  We present the thermodynamic Bethe ansatz as a way to factorize the partition
function of a 2d field theory, in particular, a conformal field theory and we
compare it with another approach to factorization due to K. Schoutens which
consists of diagonalizing matrix recursion relations between the partition
functions at consecutive levels. We prove that both are equivalent, taking as
examples the SU(2) spinons and the 3-state Potts model. In the latter case we
see that there are two different thermodynamic Bethe ansatz equation systems
with the same physical content, of which the second is new and corresponds to a
one-quasiparticle representation, as opposed to the usual two-quasiparticle
representation. This new thermodynamic Bethe ansatz system leads to a new
dilogarithmic formula for the central charge of that model.
