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Is there anything known about mixing time of Markov chains for three particle systems?

It is proved here http://www.ams.org/journals/tran/2005-357-08/S0002-9947-05-03610-X/S0002-9947-05-03610-X.pdf that the mixing time of an exclusion process is O(n). We can think if a exclusion process, a Markov chain on permutations of two particles (0,1). I wonder if there is any result such as that for three particles.

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    $\begingroup$ Some more detail of exactly what you're asking about would be helpful. Also, I think this problem is probably better suited to Mathematics: of course, many computer scientists use and study Markov chains but there doesn't seem to be any computational aspect to this question. $\endgroup$ – David Richerby May 28 '15 at 21:51

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