One-Dimensional Particle Clouds with Elastic Collisions

Abstract

We study an interacting particle system of a finite number of labelled particles on the integer lattice, in which particles have intrinsic masses and left/right jump rates. If a particle is the minimal-label particle at its site when it tries to jump left, the jump is executed. If not, ‘momentum’ is transferred to increase the rate of jumping left of the minimal-label particle. Similarly for jumps to the right. The collision rule is ‘elastic’ in the sense that the net rate of flow of mass is independent of the present configuration, in contrast to the exclusion process, for example. We show that the particle masses and jump rates determine explicitly, via a concave majorant of a simple ‘potential’ function associated to the masses and jump rates, a unique partition of the system into maximal stable subsystems. The internal configuration of each stable subsystem remains tight, while the location of each stable subsystem obeys a strong law of large numbers with an explicit speed. We indicate connections to adjacent models, including diffusions with rank-based coefficients, where, to the authors’ knowledge, the problem of identifying the analogous decomposition into stable sub-systems is yet to be fully solved.