Invariant measures of Mass Migration Processes

Monotonicity and condensation in homogeneous stochastic particle systems

“…Misanthrope processes include many well-known examples of interacting particle systems, such as zero-range processes [1], the inclusion process [40,41], and the explosive condensation model [42]. It is known [2,27] that misanthrope processes with translation invariant dynamics p(x, y) = q(x − y) exhibit stationary product measures if and only if the rates fulfil…”

“…, L} and note that the process preserves particle number x η x = N . It is known [27,43] that these processes exhibit stationary product measures if and only if the jump rates have the explicit form…”

“…We note these conditions arise from a special case of the results in [17,Theorem 2.11] on generalised misanthrope models, since α k (n) depends only on the occupation of the exit site and not the entry site. In this class, which is also discussed in detail in [27], condensing processes which are monotone, homogeneous, and have stationary product measures with a power tail w(n) ∼ n −b with b ∈ (1, 3/2] are conjectured to exist. As an example, consider the gZRP with rates given by…”

“…When the tail of the stationary measure is a power law and decays faster than n −3/2 with the occupation number n, we prove that the process is still non-monotone. In Section 5 we present preliminary results for tails that decay slower than n −3/2 , which strongly suggests that a monotone and condensing particle system exists (see [27] for further discussion).…”