Transport coefficient

A transport coefficient [math]\displaystyle{ \gamma }[/math] measures how rapidly a perturbed system returns to equilibrium. The transport coefficients occur in transport phenomenon with transport laws

[math]\displaystyle{ \mathbf{J}_k = \gamma_k \mathbf{X}_k }[/math]

where:

[math]\displaystyle{ \mathbf{J}_k }[/math] is a flux of the property [math]\displaystyle{ k }[/math]

the transport coefficient [math]\displaystyle{ \gamma _k }[/math] of this property [math]\displaystyle{ k }[/math]

[math]\displaystyle{ \mathbf{X}_k }[/math], the gradient force which acts on the property [math]\displaystyle{ k }[/math].

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

[math]\displaystyle{ \gamma = \int_0^\infty \left\langle \dot{A}(t) \dot{A}(0) \right\rangle \, dt, }[/math]

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

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

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 [math]\displaystyle{ \eta = \frac{1}{k_BT V} \int_0^\infty dt \, \langle \sigma_{xy}(0) \sigma_{xy} (t) \rangle }[/math], where [math]\displaystyle{ \sigma }[/math] 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

References

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