscopic system. This is done by solving the Poisson–Boltzmann equation analytically in the three-dimensional case. Solving this results in expressions of the distribution function for the Boltzmann equation and self-consistent average potential for the Poisson equation. These expressions are useful for analyzing quantum transport in a mesoscopic system. In metal-insulator semiconductor tunneling junctions, the electrons can build up close to the interface between layers and as a result the quantum transport of the system will be affected by the electron-electron interactions. Certain transport properties such as electric current and electronic density can be known by solving for self-consistent Coulombic average potential from the electron-electron interactions, which is related to electronic distribution. Therefore, it is essential to analytically solve the Poisson–Boltzmann equation in order to obtain the analytical quantities in the MIS tunneling junctions. Applying the following analytical solution of the Poisson–Boltzmann equation (see section 2) to MIS tunneling junctions, the following expression can be formed to express electronic transport quantities such as electronic density and electric current f 1 f 0 − − f 0 + e E z τ τ 0 m ∂ ∂ f 0 ∂ ∂ v z ( 1 − − e − − τ τ τ τ 0 ) − − ∫ ∫ 0 t e m e t − − τ τ ′ τ τ 0 ∇ ∇ ρ ρ [ r − − v ( t − − t ′ ) ] × × ∂ ∂ f 0 ∂ ∂ v d t ′