Transport coefficient
# Transport coefficient

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

The transport coefficients occur in transport phenomenon with transport laws

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

where:

    $\mathbf {J} _{k}$ is a flux of the property $k$
    the transport coefficient $\gamma _{k}$ of this property $k$
    $\mathbf {X} _{k}$, the gradient force which acts on the property $k$.

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

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

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

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

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 $\eta ={\frac {1}{k_{B}TV}}\int _{0}^{\infty }dt\,\langle \sigma _{xy}(0)\sigma _{xy}(t)\rangle$, where $\sigma$ 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).

