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We give a lower bound on the spectral gap for a class of binary collisionprocesses. In 2008, Caputo showed that, for a class of binary collisionprocesses given by simple averages on the complete graph, the analysis of thespectral gap of an $N$-component system is reduced to that of the same systemfor N=3 In this paper, we give a comparison technique to reduce the analysis ofthe spectral gap of binary collision processes given by simple averages on$d$-dimensional lattice to that on the complete graph. We also give acomparison technique to reduce the analysis of the spectral gap of binarycollision processes which are not given by simple averages to that given bysimple averages. Combining them with Caputo's result, we give a new andelementary method to obtain spectral gap estimates. The method applies to anumber of binary collision processes on the complete graph and also ond-dimensional lattice, including a class of energy exchange models which wasrecently introduced in by Grigo et al., and zero-range processes.

On the spectral gap of the kac walk and other binary collision processes on $d$-dimensional lattice