Nathan T. Moore

Topologically driven swelling of a polymer loop

Numerical studies of the average size of trivially knotted polymer loops with no excluded volume were undertaken. Topology was identified by Alexander and Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration radius, and probability density distributions as functions of gyration radius were generated for loops of up to N = 3,000 segments. Gyration radii of trivially knotted loops were found to follow a power law similar to that of self-avoiding walks consistent with earlier theoretical predictions.

Limits of analogy between self-avoidance and topology-driven swelling of polymer loops

The work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length, where the effects of knotting are small. A simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data. Contrast between this expression and the well-known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots, which may be useful for understanding where the analogy between the two physical forms is valid. The work also includes description of a new method for the computational generation of polymer loops via conditional probability. Although computationally intensive, this method generates loops without statistical bias, and thus is preferable to other loop generation routines in the region of interest.

Small Oscillations via Conservation of Energy.

The work describes an analogy-based small oscillations analysis of a standard static equilibrium lab problem. In addition to force analysis, a potential energy function for the system is developed, and by drawing out mathematical similarities to the simple harmonic oscillator, we are able to describe (and verify) the period of small oscillations about the static equilibrium state. The problem was developed and implemented in a standard University Physics course at Winona State University.