In this paper, we investigate the equilibrium and non-equilibrium properties of a model that shares several important characteristics with charged particles interacting in an Electric Curtain (EC) device. An EC comprises a periodic array of parallel electrodes, applied to each is an alternating electric potential. Depending on the applied potentials and the geometry of the electrodes, a wide variety of field structures above the plane of the electrodes are possible. The EC has multiple applications in the control and manipulation of small particles, but is under utilized in industry and science because of difficulties in predicting and understanding the particle dynamics. One particular challenge in understanding the dynamics is the many-body coulomb interactions. To better understand the role of the interactions, we study a one-dimensional analytically tractable model that encapsulates their long-range nature. Specifically, we study a Hamiltonian similar to that of the Hamiltonian mean field model but with the inclusion of an index dependent phase in the interaction term that, as we show, reflects the periodic structure of an EC field. We solve for the canonical partition function and also investigate some of the non-equilibrium behaviors. In the study of the non-equilibrium behaviors, we find an interesting property, namely that a quasistationary (lifetime diverges as the number of particles is increased) clustered state can exist when an initial configuration is ordered by the particle indices.
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Myers O, Del Maestro A, Wu J, Marshall JS. Long-range interacting pendula: A simple model for understanding complex dynamics of charged particles in an electronic curtain device. Journal of Applied Physics. 2017 Apr 21;121(15):154501.