Directed polymer in a random medium of dimension 1+1 and 1+3: weights statistics in the low-temperature phase

We consider the low-temperature $T<T_c$ disorder-dominated phase of the directed polymer in a random potentiel in dimension 1+1 (where $T_c=\infty$) and 1+3 (where $T_c<\infty$). To characterize the localization properties of the polymer of length $L$, we analyse the statistics of the weights $w_L(\vec r)$ of the last monomer as follows. We numerically compute the probability distributions $P_1(w)$ of the maximal weight $w_L^{max}= max_{\vec r} [w_L(\vec r)]$, the probability distribution $\Pi(Y_2)$ of the parameter $Y_2(L)= \sum_{\vec r} w_L^2(\vec r)$ as well as the average values of the higher order moments $Y_k(L)= \sum_{\vec r} w_L^k(\vec r)$. We find that there exists a temperature $T_{gap}<T_c$ such that (i) for $T<T_{gap}$, the distributions $P_1(w)$ and $\Pi(Y_2)$ present the characteristic Derrida-Flyvbjerg singularities at $w=1/n$ and $Y_2=1/n$ for $n=1,2..$. In particular, there exists a temperature-dependent exponent $\mu(T)$ that governs the main singularities $P_1(w) \sim (1-w)^{\mu(T)-1}$ and $\Pi(Y_2) \sim (1-Y_2)^{\mu(T)-1}$ as well as the power-law decay of the moments $\bar{Y_k(i)} \sim 1/k^{\mu(T)}$. The exponent $\mu(T)$ grows from the value $\mu(T=0)=0$ up to $\mu(T_{gap}) \sim 2$. (ii) for $T_{gap}<T<T_c$, the distribution $P_1(w)$ vanishes at some value $w_0(T)<1$, and accordingly the moments $\bar{Y_k(i)}$ decay exponentially as $(w_0(T))^k$ in $k$. The histograms of spatial correlations also display Derrida-Flyvbjerg singularities for $T<T_{gap}$. Both below and above $T_{gap}$, the study of typical and averaged correlations is in full agreement with the droplet scaling theory.Comment: 13 pages, 29 figure

Lyapunov exponents as a dynamical indicator of a phase transition

We study analytically the behavior of the largest Lyapunov exponent $\lambda_1$ for a one-dimensional chain of coupled nonlinear oscillators, by combining the transfer integral method and a Riemannian geometry approach. We apply the results to a simple model, proposed for the DNA denaturation, which emphasizes a first order-like or second order phase transition depending on the ratio of two length scales: this is an excellent model to characterize $\lambda_1$ as a dynamical indicator close to a phase transition.Comment: 8 Pages, 3 Figure

Statistical physics of the melting of inhomogeneous DNA

We studied how the inhomogeneity of a sequence affects the phase transition that takes place at DNA melting. Unlike previous works, which considered thermodynamic quantities averaged over many different inhomogeneous sequences, we focused on precise sequences and investigated the succession of local openings that lead to their dissociation. For this purpose, we performed Transfer Integral type calculations with two different dynamical models, namely the heterogeneous Dauxois-Peyrard-Bishop model and the model based on finite stacking enthalpies we recently proposed. It appears that, for both models, the essential effect of heterogeneity is to let different portions of the investigated sequences open at slightly different temperatures. Besides this macroscopic effect, the local aperture of each portion indeed turns out to be very similar to that of a homogeneous sequence with the same length. Rounding of each local opening transition is therefore merely a size effect. For the Dauxois-Peyrard-Bishop model, sequences with a few thousands base pairs are still far from the thermodynamic limit, so that it is inappropriate, for this model, to discuss the order of the transition associated with each local opening. In contrast, sequences with several hundreds to a few thousands base pairs are pretty close to the thermodynamic limit for the model we proposed. The temperature interval where power laws holds is consequently broad enough to enable the estimation of critical exponents. On the basis of the few examples we investigated, it seems that, for our model, disorder does not necessarily induce a decrease of the order of the transition

Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

We consider the non-equilibrium dynamics of disordered systems as defined by a master equation involving transition rates between configurations (detailed balance is not assumed). To compute the important dynamical time scales in finite-size systems without simulating the actual time evolution which can be extremely slow, we propose to focus on first-passage times that satisfy 'backward master equations'. Upon the iterative elimination of configurations, we obtain the exact renormalization rules that can be followed numerically. To test this approach, we study the statistics of some first-passage times for two disordered models : (i) for the random walk in a two-dimensional self-affine random potential of Hurst exponent $H$, we focus on the first exit time from a square of size $L \times L$ if one starts at the square center. (ii) for the dynamics of the ferromagnetic Sherrington-Kirkpatrick model of $N$ spins, we consider the first passage time $t_f$ to zero-magnetization when starting from a fully magnetized configuration. Besides the expected linear growth of the averaged barrier $\bar{\ln t_{f}} \sim N$, we find that the rescaled distribution of the barrier $(\ln t_{f})$ decays as $e^{- u^{\eta}}$ for large $u$ with a tail exponent of order $\eta \simeq 1.72$. This value can be simply interpreted in terms of rare events if the sample-to-sample fluctuation exponent for the barrier is $\psi_{width}=1/3$.Comment: 8 pages, 4 figure

