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

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

The Theoretical Astrophysical Observatory: Cloud-Based Mock Galaxy Catalogues

We introduce the Theoretical Astrophysical Observatory (TAO), an online virtual laboratory that houses mock observations of galaxy survey data. Such mocks have become an integral part of the modern analysis pipeline. However, building them requires an expert knowledge of galaxy modelling and simulation techniques, significant investment in software development, and access to high performance computing. These requirements make it difficult for a small research team or individual to quickly build a mock catalogue suited to their needs. To address this TAO offers access to multiple cosmological simulations and semi-analytic galaxy formation models from an intuitive and clean web interface. Results can be funnelled through science modules and sent to a dedicated supercomputer for further processing and manipulation. These modules include the ability to (1) construct custom observer light-cones from the simulation data cubes; (2) generate the stellar emission from star formation histories, apply dust extinction, and compute absolute and/or apparent magnitudes; and (3) produce mock images of the sky. All of TAO's features can be accessed without any programming requirements. The modular nature of TAO opens it up for further expansion in the future.Comment: 17 pages, 11 figures, 2 tables; accepted for publication in ApJS. The Theoretical Astrophysical Observatory (TAO) is now open to the public at https://tao.asvo.org.au/. New simulations, models and tools will be added as they become available. Contact [email protected] if you have data you would like to make public through TAO. Feedback and suggestions are very welcom

Semi-Analytic Galaxy Evolution (SAGE): Model Calibration and Basic Results

This paper describes a new publicly available codebase for modelling galaxy formation in a cosmological context, the "Semi-Analytic Galaxy Evolution" model, or SAGE for short. SAGE is a significant update to that used in Croton et al. (2006) and has been rebuilt to be modular and customisable. The model will run on any N-body simulation whose trees are organised in a supported format and contain a minimum set of basic halo properties. In this work we present the baryonic prescriptions implemented in SAGE to describe the formation and evolution of galaxies, and their calibration for three N-body simulations: Millennium, Bolshoi, and GiggleZ. Updated physics include: gas accretion, ejection due to feedback, and reincorporation via the galactic fountain; a new gas cooling--radio mode active galactic nucleus (AGN) heating cycle; AGN feedback in the quasar mode; a new treatment of gas in satellite galaxies; and galaxy mergers, disruption, and the build-up of intra-cluster stars. Throughout, we show the results of a common default parameterization on each simulation, with a focus on the local galaxy population.Comment: 15 pages, 9 figures, accepted for publication in ApJS. SAGE is a publicly available codebase for modelling galaxy formation in a cosmological context, available at https://github.com/darrencroton/sage Questions and comments can be sent to Darren Croton: [email protected]

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

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

Reply to Comment on "Criterion that Determines the Foldability of Proteins"

We point out that the correlation between folding times and $\sigma = (T_{\theta } - T_{f})/T_{\theta }$ in protein-like heteropolymer models where $T_{\theta }$ and $T_{f}$ are the collapse and folding transition temperatures was already established in 1993 before the other presumed equivalent criterion (folding times correlating with $T_{f}$ alone) was suggested. We argue that the folding times for these models show no useful correlation with the energy gap even if restricted to the ensemble of compact structures as suggested by Karplus and Shakhnovich (cond-mat/9606037).Comment: 6 pages, Latex, 2 Postscript figures. Plots explicitly showing the lack of correlation between folding time and energy gap are adde

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

Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension $1+d$ with $0<d<2$ involves at the same time (i) a confinement in a favorable tube of radius $R_S \sim L^{\nu_S}$ with $\nu_S=1/(4-d)<1/2$ (ii) a superdiffusive behavior $R \sim L^{\nu}$ with $\nu=(3-d)/(4-d)>1/2$ for the wandering of the best favorable tube available. The corresponding free-energy then scales as $F \sim L^{\omega}$ with $\omega=2 \nu-1$ and the left tail of the probability distribution involves a stretched exponential of exponent $\eta= (4-d)/2$. These results generalize the well known exact exponents $\nu=2/3$, $\omega=1/3$ and $\eta=3/2$ in $d=1$, where the subleading transverse length $R_S \sim L^{1/3}$ is known as the typical distance between two replicas in the Bethe Ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent $\alpha$ and/or to manifolds in dimension $D+d=d_{t}$ with $0<D<2$. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ($\alpha<0$), whereas it is not for growing correlations ($\alpha>0$). In particular, for an interface of dimension $(d_t-1)$ in a space of total dimension $5/3<d_t<3$ with random-bond disorder, our approach yields the confinement exponent $\nu_S = (d_t-1)(3-d_t)/(5d_t-7)$. Finally, we study the exponents in the presence of an algebraic tail $1/V^{1+\mu}$ in the disorder distribution, and obtain various regimes in the $(\mu,d)$ plane.Comment: 19 page

Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions

For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength $W$ and the number $N$ of generations. We first consider the Landauer transmission $T_N$. In the localized phase, its logarithm follows the traveling wave form $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{\xi_{loc}}$ and the localization length diverges as $\xi_{loc} \sim (W-W_c)^{-\nu_{loc}}$ with $\nu_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(t^*) \sim 1/(t^*)^{1+\beta(W)}$ for large $t^*$ with $0<\beta(W) \leq 1/2$, so that all integer moments of $T_N$ are governed by rare events. In the delocalized phase, the transmission $T_N$ remains a finite random variable as $N \to \infty$, and we measure near criticality the essential singularity $\bar{\ln (T)} \sim - | W_c-W |^{-\kappa_T}$ with $\kappa_T \sim 0.25$. We then consider the statistical properties of normalized eigenstates, in particular the entropy and the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical entropy diverges as $(W-W_c)^{- \nu_S}$ with $\nu_S \sim 1.5$, whereas it grows linearly in $N$ in the delocalized phase. Finally for the I.P.R., we explain how closely related variables propagate as traveling waves in the delocalized phase. In conclusion, both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions, and the Anderson localization/delocalization transition then corresponds to a traveling/non-traveling critical point. Moreover, our results point towards the existence of several exponents $\nu$ at criticality.Comment: 28 pages, 21 figures, comments welcom