  In both quantum optics and cold atom physics, the behaviour of bosonic
photons and atoms is often treated using phase space methods, where mode
annihilation and creation operators are represented by c-number phase space
variables, with the density operator equivalent to a distribution function of
these variables. The anti-commutation rules for fermion annihilation, creation
operators suggests the possibility of using anti-commuting Grassmann variables
to represent these operators. However, in spite of the seminal work by Cahill
and Glauber and a few applications, the use of Grassmann phase space methods in
quantum - atom optics to treat fermionic systems is rather rare, though fermion
coherent states using Grassmann variables are widely used in particle physics.
  This paper presents a phase space theory for fermion systems based on
distribution functionals, which replace the density operator and involve
Grassmann fields representing anti-commuting fermion field annihilation,
creation operators. It is an extension of a previous phase space theory paper
for fermions (Paper I) based on separate modes, in which the density operator
is replaced by a distribution function depending on Grassmann phase space
variables which represent the mode annihilation and creation operators. This
further development of the theory is important for the situation when large
numbers of fermions are involved, resulting in too many modes to treat
separately. Here Grassmann fields, distribution functionals, functional
Fokker-Planck equations and Ito stochastic field equations are involved.
Typical applications to a trapped Fermi gas of interacting spin 1/2 fermionic
atoms and to multi-component Fermi gases with non-zero range interactions are
presented, showing that the Ito stochastic field equations are local in these
cases. Solutions for spin 1/2 fermions either free or in lattices are found.
