EinsteinRelation

The Einstein relation, also known as the Einstein-Smoluchowski relation, is a fundamental principle in statistical physics that links the diffusion coefficient D of Brownian particles to their mobility μ in a fluid at temperature T. In its common form for neutral particles, D = μ k_B T, where k_B is Boltzmann's constant and T is the absolute temperature. For charged carriers, the relation is often written as D/μ = k_B T / q, with q the carrier charge, implying D = μ (k_B T)/q.

Origin and derivation: The relation was derived by Albert Einstein in 1905 to explain Brownian motion and to connect spontaneous thermal fluctuations with dissipative response. It can be derived from the Langevin equation or from the Fokker-Planck equation and linear response theory, and it is a consequence of the fluctuation-dissipation theorem.

Applications and scope: The formula holds for dilute, near-equilibrium systems in which diffusion is Fickian and the mobility is well defined. It underpins experimental estimates of mobility from diffusion measurements and is widely used in colloidal science, electrochemistry, and semiconductor physics, with appropriate generalizations to quantum or non-equilibrium settings. Limitations arise in regimes of strong interactions, high concentrations, anomalous diffusion, or far-from-equilibrium conditions, where deviations from the simple relation occur and corrections may be required.