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 $$\eta = \frac{1}{k_BT V} \int_0^\infty dt \, \langle \sigma_{xy}(0) \sigma_{xy} (t) \rangle$$, 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).