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:

$\gamma =\underset{0}{\overset{\mathrm{\infty }}{\int }}⟨\dot{A}(t)\dot{A}(0)⟩dt,$

where $A$ is an observable occurring in a perturbed Hamiltonian, $⟨\cdot ⟩$ 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:

$2t\gamma =⟨|A(t)-A(0){|}^2⟩.$

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 $\eta =\frac{1}{k_{B}TV}\underset{0}{\overset{\mathrm{\infty }}{\int }}dt⟨{\sigma }_{xy}(0){\sigma }_{xy}(t)⟩$, 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).

See also

• Linear response theory
• Onsager reciprocal relations