Entropy production moment closures and effective transport coefficients

If transport of a given (classical, fermionic, or bosonic) particle species in media is described by a Boltzmann transport equation (BTE), it is often expedient to solve this BTE in the framework of a moment expansion of the particle distribution function, while an exact solution or simulation of the problem with real material properties and complex geometries is unpractical or even unfeasible. Whereas for local thermal equilibrium (LTE) the well-known hydrodynamic equations for the densities of the conserved quantities are derived from the BTE, for non-LTE it is not obvious how to define moments and to close the truncated hierarchy of partial differential equations for these moments. This paper reviews a closure based on entropy production rate minimization, which is applicable to incoherent transport of independent particles in non-LTE interacting with an LTE-medium. The BTE is then linear, includes emission-absorption and elastic scattering processes, and is equivalent to radiative transfer equations. In a large range from diffusive (opaque media) to ballistic (transparent media) transport behaviour, the closure provides useful mean transport coefficients that are exact in the LTE limit, in contrast to the often used maximum entropy moment closure. After an introduction into the underlying theory for massive and wave-like particles, two illustrative examples are discussed. First, the two-moment approximation of radiative heat transfer is reviewed and effective absorption coefficients and the Eddington factor are calculated for a real absorption spectrum. Secondly, the approach is applied to semi-classical electric transport in mesoscopic systems and is shown to provide the correct conductance of a quasi-one-dimensional ballistic conductor with elastic scattering.