Einstein relation (kinetic theory)

The Einstein relation, in the context of kinetic theory and specifically for Brownian motion and diffusion, connects the diffusion coefficient (D) to the mobility (μ) of particles. It states that the ratio of the diffusion coefficient to the mobility is directly proportional to the thermal energy scale, which is represented by the product of Boltzmann's constant (k_B) and the absolute temperature (T).

Mathematically, the Einstein relation is expressed as:

D = μ k_B T

Where:

• D is the diffusion coefficient, a measure of how quickly particles spread out due to random motion.
• μ is the mobility, a measure of the particle's velocity in response to an applied force (e.g., electric field or concentration gradient).
• k_B is Boltzmann's constant, approximately 1.38 × 10^-23 J/K.
• T is the absolute temperature in Kelvin.

This relationship is significant because it demonstrates a fundamental connection between the random thermal motion of particles (which leads to diffusion) and their response to an external force. It reveals that diffusion and mobility are not independent phenomena but are intrinsically linked by the thermal energy of the system.

The Einstein relation holds true under certain conditions, primarily for systems in thermal equilibrium and where the particles are sufficiently small to experience Brownian motion. It is a cornerstone of understanding transport phenomena in various physical and chemical systems, including electrolyte solutions, semiconductors, and colloids. Deviations from the Einstein relation may occur in systems that are far from equilibrium or when particle interactions become significant. The original derivation considered spherical particles in a viscous fluid, but more general forms exist for anisotropic systems.