Elasticity of a contact-line and avalanche-size distribution at depinning

Motivated by recent experiments, we extend the Joanny-deGennes calculation of the elasticity of a contact line to an arbitrary contact angle and an arbitrary plate inclination in presence of gravity. This requires a diagonalization of the elastic modes around the non-linear equilibrium profile, which is carried out exactly. We then make detailed predictions for the avalanche-size distribution at quasi-static depinning: we study how the universal (i.e. short-scale independent) rescaled size distribution and the ratio of moments of local to global avalanches depend on the precise form of the elastic kernel.Comment: 15 pages, 11 figure

Size distributions of shocks and static avalanches from the Functional Renormalization Group

Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d=4 this yields the mean-field distribution P(S) ~ S^(-3/2) exp(-S/[4 S_m]) where S_m is a large-scale cutoff, in some cases calculable. Resumming all 1-loop contributions, we find P(S) ~ S^(-tau) exp(C (S/S_m)^(1/2) -B/4 (S/S_m)^delta) where B, C, delta, tau are obtained to first order in epsilon=4-d. Our result is consistent to O(epsilon) with the relation tau = 2-2/(d+zeta), where zeta is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field and random-periodic disorder. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with tau=2-1/(d+zeta) to O(epsilon=2-d). We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the above relations for tau be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyper-plane of co-dimension one is in mean-field (valid close to and above d=4) given by P(S) ~ K_{1/3}(S)/S, where K is the Bessel-K function, thus tau=4/3 for the hyper plane.Comment: 34 pages, 30 figure

Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions

We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e. below the critical temperature $T_c$, the disorder and thermally averaged correlation function $B(r)$ of the phase field $\theta(x)$, $B(r) = \bar{}$ behaves, for $r \gg a$, as $B(r) \simeq A(\tau) \ln^2 (r/a)$ where $r = |r|$ and $a$ is a microscopic length scale. We derive the RG equations up to cubic order in $\tau = (T_c-T)/T_c$ and predict the universal amplitude ${A}(\tau) = 2\tau^2-2\tau^3 + {\cal O}(\tau^4)$. The universality of $A(\tau)$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute ${A}(\tau)$ numerically and obtain a remarkable agreement with our analytical prediction, up to $\tau \approx 0.5$.Comment: 5 pages, 3 figure

Universal distribution of threshold forces at the depinning transition

We study the distribution of threshold forces at the depinning transition for an elastic system of finite size, driven by an external force in a disordered medium at zero temperature. Using the functional renormalization group (FRG) technique, we compute the distribution of pinning forces in the quasi-static limit. This distribution is universal up to two parameters, the average critical force, and its width. We discuss possible definitions for threshold forces in finite-size samples. We show how our results compare to the distribution of the latter computed recently within a numerical simulation of the so-called critical configuration.Comment: 12 pages, 7 figures, revtex

First-principle derivation of static avalanche-size distribution

We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e. mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as a the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian Force model (BFM), the static version of the so-called ABBM model in the non-equilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Levy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.Comment: 33 pages, 24 figure

Ebbie: automated analysis and storage of small RNA cloning data using a dynamic web server

BACKGROUND: DNA sequencing is used ubiquitously: from deciphering genomes[1] to determining the primary sequence of small RNAs (smRNAs) [2-5]. The cloning of smRNAs is currently the most conventional method to determine the actual sequence of these important regulators of gene expression. Typical smRNA cloning projects involve the sequencing of hundreds to thousands of smRNA clones that are delimited at their 5' and 3' ends by fixed sequence regions. These primers result from the biochemical protocol used to isolate and convert the smRNA into clonable PCR products. Recently we completed a smRNA cloning project involving tobacco plants, where analysis was required for ~700 smRNA sequences[6]. Finding no easily accessible research tool to enter and analyze smRNA sequences we developed Ebbie to assist us with our study. RESULTS: Ebbie is a semi-automated smRNA cloning data processing algorithm, which initially searches for any substring within a DNA sequencing text file, which is flanked by two constant strings. The substring, also termed smRNA or insert, is stored in a MySQL and BlastN database. These inserts are then compared using BlastN to locally installed databases allowing the rapid comparison of the insert to both the growing smRNA database and to other static sequence databases. Our laboratory used Ebbie to analyze scores of DNA sequencing data originating from an smRNA cloning project[6]. Through its built-in instant analysis of all inserts using BlastN, we were able to quickly identify 33 groups of smRNAs from ~700 database entries. This clustering allowed the easy identification of novel and highly expressed clusters of smRNAs. Ebbie is available under GNU GPL and currently implemented on CONCLUSION: Ebbie was designed for medium sized smRNA cloning projects with about 1,000 database entries [6-8].Ebbie can be used for any type of sequence analysis where two constant primer regions flank a sequence of interest. The reliable storage of inserts, and their annotation in a MySQL database, BlastN[9] comparison of new inserts to dynamic and static databases make it a powerful new tool in any laboratory using DNA sequencing. Ebbie also prevents manual mistakes during the excision process and speeds up annotation and data-entry. Once the server is installed locally, its access can be restricted to protect sensitive new DNA sequencing data. Ebbie was primarily designed for smRNA cloning projects, but can be applied to a variety of RNA and DNA cloning projects[2,3,10,11]

Interacting crumpled manifolds

In this article we study the effect of a delta-interaction on a polymerized membrane of arbitrary internal dimension D. Depending on the dimensionality of membrane and embedding space, different physical scenarios are observed. We emphasize on the difference of polymers from membranes. For the latter, non-trivial contributions appear at the 2-loop level. We also exploit a ``massive scheme'' inspired by calculations in fixed dimensions for scalar field theories. Despite the fact that these calculations are only amenable numerically, we found that in the limit of D to 2 each diagram can be evaluated analytically. This property extends in fact to any order in perturbation theory, allowing for a summation of all orders. This is a novel and quite surprising result. Finally, an attempt to go beyond D=2 is presented. Applications to the case of self-avoiding membranes are mentioned

Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model

We obtain an exact solution for the motion of a particle driven by a spring in a Brownian random-force landscape, the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model. Many experiments on quasi-static driving of elastic interfaces (Barkhausen noise in magnets, earthquake statistics, shear dynamics of granular matter) exhibit the same universal behavior as this model. It also appears as a limit in the field theory of elastic manifolds. Here we discuss predictions of the ABBM model for monotonous, but otherwise arbitrary, time-dependent driving. Our main result is an explicit formula for the generating functional of particle velocities and positions. We apply this to derive the particle-velocity distribution following a quench in the driving velocity. We also obtain the joint avalanche size and duration distribution and the mean avalanche shape following a jump in the position of the confining spring. Such non-stationary driving is easy to realize in experiments, and provides a way to test the ABBM model beyond the stationary, quasi-static regime. We study extensions to two elastically coupled layers, and to an elastic interface of internal dimension d, in the Brownian force landscape. The effective action of the field theory is equal to the action, up to 1-loop corrections obtained exactly from a functional determinant. This provides a connection to renormalization-group methods.Comment: 18 pages, 3 figure