Transport coefficient

A transport coefficient ${\displaystyle \gamma }$ measures how rapidly a perturbed system returns to equilibrium.

The transport coefficients occur in transport phenomenon with transport laws

${\displaystyle \mathbf {J} _{k}=\gamma _{k}\mathbf {X} _{k}}$

where:

${\displaystyle \mathbf {J} _{k}}$ is a flux of the property ${\displaystyle k}$

the transport coefficient ${\displaystyle \gamma _{k}}$ of this property ${\displaystyle k}$

${\displaystyle \mathbf {X} _{k}}$, the gradient force which acts on the property ${\displaystyle k}$.

Transport coefficients can be expressed via a Green–Kubo relation:

${\displaystyle \gamma =\int _{0}^{\infty }\left\langle {\dot {A}}(t){\dot {A}}(0)\right\rangle \,dt,}$

where ${\displaystyle A}$ is an observable occurring in a perturbed Hamiltonian, ${\displaystyle \langle \cdot \rangle }$ is an ensemble average and the dot above the A denotes the time derivative.^[1] For times ${\displaystyle t}$ that are greater than the correlation time of the fluctuations of the observable the transport coefficient obeys a generalized Einstein relation:

${\displaystyle 2t\gamma =\left\langle |A(t)-A(0)|^{2}\right\rangle .}$

In general a transport coefficient is a tensor.

Examples

• Diffusion constant, relates the flux of particles with the negative gradient of the concentration (see Fick's laws of diffusion)
• Thermal conductivity (see Fourier's law)
• Ionic conductivity
• Mass transport coefficient
• Shear viscosity ${\displaystyle \eta ={\frac {1}{k_{B}TV}}\int _{0}^{\infty }dt\,\langle \sigma _{xy}(0)\sigma _{xy}(t)\rangle }$, where ${\displaystyle \sigma }$ is the viscous stress tensor (see Newtonian fluid)
• Electrical conductivity

Transport coefficients of higher order

For strong gradients the transport equation typically has to be modified with higher order terms (and higher order Transport coefficients).^[2]

See also

• Linear response theory
• Onsager reciprocal relations