An Exchangeable Pair-Hawkes Reference Model and a Cumulant Obstruction for Hard-Sphere Collision Statistics
Collision counts in a dilute gas of hard particles reflect the dependence between successive encounters. We study whether these fluctuations can be represented by a positive linear Hawkes process, in which each event adds a nonnegative contribution to the rate of future events. We construct such a model for particle pairs, require invariance under particle relabeling, and derive its stationary mean and covariance. We prove that, for a stationary positive Hawkes process with finitely many event types and finite expected total family sizes, the long-time growth rate of the third centered moment of any count formed by selecting event types is at least that of its variance. More generally, the long-time growth rates are nondecreasing with cumulant order. These predicted growth rates therefore cannot be matched simultaneously to the ones found in literature in approximations that follow velocity and collision count for a particle in an equilibrium gas. Simulations support the covariance predictions and illustrate the dependence of the physical comparison on the observation window. This restriction motivates models that retain the state left by a collision or allow past events to reduce future rates.