A Counter Example to Cercignani’s Conjecture for the d Dimensional Kac Model

Abstract

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 the Maxwellian case in a series of papers by authors such as Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, but the hope to get a limiting relaxation rate for the Boltzmann equation with this linear method was already shown to be unrealistic (although the problem is still important and interesting due to its connection with the linearized Boltzmann operator). 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, Γ _N , still yields a very strong dependency in the number of particles of the problem when we consider the general case.

Appendix: A Fubini Type Theorem

This appendix contains the proof to Theorem 2, which we felt would have encumbered the main article, but pose a necessary step in the proof of our main result.

Proof of Theorem 2

The proof relies heavily on the transformation (16) and the following Fubini-like formula for spheres (which can be found in [7]):

$$ \everymath{\displaystyle }\begin{array}[b]{@{}lll} \int _{\mathbb{S}^{m-1}(r)}fd\gamma^m_r &=& \frac{ \vert\mathbb{S}^{m-j-1} \vert}{ \vert\mathbb{S}^{m-1} \vert r^{m-2}} \int_{\sum_{i=1}^j x_i^2 \leq r^2}dx_1 \dots dx_j \Biggl(r^2-\sum_{i=1}^j x_i^2 \Biggr)^{\frac{m-j-2}{2}} \cr\noalign{\vspace{3pt}} &&{} \times \int_{\mathbb{S}^{m-j-1} (\sqrt {r^2-\sum _{i=1}^j x_i^2} )}fd \gamma^{m-j}_{\sqrt{r^2-\sum_{i=1}^j x_i^2}}, \end{array} $$

(61)

where $d\gamma^{m}_{r}$ is the uniform probability measure on the appropriate sphere.

We start by defining the new variables

where R ₁,R ₂ are transformation like (16). We notice that under the above transformation the domain

$$\sum_{i=1}^N |v_i|^2=E, \qquad \sum_{i=1}^N v_i=z $$

transforms into

$$ \sum_{i=1, i\neq j}^{N-1} | \xi_i|^2+\frac{N}{N-j} \biggl(\xi_j- \frac{\sqrt{j}z}{N} \biggr)^2=E-\frac{|z|^2}{N} . $$

(62)

Denoting by $\widetilde{\xi_{j}}=\sqrt{\frac{N}{N-j}} (\xi_{j}-\frac {\sqrt {j}z}{N} )$ and using the fact that R=R ₁⊗R ₂ is orthogonal along with (61) we find that

and since

$$E-\frac{|z|^2}{N}-\sum_{i=1}^{j-1} | \xi_i|^2 -|\widetilde{\xi_j}|^2=E- \sum_{i=1}^j |v_i|^2 - \frac{|z-\sum_{i=1}^j v_i|^2}{N-j}, $$

the result follows. □