Therefore, Φ Φ is a mapping from the ( s − − 1 ) - unit simplex into itself, where s stands for the cardinality of the set S. When s is too large, solving equation (1) is intractable or computationally very costly. One natural way to approximate these evolution equations is to reduce sequentially the state space using a mean field particle model. One of the simplest mean field simulation scheme is defined by the Markov chain