a relation of the form between the density of holes or electrons and the corresponding quasi Fermi level (or electrochemical potential) , the Einstein relation is where is the electrical mobility (see for a proof of this relation). An example assuming a parabolic dispersion relation for the density of states and the Maxwell–Boltzmann statistics, which is often used to describe inorganic semiconductor materials, one can compute (see density of states): where is the total density of available energy states, which gives the simplified relation: Nernst–Einstein equation By replacing the diffusivities in the expressions of electric ionic mobilities of the cations and anions from the expressions of the equivalent conductivity of an electrolyte the Nernst–Einstein equation is derived: Proof of the general case The proof of the Einstein relation can be found in many references, for example see Kubo. Suppose some fixed, external potential energy generates a conservative force (for example, an electric force) on a particle located at a given position . We assume that the particle would respond by moving with velocity (see Drag (physics)). Now assume that there are a large number of such particles, with local concentration as a function of the position. After some time, equilibrium will be established: particles will pile up around the areas with lowest potential energy , but still will be spread out to some extent because of diffusion. At equilibrium, there is no net flow of particles: the tendency of particles to get pulled towards lower , called the drift current, perfectly balances the tendency of particles to spread out due to diffusion, called the diffusion current (see drift-diffusion equation). The net flux of particles due to the drift current is i.e., the number of particles flowing past a given position equals the particle concentration times the average velocity. The flow of particles due to the diffusion current is, by Fick's law, where the minus sign means that particles flow from higher to lower concentration. Now consider the equilibrium condition. First, there is no net flow, i.e. . Second, for non-interacting point particles, the equilibrium density is solely a function of the local potential energy , i.e. if two locations have the same then they will also have the same (e.g. see Maxwell-Boltzmann statistics as discussed below.) That means, applying the chain rule, Therefore, at equilibrium: As this expression holds at every position , it implies the general form of the