Einstein relation (kinetic theory)

In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected^{[clarification needed]} connection revealed independently by William Sutherland in 1904,^[1][2][3] Albert Einstein in 1905,^[4] and by Marian Smoluchowski in 1906^[5] in their works on Brownian motion. The more general form of the equation in the classical case is^[6]

$${\displaystyle D=\mu \,k_{\text{B}}T,}$$

• D is the diffusion coefficient;
• μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = v_d/F;
• k_B is the Boltzmann constant;
• T is the absolute temperature.

This equation is an early example of a fluctuation-dissipation relation.^[7] Note that the equation above describes the classical case and should be modified when quantum effects are relevant.

Two frequently used important special forms of the relation are:

• Einstein–Smoluchowski equation, for diffusion of charged particles:^[8]
  $${\displaystyle D={\frac {\mu _{q}\,k_{\text{B}}T}{q}}}$$

  
• Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with low Reynolds number:
  $${\displaystyle D={\frac {k_{\text{B}}T}{6\pi \,\eta \,r}}}$$

  

Here

• q is the electrical charge of a particle;
• μ_q is the electrical mobility of the charged particle;
• η is the dynamic viscosity;
• r is the radius of the spherical particle.