  We introduce the notion of order of magnitude reversibility
(OM-reversibility) in Markov chains that are parametrized by a positive
parameter $\ep$. OM-reversibility is a weaker condition than reversibility, and
requires only the knowledge of order of magnitude of the transition
probabilities. For an irreducible, OM-reversible Markov chain on a finite state
space, we prove that the stationary distribution satisfies order of magnitude
detailed balance (analog of detailed balance in reversible Markov chains). The
result characterizes the states with positive probability in the limit of the
stationary distribution as $\ep \to 0$, which finds an important application in
the case of singularly perturbed Markov chains that are reducible for $\ep=0$.
We show that OM-reversibility occurs naturally in macroscopic systems,
involving many interacting particles. Clustering is a common phenomenon in
biological systems, in which particles or molecules aggregate at one location.
We give a simple condition on the transition probabilities in an interacting
particle Markov chain that characterizes clustering. We show that such
clustering processes are OM-reversible, and we find explicitly the order of
magnitude of the stationary distribution. Further, we show that the single pole
states, in which all particles are at a single vertex, are the only states with
positive probability in the limit of the stationary distribution as the rate of
diffusion goes to zero.
