Transport coefficients

A transport coefficient <math>\gamma</math> measures how rapidly a perturbed system returns to equilibrium.

The transport coefficients occur in transport phenomenon with transport laws <math> {\mathbf{J} {_k}} \, = \, \gamma_k \, \mathbf{X} {_k}</math> where:

<math> {\mathbf{J}{_k}} </math> is a flux of the property <math> k </math>

the transport coefficient <math> \gamma _k </math> of this property <math> k </math>

<math> {\mathbf{X}{_k}} </math>, the gradient force which acts on the property <math> k </math>.

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

<math>\gamma= \int_0^\infty \langle \dot{A}(t) \dot{A}(0) \rangle \, dt,</math>

where <math>A</math> is an observable occurring in a perturbed Hamiltonian, <math>\langle \cdot \rangle</math> is an ensemble average and the dot above the A denotes the time derivative.<ref>Water in Biology, Chemistry, and Physics: Experimental Overviews and Computational Methodologies, G. Wilse Robinson, Template:ISBN, p. 80, Google Books</ref> For times <math>t</math> that are greater than the correlation time of the fluctuations of the observable the transport coefficient obeys a generalized Einstein relation:

<math>2t\gamma=\langle |A(t)-A(0)|^2 \rangle.</math>

In general a transport coefficient is a tensor.

Transport coefficients, like shear viscosity and bulk viscosity, are important for the description of fireball expansion in heavy-ion collisions.

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)
• Mass transport coefficient
• Shear viscosity <math>\eta = \frac{1}{k_BT V} \int_0^\infty dt \, \langle \sigma_{xy}(0) \sigma_{xy} (t) \rangle</math>, where <math>\sigma</math> is the viscous stress tensor (see Newtonian fluid)
• Electrical conductivity

See also

• Linear response theory
• Onsager reciprocal relations

References

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