Quasi-one-dimensional models

Quasi-one-dimensional models (Q1D)<ref name ="quasipaper"/> are particle dynamics simulation models developed at Yale University by Prasanta Pal, Corey O'Hern, and Jerzy Blawzdziewicz. These models elucidate the relationship between kinetic arrest, dimensional coupling, close packing and diffusion process. Q1D is one of the few exactly solvable kinetic models in the context of soft condensed matter Physics.
Model Definition
Model Properties
The model consists of quasi-one-dimensional geometry wherein the dynamics of interacting particles are constrained by this geometry.
The abstract form of quasi-one-dimensional geometry is the '+' (plus) symbol. The center of the '+' is called the intersection or junction (J). In this model, the two fundamental constituents of quasi-one-dimensional geometry are:
# A one-dimensional line twisted one or more times to create junctions.
# Particles (hard or soft) moving under the geometrical and inter-particle interaction constraints.
The complexity of the intersections and topologies are described in knot theory.<ref name="KnotBook"/><ref name="KnotAtlas"/> If a point moves on a quasi-one-dimensional geometry it is called motion in quasi-one-dimensional models.<ref name="figure8paper"/> The characteristic motion of a set of particles in a quasi-one-dimensional geometry is controlled by the dynamics of the interacting particles and the dynamical control mechanism associated with the junction or junction dynamics. The simplest form of junction dynamics is that of a traffic junction, where motion along only one particular direction of the junction is allowed at any instant of time. The occupation of the junction in one direction forbids flow through the junction in the perpendicular (conjugate) direction.
Elementary Form
The elementary form of the quasi-one-dimensional geometry is a twisted circle that looks like the shape of figure-8 or lemniscate of Bernoulli. Quasi-one-dimensional geometry possesses both one-dimensional (far away from the junction) and two dimensional (near the junction) features. In nature, quasi-one-dimensional geometry is observed in the geometry of the DNA supercoil, nephron, circulatory system, etc. This geometry mimics the motion of particles in a dense environment <ref name="figure8paper"/> by virtue of the junctions.
As the number of junctions grow, the geometry and associated dynamics start showing higher-dimensional features.<ref name ="quasipaper"/>
Dynamics
The system dynamics is controlled by the traffic rule at the junction. Although the model has been developed as a continuous system, it can be mapped into a directed graph that can be used to calculate the dynamical timescale of the system.
The initial models were developed to study single file hard particle diffusive dynamics, however other kinds of dynamics, e.g., motion under Lennard-Jones potential, are still open problems. This models can be extended to the 3D quasi models to understand 3D motion.
Applications
Some of the applications of Q1D models include the studies of solvable glassy dynamics<ref name="glassydynamics"/> and the glass transition, the high dimensional-Wiener process, chaotic dynamical systems, close packing problems and microfluidics . This is one of the first few examples of a kinetically constrained model<ref name="kineticallyconstrained"/> that can be simulated on scalable computer to study very long time scale rearrangement dynamics.

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