Long Range Interacting Systems PDF Download

It is well known that some driven systems undergo transitions when a system parameter is changed adiabatically around a critical value. This transition can be the result of a fundamental change in the structure of the phase space, called a bifurcation. Most of these transitions are well classified in the theory of bifurcations. Among the driven systems, spatiotemporally periodic (STP) potentials are noteworthy due to the intimate coupling between their time and spatial components. A paradigmatic example of such a system is the Kapitza pendulum, which is a pendulum with an oscillating suspension point. The Kapitza pendulum has the strange property that it will stand stably in the inverted position for certain driving frequencies and amplitudes. A particularly interesting and useful STP system is an array of parallel electrodes driven with an AC electrical potential such that adjacent electrodes are 180 degrees out of phase. Such an electrode array embedded in a surface is called an Electric Curtain (EC). As we will show, by using two ECs and a quadrupole trap it is posible to produce an electric potential simular in form to that of the Kapitza pendulum. Here I will present the results of four related pieces of work, each focused on understanding the behaviors STP systems, long-range interacting particles, and long-range interacting particles in STP systems. I will begin with a discussion on the experimental results of the EC as applied to the cleaning of solar panels in extraterrestrial environments, and as a way to produce a novel one-dimensional multiparticle STP potential. Then I will present a numerical investigation and dynamical systems analysis of the dynamics that may be possible in an EC. Moving to a simpler model in order to explore the rudimentary physics of coulomb interactions in a STP potential, I will show that the tools of statistical mechanics may be important to the study of such systems to understand transitions that fall outside of bifurcation theory. Though the Coulomb and, similarly, gravitational interactions of particles are prevalent in nature, these long-range interactions are not well understood from a statistical mechanics perspective because they are not extensive or additive. Finally, I will present a simple model for understanding long-range interacting pendula, finding interesting non-equilibrium behavior of the pendula angles. Namely, that a quasistationary clustered state can exist when the angles are initially ordered by their index.