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A transport coefficient γ γ {\displaystyle \gamma } measures how rapidly a perturbed system returns to equilibrium.

The transport coefficients occur in transport phenomenon with transport laws

J k = γ γ k X k {\displaystyle \mathbf {J} _{k}=\gamma _{k}\mathbf {X} _{k}}

where:

J k {\displaystyle \mathbf {J} _{k}} is a flux of the property k {\displaystyle k}

the transport coefficient γ γ k {\displaystyle \gamma _{k}} of this property k {\displaystyle k}

X k {\displaystyle \mathbf {X} _{k}} , the gradient force which acts on the property k {\displaystyle k} .

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

γ γ = ∫ ∫ 0 ∞ ∞ ⟨ A ˙ ˙ ( t ) A ˙ ˙ ( 0 ) ⟩ d t , {\displaystyle \gamma =\int _{0}^{\infty }\left\langle {\dot {A}}(t){\dot {A}}(0)\right\rangle \,dt,}

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

2 t γ γ = ⟨ | A ( t ) − − A ( 0 ) | 2 ⟩ . {\displaystyle 2t\gamma =\left\langle |A(t)-A(0)|^{2}\right\rangle .}

In general a transport coefficient is a tensor.

• Examples

• Transport coefficients of higher order

• See also

• References

• 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 η = 1 k B T V ∫ ∫ 0 ∞ ∞ d t ⟨ ⟨ σ σ x y ( 0 ) σ σ x y ( t ) ⟩ ⟩ {\displaystyle \eta ={\frac {1}{k_{B}TV}}\int _{0}^{\infty }dt\,\langle \sigma _{xy}(0)\sigma _{xy}(t)\rangle } , where σ σ {\displaystyle \sigma } is the viscous stress tensor (see Newtonian fluid)

• Electrical conductivity

For strong gradients the transport equation typically has to be modified with higher order terms (and higher order Transport coefficients).cite-ref-2[2]

• Linear response theory

• Onsager reciprocal relations

cite-note-11. ↑ Water in Biology, Chemistry, and Physics: Experimental Overviews and Computational Methodologies, G. Wilse Robinson, ISBN 9789810224516, p. 80, Google Books

cite-note-22. ↑ Kockmann, N. (2007). Transport Phenomena in Micro Process Engineering. Deutschland: Springer Berlin Heidelberg, page 66, Google books