  Kac's $d$ dimensional model gives a linear, many particle, binary collision
model from which, under suitable conditions, the celebrated Boltzmann equation,
in its spatially homogeneous form, arise as a mean field limit. The ergodicity
of the evolution equation leads to questions about the relaxation rate, in hope
that such a rate would pass on the Boltzmann equation as the number of
particles goes to infinity. This program, starting with Kac and his one
dimensional 'Spectral Gap Conjecture' at 1956, finally reached its conclusion
in a series of papers by authors such as Janvresse, Maslen, Carlen, Carvalho,
Loss and Geronimo, but the hope to get a a limiting relaxation rate for the
Boltzmann equation with this linear method was already shown to be unrealistic.
A less linear approach, via a many particle version of Cercignani's conjecture,
is the grounds for this paper. In our paper, we extend recent results by the
author from the one dimensional Kac model to the $d$ dimensional one, showing
that the entropy-entropy production ratio, $\Gamma_N$, still yields a very
strong dependency in the number of particles of the problem when we consider
the general case.
