# The Cluster's Transport Phenomena

A page about transport phenomena — the three parallel transport processes: viscosity, thermal conductivity, and electrical conductivity.

## The three transport coefficients

Transport phenomena study the flow of extensive quantities (momentum, energy, charge) in response to gradients:
- **Viscosity (eta)**: Flow of momentum in response to a velocity gradient (shear stress = eta du/dy)
- **Thermal conductivity (k)**: Flow of heat in response to a temperature gradient (Fourier's law: j_q = -k nabla T)
- **Electrical conductivity (sigma)**: Flow of charge in response to an electric field (Ohm's law: j = sigma E)

In the cluster, the edit three transport coefficients give an edit momentum flow.

## The kinetic theory results

From kinetic theory of gases (assuming Maxwell-Boltzmann distribution, mean free path lambda):
- eta = (1/3) n m v_bar lambda (independent of density!)
- k = (1/3) n c_v v_bar lambda (where c_v is the specific heat per molecule)
- sigma = n e^2 tau / m (Drude model, where tau = lambda / v_bar)

The striking prediction: eta ~ T^{1/2} (increases with temperature) and is independent of density at moderate pressures. This is experimentally verified for gases.

In the cluster, the edit kinetic theory results give an edit density prediction.

## The Wiedemann-Franz law

For metals, the ratio of thermal to electrical conductivity is:
k / (sigma T) = (pi^2 / 3) (k_B / e)^2 = L_0 = 2.44 x 10^{-8} W Ohm / K^2
This universal ratio L_0 is the Lorenz number. The Wiedemann-Franz law follows because both heat and charge are carried by free electrons.

In the cluster, the edit Wiedemann-Franz law gives an edit universal ratio.

## The Einstein relation

The Einstein relation connects the diffusion coefficient D to the mobility mu:
D / mu = k_B T / q
where q is the charge of the diffusing particle. For an electric charge carrier: D / mu = k_B T / e. This relates the random walk (diffusion) to the directed motion (mobility).

In the cluster, the edit Einstein relation connects an edit diffusion coefficient.

## Applications

- **Engineering**: Design of heat exchangers, lubrication systems, electrical circuits
- **Materials science**: Thermal management of electronic devices, designing thermoelectric materials (low k / high sigma)
- **Geophysics**: Mantle convection, magma transport
- **Astrophysics**: Stellar interiors, accretion disks
- **Biophysics**: Nerve impulse propagation (action potentials)

In the cluster, edit applications include:
- edit Engineering
- edit Materials science
- edit Geophysics
- edit Astrophysics
- edit Biophysics

## This phenomena

This page is about transport phenomena. eta = (1/3) n m v_bar lambda. k / (sigma T) = L_0. D / mu = k_B T / q. The phenomena is real.