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:

${\displaystyle \gamma ={\int }_0^{\mathrm{\infty }}⟨\dot{A}(t)\dot{A}(0)⟩dt,}$

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

${\displaystyle 2t\gamma =⟨|A(t)-A(0){|}^2⟩.}$

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 ${\displaystyle \eta =\frac{1}{k_{B}TV}{\int }_0^{\mathrm{\infty }}dt⟨{\sigma }_{xy}(0){\sigma }_{xy}(t)⟩}$, 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

References

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