Thermodynamic behaviour of two-dimensional vesicles revisited

We study pressurised self-avoiding ring polymers in two dimensions using Monte Carlo simulations, scaling arguments and Flory-type theories, through models which generalise the model of Leibler, Singh and Fisher [Phys. Rev. Lett. Vol. 59, 1989 (1987)]. We demonstrate the existence of a thermodynamic phase transition at a non-zero scaled pressure $\tilde{p}$, where $\tilde{p} = Np/4\pi$, with the number of monomers $N \rightarrow \infty$ and the pressure $p \rightarrow 0$, keeping $\tilde{p}$ constant, in a class of such models. This transition is driven by bond energetics and can be either continuous or discontinuous. It can be interpreted as a shape transition in which the ring polymer takes the shape, above the critical pressure, of a regular N-gon whose sides scale smoothly with pressure, while staying unfaceted below this critical pressure. In the general case, we argue that the transition is replaced by a sharp crossover. The area, however, scales with $N^2$ for all positive $p$ in all such models, consistent with earlier scaling theories.Comment: 6 pages, 4 figures, EPL forma

Non-monotonic behavior of timescales of passage in heterogeneous media: Dependence on the nature of barriers

Usually time of passage across a region may be expected to increase with the number of barriers along the path. Can this intuition fail depending on the special nature of the barrier? We study experimentally the transport of a robotic bug which navigates through a spatially patterned array of obstacles. Depending on the nature of the obstacles we call them either entropic or energetic barriers. For energetic barriers we find that the timescales of first passage vary non-monotonically with the number of barriers, while for entropic barriers first passage times increase monotonically. We perform an exact analytic calculation to derive closed form solutions for the mean first passage time for different theoretical models of diffusion. Our analytic results capture this counter-intuitive non-monotonic behaviour for energetic barriers. We also show non-monotonic effective diffusivity in the case of energetic barriers. Finally, using numerical simulations, we show this non-monotonic behaviour for energetic barriers continues to hold true for super-diffusive transport. These results may be relevant for timescales of intra-cellular biological processes

Asymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter

We study the inflated phase of two dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight \mu^t \exp[- Jb] is associated to a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity \mu and the bending rigidity J. In the limit \mu -> 0, the mean perimeter has the asymptotic behaviour \avg{t}/4 \sqrt{A} \simeq 1 - K(J)/(\ln \mu)^2 + O (\mu/ \ln \mu) . The constant K(J) is found to be the same for both types of polygons, suggesting that self-avoiding polygons should also exhibit the same asymptotic behaviour.Comment: 10 pages, 3 figure

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter $N$ and variable area, associating a weight $\exp[pA - Jb ]$ to a polygon with area $A$ and $b$ bends. For convex and column-convex polygons, we show that $/A_{max} = 1 - K(J)/\tilde{p}^2 + \mathcal{O}(\rho^{-\tilde{p}})$, where $\tilde{p}=pN \gg 1$, and $\rho<1$. The constant $K(J)$ is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for $J \neq 0$. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.Comment: 7 page