A transport coefficient 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 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 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 = ∫ 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. 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 = ⟨ | 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 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 = 1 k B T V ∫ 0 ∞ d t ⟨ σ x y ( 0 ) 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 {\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).
See also Linear response theory Onsager reciprocal relations
References