Single Chain Force Spectroscopy: Sequence Dependence

We study the elastic properties of a single A/B copolymer chain with a specific sequence. We predict a rich structure in the force extension relations which can be addressed to the sequence. The variational method is introduced to probe local minima on the path of stretching and releasing. At given force, we find multiple configurations which are separated by energy barriers. A collapsed globular configuration consists of several domains which unravel cooperatively. Upon stretching, unfolding path shows stepwise pattern corresponding to the unfolding of each domain. While releasing, several cores can be created simultaneously in the middle of the chain resulting in a different path of collapse.Comment: 6 pages 3 figure

Bubble dynamics in DNA

The formation of local denaturation zones (bubbles) in double-stranded DNA is an important example for conformational changes of biological macromolecules. We study the dynamics of bubble formation in terms of a Fokker-Planck equation for the probability density to find a bubble of size n base pairs at time t, on the basis of the free energy in the Poland-Scheraga model. Characteristic bubble closing and opening times can be determined from the corresponding first passage time problem, and are sensitive to the specific parameters entering the model. A multistate unzipping model with constant rates recently applied to DNA breathing dynamics [G. Altan-Bonnet et al, Phys. Rev. Lett. 90, 138101 (2003)] emerges as a limiting case.Comment: 9 pages, 2 figure

THERMODYNAMICS OF A BROWNIAN BRIDGE POLYMER MODEL IN A RANDOM ENVIRONMENT

We consider a directed random walk making either 0 or $+1$ moves and a Brownian bridge, independent of the walk, conditioned to arrive at point $b$ on time $T$. The Hamiltonian is defined as the sum of the square of increments of the bridge between the moments of jump of the random walk and interpreted as an energy function over the bridge connfiguration; the random walk acts as the random environment. This model provides a continuum version of a model with some relevance to protein conformation. The thermodynamic limit of the specific free energy is shown to exist and to be self-averaging, i.e. it is equal to a trivial --- explicitly computed --- random variable. An estimate of the asymptotic behaviour of the ground state energy is also obtained.Comment: 20 pages, uuencoded postscrip

To maximize or not to maximize the free energy of glassy systems, !=?

The static free energy of glassy systems can be expressed in terms of the Parisi order parameter function. When this function has a discontinuity, the location of the step is determined by maximizing the free energy. In dynamics a transition is found at larger temperature, while the location of the step satisfies a marginality criterion. It is shown here that in a replica calculation this criterion minimizes the free energy. This leads to first order phase transitions at the dynamic transition point. Though the order parameter function is the same as in the long-time limit of a dynamical analysis, thermodynamics is different.Comment: 4 pages PostScript, one figur

The SPHINX cosmological simulations of the first billion years: The impact of binary stars on reionization

We present the SPHINX suite of cosmological adaptive mesh refinement simulations, the first radiation-hydrodynamical simulations to simultaneously capture large-scale reionization and the escape of ionizing radiation from thousands of resolved galaxies. Our $5$ and $10$ co-moving Mpc volumes resolve haloes down to the atomic cooling limit and model the inter-stellar medium with better than $\approx10$ pc resolution. The project has numerous goals in improving our understanding of reionization and making predictions for future observations. In this first paper we study how the inclusion of binary stars in computing stellar luminosities impacts reionization, compared to a model that includes only single stars. Owing to the suppression of galaxy growth via strong feedback, our galaxies are in good agreement with observational estimates of the galaxy luminosity function. We find that binaries have a significant impact on the timing of reionization: with binaries, our boxes are $99.9$ percent ionized by volume at $z\approx 7$, while without them our volumes fail to reionize by $z=6$. These results are robust to changes in volume size, resolution, and feedback efficiency. The escape of ionizing radiation from individual galaxies varies strongly and frequently. On average, binaries lead to escape fractions of $\approx 7-10$ percent, about $3.5$ times higher than with single stars only. The higher escape fraction is a result of a shallower decline in ionizing luminosity with age, and is the primary reason for earlier reionization, although the higher integrated luminosity with binaries also plays a sub-dominant role

Roles of stiffness and excluded volume in DNA denaturation

The nature and the universal properties of DNA thermal denaturation are investigated by Monte Carlo simulations. For suitable lattice models we determine the exponent c describing the decay of the probability distribution of denaturated loops of length l, $P \sim l^{-c}$. If excluded volume effects are fully taken into account, c= 2.10(4) is consistent with a first order transition. The stiffness of the double stranded chain has the effect of sharpening the transition, if it is continuous, but not of changing its order and the value of the exponent c, which is also robust with respect to inclusion of specific base-pair sequence heterogeneities.Comment: RevTeX 4 Pages and 4 PostScript figures included. Final version as publishe