Transport coefficient explained

A transport coefficient

\gamma measures how rapidly a perturbed system returns to equilibrium.

The transport coefficients occur in transport phenomenon with transport laws

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

where:

J _{k} is a flux of the property k

the transport coefficient

\gamma _{k} of this property k X _{k} , the gradient force which acts on the property k .

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

\gamma = \int ^{inf ty} _{0} \left\langle • A (t) • A ( 0 ) \right\rangle dt,

where

A is an observable occurring in a perturbed Hamiltonian, \langle ⋅ \rangle is an ensemble average and the dot above the A denotes the time derivative.^[1] For times t that are greater than the correlation time of the fluctuations of the observable the transport coefficient obeys a generalized Einstein relation:2 t\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

η = (1)/(k _{BT} V) \int ^{inf ty} _{0} 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).^[2]

See also

• Linear response theory
• Onsager reciprocal relations

References

1. Water in Biology, Chemistry, and Physics: Experimental Overviews and Computational Methodologies, G. Wilse Robinson,, p. 80, Google Books
2. Kockmann, N. (2007). Transport Phenomena in Micro Process Engineering. Deutschland: Springer Berlin Heidelberg, page 66, Google books