Self-Assembling Families of LC Ladder Circuits With Frequency-Controlled Growth

A family of LC ladder circuits are analyzed as an abstract model for growth in a diverse set of systems, with possible applications to biological organisms, self-assembly of nanostructures, models of topological insulators, and classical simulation of quantum circuits. In the LC circuit tile assembly model (lc-CTAM), tiles of inductors and capacitors attach to a growing assembly if the magnitude of the electric node potential at the tip is greater than a threshold. The frequency response, including poles and zeroes, and time and space behavior of the node potentials are characterized as a function of length and the circuit parameters. When a resistance is present, growth is always bounded. As the resistance goes to zero, growth is bounded for low frequencies. At higher frequencies, growth can be both bounded or unbounded, depending on frequency and circuit parameters. In some instances, the distribution of node potentials as a function of length exhibits complex, aperiodic behavior. Both the lc-CTAM model and the methods used to analyze it, including Chebyshev polynomials, are a novel application of lumped circuit analysis. Through exact mathematical characterization, the lc-CTAM demonstrates how complex behavior can arise from simple systems, has the potential to inform investigations of biological growth, and may assist in the design of new materials and computational paradigms.