Z. Angew. Math. Phys. (2017) 68:100
2017 Springer International Publishing AG
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Kershaw-type transport equations for fermionic radiation
and Wieslaw Larecki
Abstract. Besides the maximum entropy closure procedure, other procedures can be used to close the systems of spectral
moment equations. In the case of classical and bosonic radiation, the closed-form analytic Kershaw-type and B-distribution
closure procedures have been used. It is shown that the Kershaw-type closure procedure can also be applied to the spectral
moment equations of fermionic radiation. First, a description of the Kershaw-type closure for the system consisting of
an arbitrary number of one-dimensional moment equations is presented. Next, the Kershaw-type two-ﬁeld and three-ﬁeld
transport equations for fermionic radiation are analyzed. In the ﬁrst case, the independent variables are the energy density
and the heat ﬂux. The second case includes additionally the ﬂux of the heat ﬂux as an independent variable. The general-
ization of the former two-ﬁeld case to three space dimensions is also presented. The fermionic Kershaw-type closures diﬀer
from those previously derived for classical and bosonic radiation. It is proved that the obtained one-dimensional systems of
transport equations are strictly hyperbolic and causal. The fermionic Kershaw-type closure functions behave qualitatively
in the same way as the fermionic maximum entropy closure functions, but attain diﬀerent numerical values.
Mathematics Subject Classiﬁcation. 85A25, 82C40, 26D15, 35L02, 35L40.
Keywords. Fermionic radiation, Moment equations, Moment realizability problem, Kershaw-type closure, Three-moment
In the kinetic theory of radiation, various types of moments and hence diﬀerent types of moment equa-
tions have been introduced depending on what pattern of integration of the radiative transfer equation
multiplied by the prescribed weight functions over the space of a particle wave vector k is used. Here, the
spectral (frequency-dependent) type of moment equations is considered. It is obtained by decomposing
the inﬁnitesimal wave-vector volume element d
k as d
k = k
g, where k is the magnitude of k and
g is the unit vector in the direction of k and subsequently performing only the integration over g ∈ S
Most of the literature dealing with radiation transport is focused on classical and bosonic radiation.
Our paper focuses on fermionic radiation [1–10]. Fermionic radiation may be considered as a gas of
massless fermions. The fundamental object for describing this gas is a distribution function, denoted f.
The Pauli exclusion principle requires that f must satisfy the condition 0 ≤ f ≤ 1, which imposes various
restrictions on the admissible values of the moments of f. These restrictions are not the same as those
encountered in the case of classical and bosonic radiation, where f obeys the condition 0 ≤ f ≤∞.
For the investigation of a radiation ﬁeld, both the direct numerical methods of integration of the
radiative transfer equation and the moment methods can be used. Concerning the moment methods,
the form of the closure of the ﬁnite system of moment equations plays a crucial role, and various closure
procedures have been proposed. One of them, the linear closure procedure has several drawbacks described
in Ref. . As to the nonlinear closure procedures which do not have these drawbacks, besides the most
commonly used maximum entropy closure, the Kershaw-type and B-distribution closures have been
applied to the moment equations in slab and higher-dimensional geometries to describe classical and