Non-linear Response of the trap model in the aging regime : Exact results in the strong disorder limit

We study the dynamics of the one dimensional disordered trap model presenting a broad distribution of trapping times $p(\tau) \sim 1/\tau^{1+\mu}$, when an external force is applied from the very beginning at $t=0$, or only after a waiting time $t_w$, in the linear as well as in the non-linear response regime. Using a real-space renormalization procedure that becomes exact in the limit of strong disorder $\mu \to 0$, we obtain explicit results for many observables, such as the diffusion front, the mean position, the thermal width, the localization parameters and the two-particle correlation function. In particular, the scaling functions for these observables give access to the complete interpolation between the unbiased case and the directed case. Finally, we discuss in details the various regimes that exist for the averaged position in terms of the two times and the external field.Comment: 27 pages, 1 eps figur

Localization of thermal packets and metastable states in Sinai model

We consider the Sinai model describing a particle diffusing in a 1D random force field. As shown by Golosov, this model exhibits a strong localization phenomenon for the thermal packet: the disorder average of the thermal distribution of the relative distance y=x-m(t), with respect to the (disorder-dependent) most probable position m(t), converges in the limit of infinite time towards a distribution P(y). In this paper, we revisit this question of the localization of the thermal packet. We first generalize the result of Golosov by computing explicitly the joint asymptotic distribution of relative position y=x(t)-m(t) and relative energy u=U(x(t))-U(m(t)) for the thermal packet. Next, we compute in the infinite-time limit the localization parameters Y_k, representing the disorder-averaged probabilities that k particles of the thermal packet are at the same place, and the correlation function C(l) representing the disorder-averaged probability that two particles of the thermal packet are at a distance l from each other. We moreover prove that our results for Y_k and C(l) exactly coincide with the thermodynamic limit of the analog quantities computed for independent particles at equilibrium in a finite sample of length L. Finally, we discuss the properties of the finite-time metastable states that are responsible for the localization phenomenon and compare with the general theory of metastable states in glassy systems, in particular as a test of the Edwards conjecture.Comment: 17 page

Localization properties of the anomalous diffusion phase x tμx ~ t^{\mu} in the directed trap model and in the Sinai diffusion with bias

We study the anomalous diffusion phase $x ~ t^{\mu}$ with $0<\mu<1$ which exists both in the Sinai diffusion at small bias, and in the related directed trap model presenting a large distribution of trapping time $p(\tau) \sim 1/\tau^{1+\mu}$. Our starting point is the Real Space Renormalization method in which the whole thermal packet is considered to be in the same renormalized valley at large time : this assumption is exact only in the limit $\mu \to 0$ and corresponds to the Golosov localization. For finite $\mu$, we thus generalize the usual RSRG method to allow for the spreading of the thermal packet over many renormalized valleys. Our construction allows to compute exact series expansions in $\mu$ of all observables : at order $\mu^n$, it is sufficient to consider a spreading of the thermal packet onto at most $(1+n)$ traps in each sample, and to average with the appropriate measure over the samples. For the directed trap model, we show explicitly up to order $\mu^2$ how to recover the diffusion front, the thermal width, and the localization parameter $Y_2$. We moreover compute the localization parameters $Y_k$ for arbitrary $k$, the correlation function of two particles, and the generating function of thermal cumulants. We then explain how these results apply to the Sinai diffusion with bias, by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models.Comment: 33 pages, 3 eps figure

Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-dimensional Random Environments

Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields asymptotically exact long time results. The distribution of the position of a particle and the probability of it not returning to the origin are obtained, as well as the two-time distribution which exhibits "aging" with $\frac{\ln t}{\ln t'}$ scaling and a singularity at $\ln t =\ln t'$. The effects of a small uniform force are also studied. Extension to motion of many domain walls yields non-equilibrium time dependent correlations for the 1D random field Ising model with Glauber dynamics and "persistence" exponents of 1D reaction-diffusion models with random forces.Comment: 5 pages, 1 figures, RevTe

Random walks and polymers in the presence of quenched disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both `random walk models', where the random walk is a dynamical process generated by local transition rules, and on `polymer models', where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points : thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. We describe the results of this analysis for the bidimensional wetting and for the Poland-Scheraga model of DNA denaturation.Comment: 17 pages, Conference Proceedings "Mathematics and Physics", I.H.E.S., France, November 200

Fluctuation-Dissipation relations far from Equilibrium

In this Article we review some recent progresses in the field of non-equilibrium linear response theory. We show how a generalization of the fluctuation-dissipation theorem can be derived for Markov processes, and discuss the Cugliandolo-Kurchan \cite{Cugliandolo93} fluctuation dissipation relation for aging systems and the theorem by Franz {\it et. al.} \cite{Franz98} relating static and dynamic properties. We than specialize the subject to phase-ordering systems examining the scaling properties of the linear response function and how these are determined by the behavior of topological defects. We discuss how the connection between statics and dynamics can be violated in these systems at the lower critical dimension or as due to stochastic instability.Comment: 18 pages, 10 figure

Single Molecule Statistics and the Polynucleotide Unzipping Transition

We present an extensive theoretical investigation of the mechanical unzipping of double-stranded DNA under the influence of an applied force. In the limit of long polymers, there is a thermodynamic unzipping transition at a critical force value of order 10 pN, with different critical behavior for homopolymers and for random heteropolymers. We extend results on the disorder-averaged behavior of DNA's with random sequences to the more experimentally accessible problem of unzipping a single DNA molecule. As the applied force approaches the critical value, the double-stranded DNA unravels in a series of discrete, sequence-dependent steps that allow it to reach successively deeper energy minima. Plots of extension versus force thus take the striking form of a series of plateaus separated by sharp jumps. Similar qualitative features should reappear in micromanipulation experiments on proteins and on folded RNA molecules. Despite their unusual form, the extension versus force curves for single molecules still reveal remnants of the disorder-averaged critical behavior. Above the transition, the dynamics of the unzipping fork is related to that of a particle diffusing in a random force field; anomalous, disorder-dominated behavior is expected until the applied force exceeds the critical value for unzipping by roughly 5 pN.Comment: 40 pages, 18 figure

Strong disorder RG approach – a short review of recent developments

The strong disorder RG approach for random systems has been extended in many new directions since our previous review of 2005 [F. Igloi, C. Monthus, Phys. Rep. 412, 277 (2005)]. The aim of the present colloquium paper is thus to give an overview of these various recent developments. In the field of quantum disordered models, recent progress concern infinite disorder fixed points for short-ranged models in higher dimensions d > 1, strong disorder fixed points for long-ranged models, scaling of the entanglement entropy in critical ground-states and after quantum quenches, the RSRG-X procedure to construct the whole set excited stated and the RSRG-t procedure for the unitary dynamics in many-body-localized phases, the Floquet dynamics of periodically driven chains, the dissipative effects induced by the coupling to external baths, and Anderson Localization models. In the field of classical disordered models, new applications include the contact process for epidemic spreading, the strong disorder renormalization procedure for general master equations, the localization properties of random elastic networks, and the synchronization of interacting non-linear dissipative oscillators. Application of the method for aperiodic (or deterministic) disorder is also mentioned