Transport coefficient

A transport coefficient γ measures how rapidly a perturbed system returns to equilibrium. The transport coefficients occur in transport phenomenon with transport laws

𝐉k=γk𝐗k

where:

𝐉k is a flux of the property k
the transport coefficient γk of this property k
𝐗k, the gradient force which acts on the property k.

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

γ=∫0∞⟨A˙(t)A˙(0)⟩dt,

where A is an observable occurring in a perturbed Hamiltonian, ⟨⋅⟩ 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:

2tγ=⟨|A(t)−A(0)|2⟩.

In general a transport coefficient is a tensor.

## Contents

* 1 Examples
* 2 Transport coefficients of higher order
* 3 See also
* 4 References

## 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 η=1kBTV∫0∞dt⟨σxy(0)σxy(t)⟩, where σ 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, ISBN 9789810224516, p. 80, Google Books
2. ↑ Kockmann, N. (2007). Transport Phenomena in Micro Process Engineering. Deutschland: Springer Berlin Heidelberg, page 66, Google books

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