Quasi-one-dimensional models

Quasi-one-dimensional models (Q1D)[1] 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

Demonstration of quasi-one-dimensional topologies with different topology parameters, e.g. intersections, average particles per lobe and number of particles.

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:

  1. A one-dimensional line twisted one or more times to create junctions.
  2. 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.[2][3] If a point moves on a quasi-one-dimensional geometry it is called motion in quasi-one-dimensional models.[4] 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.

Demonstration of a one-dimensional circle transforming into figure-8 (lemniscate) shape. Due to the junction in the figure-8 shape, the one-dimensional circular geometry, becomes quasi-one-dimensional geometry.

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 [4] by virtue of the junctions. As the number of junctions grow, the geometry and associated dynamics start showing higher-dimensional features.[1]

Dynamics

Illustration of diffusive particle motion in quasi-two dimensional models. Below a critical density, the a particle position would be completely randomized after waiting for sufficiently long time. This time scale is the rearrangement time scale of the glassy 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.

Directed graph of J=2, N=7, K=1 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.

Demonstration of 3D quasi-one-dimensional model. In three dimensions, the particles are in the shape of a tube.

Applications

Some of the applications of Q1D models include the studies of solvable glassy dynamics[5] 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[6] that can be simulated on scalable computer to study very long time scale rearrangement dynamics.

Notes

  1. ↑ 1.0 1.1 'Quasi-One Dimensional Models for Glassy Dynamics',"http://arxiv.org/pdf/1401.0960v1.pdf".
  2. ↑ The Knot Book,"http://www.amazon.com/The-Knot-Book-Colin-Adams/dp/0821836781" .
  3. ↑ The Knot Atlas,"http://katlas.math.toronto.edu/wiki/Image:Rolfsen_240.png" .
  4. ↑ 4.0 4.1 'Minimal Model for Kinetic Arrest, Phys. Rev. E, Volume 78, Issue 1, "http://pre.aps.org/abstract/PRE/v78/i1/e011111".
  5. ↑ Glassy dynamics, "http://www.scholarpedia.org/article/Glassy_dynamics".
  6. ↑ Glassy dynamics of kinetically constrained models, Advances in Physics, Volume 52, Issue 4, 2003.