Interacting Elastic Lattice Polymers: a Study of the Free-Energy of Globular Rings

We introduce and implement a Monte Carlo scheme to study the equilibrium statistics of polymers in the globular phase. It is based on a model of "interacting elastic lattice polymers" and allows a sufficiently good sampling of long and compact configurations, an essential prerequisite to study the scaling behaviour of free energies. By simulating interacting self-avoiding rings at several temperatures in the collapsed phase, we estimate both the bulk and the surface free energy. Moreover from the corresponding estimate of the entropic exponent $\alpha-2$ we provide evidence that, unlike for swollen and $\Theta$-point rings, the hyperscaling relation is not satisfied for globular rings.Comment: 8 pages; v2: typos removed, published versio

A lattice polymer study of DNA renaturation dynamics

DNA renaturation is the recombination of two complementary single strands to form a double helix. It is experimentally known that renaturation proceeds through the formation of a double stranded nucleus of several base pairs (the rate limiting step) followed by a much faster zippering. We consider a lattice polymer model undergoing Rouse dynamics and focus on the nucleation of two diffusing strands. We study numerically the dependence of various nucleation rates on the strand lengths and on an additional local nucleation barrier. When the local barrier is sufficiently high, all renaturation rates considered scale with the length as predicted by Kramers' rate theory and are also in agreement with experiments: their scaling behavior is governed by exponents describing equilibrium properties of polymers. When the local barrier is lowered renaturation occurs in a regime of genuine non-equilibrium behavior and the scaling deviates from the rate theory prediction.Comment: 13 pages, 6 figures. To appear in Journal of Statistical Mechanic

Nonequilibrium fluctuation-dissipation theorem and heat production

We use a relationship between response and correlation function in nonequilibrium systems to establish a connection between the heat production and the deviations from the equilibrium fluctuation-dissipation theorem. This scheme extends the Harada-Sasa formulation [Phys. Rev. Lett. 95, 130602 (2005)], obtained for Langevin equations in steady states, as it also holds for transient regimes and for discrete jump processes involving small entropic changes. Moreover, a general formulation includes two times and the new concepts of two-time work, kinetic energy, and of a two-time heat exchange that can be related to a nonequilibrium "effective temperature". Numerical simulations of a chain of anharmonic oscillators and of a model for a molecular motor driven by ATP hydrolysis illustrate these points.Comment: 5 pages, 3 figure

Topological and geometrical entanglement in a model of circular DNA undergoing denaturation

The linking number (topological entanglement) and the writhe (geometrical entanglement) of a model of circular double stranded DNA undergoing a thermal denaturation transition are investigated by Monte Carlo simulations. By allowing the linking number to fluctuate freely in equilibrium we see that the linking probability undergoes an abrupt variation (first-order) at the denaturation transition, and stays close to 1 in the whole native phase. The average linking number is almost zero in the denatured phase and grows as the square root of the chain length, N, in the native phase. The writhe of the two strands grows as the square root of N in both phases.Comment: 7 pages, 11 figures, revte

Ranking knots of random, globular polymer rings

An analysis of extensive simulations of interacting self-avoiding polygons on cubic lattice shows that the frequencies of different knots realized in a random, collapsed polymer ring decrease as a negative power of the ranking order, and suggests that the total number of different knots realized grows exponentially with the chain length. Relative frequencies of specific knots converge to definite values because the free energy per monomer, and its leading finite size corrections, do not depend on the ring topology, while a subleading correction only depends on the crossing number of the knots.Comment: 4 pages, 5 figure

Breakdown of thermodynamic equilibrium for DNA hybridization in microarrays

Test experiments of hybridization in DNA microarrays show systematic deviations from the equilibrium isotherms. We argue that these deviations are due to the presence of a partially hybridized long-lived state, which we include in a kinetic model. Experiments confirm the model predictions for the intensity vs. free energy behavior. The existence of slow relaxation phenomena has important consequences for the specificity of microarrays as devices for the detection of a target sequence from a complex mixture of nucleic acids.Comment: 4 pages, 4 figure

Complex networks of earthquakes and aftershocks

We invoke a metric to quantify the correlation between any two earthquakes. This provides a simple and straightforward alternative to using space-time windows to detect aftershock sequences and obviates the need to distinguish main shocks from aftershocks. Directed networks of earthquakes are constructed by placing a link, directed from the past to the future, between pairs of events that are strongly correlated. Each link has a weight giving the relative strength of correlation such that the sum over the incoming links to any node equals unity for aftershocks, or zero if the event had no correlated predecessors. A correlation threshold is set to drastically reduce the size of the data set without losing significant information. Events can be aftershocks of many previous events, and also generate many aftershocks. The probability distribution for the number of incoming and outgoing links are both scale free, and the networks are highly clustered. The Omori law holds for aftershock rates up to a decorrelation time that scales with the magnitude, m, of the initiating shock as tcutoff~10β m with β~-3/4. Another scaling law relates distances between earthquakes and their aftershocks to the magnitude of the initiating shock. Our results are inconsistent with the hypothesis of finite aftershock zones. We also find evidence that seismicity is dominantly triggered by small earthquakes. Our approach, using concepts from the modern theory of complex networks, together with a metric to estimate correlations, opens up new avenues of research, as well as new tools to understand seismicity

Correlated flares in models of a magnetized "canopy"

A model of the Lu-Hamilton kind is applied to the study of critical behavior of the magnetized solar atmosphere. The main novelty is that its driving is done via sources undergoing a diffusion. This mimics the effect of a virtual turbulent substrate forcing the system. The system exhibits power-law statistics not only in the size of the flares, but also in the distribution of the waiting times.Comment: 5 pages + 9 figures, Accepted to Physica

Nonequilibrium Linear Response for Markov Dynamics, II: Inertial Dynamics

We continue our study of the linear response of a nonequilibrium system. This Part II concentrates on models of open and driven inertial dynamics but the structure and the interpretation of the result remain unchanged: the response can be expressed as a sum of two temporal correlations in the unperturbed system, one entropic, the other frenetic. The decomposition arises from the (anti)symmetry under time-reversal on the level of the nonequilibrium action. The response formula involves a statistical averaging over explicitly known observables but, in contrast with the equilibrium situation, they depend on the model dynamics in terms of an excess in dynamical activity. As an example, the Einstein relation between mobility and diffusion constant is modified by a correlation term between the position and the momentum of the particle

A scale-free network hidden in the collapsing polymer

We show that the collapsed globular phase of a polymer accommodates a scale-free incompatibility graph of its contacts. The degree distribution of this network is found to decay with the exponent $\gamma = 1/(2-c)$ up to a cut-off degree $d_c \propto L^{2-c}$, where $c$ is the loop exponent for dense polymers ($c=11/8$ in two dimensions) and $L$ is the length of the polymer. Our results exemplify how a scale-free network (SFN) can emerge from standard criticality.Comment: 4 pages, 3 figures, address correcte