Effect of Memory and Active Forces on Transition Path Time Distributions.

An analytical expression is derived for the transition path time distribution for a one-dimensional particle crossing of a parabolic barrier. Two cases are analyzed: (i) a non-Markovian process described by a generalized Langevin equation with a power-law memory kernel and (ii) a Markovian process with a noise violating the fluctuation-dissipation theorem, modeling the stochastic dynamics generated by active forces. In case i, we show that the anomalous dynamics strongly affect the short time behavior of the distributions, but this happens only for very rare events not influencing the overall statistics. At long times the decay is always exponential, in disagreement with a recent study suggesting a stretched exponential decay. In case ii, the active forces do not substantially modify the short time behavior of the distribution but do lead to an overall decrease of the average transition path time. These findings offer some novel insights, useful for the analysis of experiments of transition path times in (bio)molecular systems.

Non-Markovian dynamics of reaction coordinate in polymer folding.

We develop a theoretical description of the critical zipping dynamics of a self-folding polymer. We use tension propagation theory and the formalism of the generalized Langevin equation applied to a polymer that contains two complementary parts which can bind to each other. At the critical temperature, the (un)zipping is unbiased and the two strands open and close as a zipper. The number of broken base pairs n(t) displays a subdiffusive motion characterized by a variance growing as 〈Δn2(t)〉 ∼ tα with α < 1 at long times. Our theory provides an estimate of both the asymptotic anomalous exponent α and of the subleading correction term, which are both in excellent agreement with numerical simulations. The results indicate that the tension propagation theory captures the relevant features of the dynamics and shed some new insights on related polymer problems characterized by anomalous dynamical behavior.

Fractional Brownian motion and the critical dynamics of zipping polymers.

We consider two complementary polymer strands of length L attached by a common-end monomer. The two strands bind through complementary monomers and at low temperatures form a double-stranded conformation (zipping), while at high temperature they dissociate (unzipping). This is a simple model of DNA (or RNA) hairpin formation. Here we investigate the dynamics of the strands at the equilibrium critical temperature T=T(c) using Monte Carlo Rouse dynamics. We find that the dynamics is anomalous, with a characteristic time scaling as τ∼L(2.26(2)), exceeding the Rouse time ∼L(2.18). We investigate the probability distribution function, velocity autocorrelation function, survival probability, and boundary behavior of the underlying stochastic process. These quantities scale as expected from a fractional Brownian motion with a Hurst exponent H=0.44(1). We discuss similarities to and differences from unbiased polymer translocation.

Criticality on networks with topology-dependent interactions.

Weighted scale-free networks with topology-dependent interactions are studied. It is shown that the possible universality classes of critical behavior, which are known to depend on topology, can also be explored by tuning the form of the interactions at fixed topology. For a model of opinion formation, simple mean field and scaling arguments show that a mapping gamma'=(gamma-mu)(1-mu) describes how a shift of the standard exponent gamma of the degree distribution can absorb the effect of degree-dependent pair interactions J(ij) proportional to (k(i)k(j))(-mu), where k(i) stands for the degree of vertex i. This prediction is verified by extensive numerical investigations using the cavity method and Monte Carlo simulations. The critical temperature of the model is obtained through the Bethe-Peierls approximation and with the replica technique. The mapping can be extended to nonequilibrium models such as those describing the spreading of a disease on a network.

Trading interactions for topology in scale-free networks.

Scale-free networks with topology-dependent interactions are studied. It is shown that the universality classes of critical behavior, which conventionally depend only on topology, can also be explored by tuning the interactions. A mapping, gamma'=(gamma-mu)/(1-mu), describes how a shift of the standard exponent gamma of the degree distribution P(q) can absorb the effect of degree-dependent pair interactions J(ij)proportional to(q(i)q(j))(-mu). The replica technique, cavity method, and Monte Carlo simulation support the physical picture suggested by Landau theory for the critical exponents and by the Bethe-Peierls approximation for the critical temperature. The equivalence of topology and interaction holds for equilibrium and nonequilibrium systems, and is illustrated with interdisciplinary applications.

Statistical mechanics of RNA folding: a lattice approach.

We propose a lattice model for the secondary structure of RNA based on a self-interacting two-tolerant trail. Self-avoidance and pseudoknots are taken into account. We investigate a simple version of the model in which the native state of RNA consists of just one hairpin. Using exact arguments and Monte Carlo simulations we determine the phase diagram for this case. We show that the denaturation transition is first order and can either occur directly or through an intermediate molten phase.

Polymer theta-point as a knot delocalization transition.

We study numerically the tightness of prime flat knots in a model of self-attracting polymers with excluded volume. We find that these knots are localized in the high temperature swollen regime, but become delocalized in the low temperature globular phase. Precisely at the collapse transition, the knots are weakly localized. Some of our results can be interpreted in terms of the theory of polymer networks, which allows one to conjecture exact exponents for the knot length probability distributions.

Generalized contact process with n absorbing states.

We investigate the critical properties of a one-dimensional stochastic lattice model with n (permutation symmetric) absorbing states. We analyze the cases with n/=3 with exponents z=nu( ||)/nu( perpendicular)=2, nu( perpendicular)=1, and beta=1. These exponents coincide with those of the multispecies (bosonic) branching annihilating random walks. For n=3 we also show that, upon breaking the symmetry to a lower one (Z2), one gets a transition either in the directed percolation, or in the parity conserving class, depending on the choice of parameters.

Directed polymers with tilted columnar disorder and Burgers-like turbulence.

The minimal energy variations of a directed polymer with tilted columnar disorder in two dimensions are shown numerically to obey multiscaling at short distances which crosses over to global simple scaling at large distances. The scenario is analogous to that of structure functions in bifractal Burgers turbulence. Some scaling properties are predicted from extreme value statistics. The multiscaling disappears for zero tilt.

Dissipative Abelian sandpiles and random walks.

We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph that consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that the correlation length exponent nu of the dissipative sandpiles always equals 1/d(w), where d(w) is the fractal dimension of the random walker. This leads to a new understanding of the known result that nu=1/2 on any Euclidean lattice. Our result is, however, more general, and as an example we also present exact data for finite Sierpinski gaskets, which fully confirm our predictions.

One-dimensional contact process: duality and renormalization.

We study the one-dimensional contact process in its quantum version using a recently proposed real-space renormalization technique for stochastic many-particle systems. Exploiting the duality and other properties of the model, we can apply the method for cells with up to 37 sites. After suitable extrapolation, we obtain exponent estimates that are comparable in accuracy with the best known in the literature.