Polygons in restricted geometries subjected to infinite forces

Abstract

We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z 3 . These polygons are subjected to a force f , parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. We obtain and prove the asymptotic form of the free energy in both limits f → ±∞. We conjecture that the f → −∞ asymptote is the same as the limiting free energy of "Hamiltonian" polygons, polygons which visit every vertex in a L × M × N box. We investigate such polygons, and in particular use a transfer-matrix methodology to establish that the conjecture is true for some small tube sizes.Dedicated to Anthony J. Guttmann on the occasion of his 70 th birthday. arXiv:1604.07465v1 [math-ph]

Characterising knotting properties of polymers in nanochannels

“…with F T ( f ) asymptotic to the lower bound for f → ∞ for any T, and for f → −∞ for small tube sizes (this is conjectured to be true for any T), see [5].…”

“…where c n is the number of n-step self-avoiding walks (SAWs) in Z 3 starting at the origin and κ is their connective constant, and c T,n is the number of these confined to T. A subset of self-avoiding polygons in T are Hamiltonian polygons: those which occupy every vertex in a s × L × M subtube of T. These serve as an idealised model of tightly packed ring polymers, in addition to being a useful lower bound for general polygons in the f < 0 compressed regime. We define the number of Hamiltonian polygons, p H T,n , to be the number of n-edge polygons in P T,n which have span s and occupy every vertex in an s × L × M subtube of T. We define W = (L + 1)(M + 1) (the number of vertices in an integer plane x = i ≥ 0 of the tube) and will assume without loss of generality that L ≥ M; note that p H T,n = 0 if n is not a multiple of W. The following limit has been proved to exist [5] (see also [7]):…”

“…Here, we will also be interested in the dual model, called the "fixed-span" model, with partition function given by For this partition function, when g 0 densely packed (in terms of number of edges per span) polygons dominate the partition function, while when g 0 polygons with very few edges per span dominate. For this model, the probability associated with a span s polygon π is given by P (sp,g) s (π) = e g|π| Q T,s (g) and the associated (limiting) free energy per span exists [5] (see also [3]):…”

Knotting statistics for polygons in lattice tubes

“…with F T (f ) asymptotic to the lower bound for f → ∞ for any T, and for f → −∞ for small tube sizes (this is conjectured to be true for any T), see [36]. More specifically, in [36] it is established that: lim…”

“…where t F T,s is the number of full s-patterns. In [36] it was established that β F T /W T = κ H T for all tubes such that 5 ≥ L ≥ M ≥ 0; this is because Hamiltonian s-patterns are the dominant class amongst full s-patterns. It is also known that for any tube dimensions lim f →−∞ F T (f ) = β F T /W T .…”