arXiv subjects

Kay Joerg Wiese

Publications and source records attributed to Kay Joerg Wiese.

At least 55 records · Page 3Linked to original sources

arXiv2011

Equilibrium avalanches in spin glasses

We study the distribution of equilibrium avalanches (shocks) in Ising spin glasses which occur at zero temperature upon small changes in the magnetic field. For the infinite-range Sherrington-Kirkpatrick model we present a detailed derivation of the density rho(Delta M) of the magnetization jumps Delta M. It is obtained by introducing a multi-component generalization of the Parisi-Duplantier equation, which allows us to compute all cumulants of the magnetization. We find that rho(Delta M) ~ (Delta M)^(-tau) with an avalanche exponent tau=1 for the SK model, originating from the marginal stability (criticality) of the model. It holds for jumps of size 1 << Delta M < N^(1/2) being provoked by changes of the external field by delta H = O(N^[-1/2]) where N is the total number of spins. Our general formula also suggests that the density of overlap q between initial and final state in an avalanche is rho(q) ~ 1/(1-q). These results show interesting similarities with numerical simulations for the out-of-equilibrium dynamics of the SK model. For finite-range models, using droplet arguments, we obtain the prediction tau= (d_f + theta)/d_m, where d_f,d_m and theta are the fractal dimension, magnetization exponent and energy exponent of a droplet, respectively. This formula is expected to apply to other glassy disordered systems, such as the random-field model and pinned interfaces. We make suggestions for further numerical investigations, as well as experimental studies of the Barkhausen noise in spin glasses.

cond-mat.dis-nn↗

arXiv2011

Distribution of velocities in an avalanche

For a driven elastic object near depinning, we derive from first principles the distribution of instantaneous velocities in an avalanche. We prove that above the upper critical dimension, d >= d_uc, the n-times distribution of the center-of-mass velocity is equivalent to the prediction from the ABBM stochastic equation. Our method allows to compute space and time dependence from an instanton equation. We extend the calculation beyond mean field, to lowest order in epsilon=d_uc-d.

cond-mat.dis-nn↗

arXiv2009

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.

cond-mat.dis-nn↗

arXiv2009

Field Theory of the RNA Freezing Transition

Folding of RNA is subject to a competition between entropy, relevant at high temperatures, and the random, or random looking, sequence, determining the low- temperature phase. It is known from numerical simulations that for random as well as biological sequences, high- and low-temperature phases are different, e.g. the exponent rho describing the pairing probability between two bases is rho = 3/2 in the high-temperature phase, and approximatively 4/3 in the low-temperature (glass) phase. Here, we present, for random sequences, a field theory of the phase transition separating high- and low-temperature phases. We establish the existence of the latter by showing that the underlying theory is renormalizable to all orders in perturbation theory. We test this result via an explicit 2-loop calculation, which yields rho approximatively 1.36 at the transition, as well as diverse other critical exponents, including the response to an applied external force (denaturation transition).

q-bio.BM↗

arXiv2009

Height fluctuations of a contact line: a direct measurement of the renormalized disorder correlator

We have measured the center-of-mass fluctuations of the height of a contact line at depinning for two different systems: liquid hydrogen on a rough cesium substrate and isopropanol on a silicon wafer grafted with silanized patches. The contact line is subject to a confining quadratic well, provided by gravity. From the second cumulant of the height fluctuations, we measure the renormalized disorder correlator Delta(u), predicted by the Functional RG theory to attain a fixed point, as soon as the capillary length is large compared to the Larkin length set by the microscopic disorder. The experiments are consistent with the asymptotic form for Delta(u) predicted by Functional RG, including a linear cusp at u=0. The observed small deviations could be used as a probe of the underlying physical processes. The third moment, as well as avalanche-size distributions are measured and compared to predictions from Functional RG.

cond-mat.dis-nn↗

arXiv2009

Avalanche-size distribution at the depinning transition: A numerical test of the theory

We calculate numerically the sizes S of jumps (avalanches) between successively pinned configurations of an elastic line (d=1) or interface (d=2), pulled by a spring of (small) strength m^2 in a random-field landscape. We obtain strong evidence that the size distribution, away from the small-scale cutoff, takes the form P(S) = p(S/S_m) /S_m^2, where S_m:= /(2 ), proportional to m^(-d-zeta), is the scale of avalanches, and zeta the roughness exponent at the depinning transition. Measurement of the scaling function f(s) := s^tau p(s) is compared with the predictions from a recent Functional RG (FRG) calculation, both at mean-field and one-loop level. The avalanche-size exponent tau is found in good agreement with the conjecture tau = 2- 2/(d+zeta), recently confirmed to one loop via the FRG. The function f(s) exhibits a shoulder and a stretched exponential decay at large s, with ln f(s) proportional to - s^delta, and delta approximately 7/6 in d=1. The function f(s), universal ratios of moments, and the generating function are found in excellent agreement with the one-loop FRG predictions. The distribution of local avalanche sizes, i.e. of the jumps of a subspace of the manifold of dimension d', is also computed and compared to our FRG predictions, and to the conjecture tau' =2- 2/(d'+zeta).

cond-mat.dis-nn↗

arXiv2008

Field theory conjecture for loop-erased random walks

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.

cond-mat.dis-nn↗

arXiv2008

Critical interfaces in the random-bond Potts model

We study geometrical properties of interfaces in the random-temperature q-states Potts model as an example of a conformal field theory weakly perturbed by quenched disorder. Using conformal perturbation theory in q-2 we compute the fractal dimension of Fortuin Kasteleyn domain walls. We also compute it numerically both via the Wolff cluster algorithm for q=3 and via transfer-matrix evaluations. We obtain numerical results for the fractal dimension of spin cluster interfaces for q=3. These are found numerically consistent with the duality kappa(spin) * kappa(FK)= 16 as expressed in putative SLE parameters.

cond-mat.dis-nn↗

arXiv2008

Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records

We review how the renormalized force correlator Delta(u), the function computed in the functional RG field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a 1-dimensional random-force landscape. The limit of small velocity allows to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Frechet. For each class we obtain analytically the universal function Delta(u), the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w. We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the ABBM model, avalanche distributions and Delta(u) can be computed for any velocity. For 2-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(u), and to extract the anisotropic scaling exponents zeta_x > zeta_y. We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered e.g. in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w.

cond-mat.dis-nn↗

arXiv2008

Statistics of static avalanches in a random pinning landscape

We study the minimum-energy configuration of a d-dimensional elastic interface in a random potential tied to a harmonic spring. As a function of the spring position, the center of mass of the interface changes in discrete jumps, also called shocks or "static avalanches''. We obtain analytically the distribution of avalanche sizes and its cumulants within an epsilon=4-d expansion from a tree and 1-loop resummation, using functional renormalization. This is compared with exact numerical minimizations of interface energies for random field disorder in d=2,3. Connections to the Burgers equation and to dynamic avalanches are discussed.

cond-mat.dis-nn↗

arXiv2008

Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together

We compute the Functional Renormalization Group (FRG) disorder- correlator function R(v) for d-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space dimension N. It measures the equilibrium response of the manifold in a quadratic potential well as the center of the well is varied from 0 to v. We find two distinct scaling regimes: (i) a "single shock" regime, v^2 ~ 1/L^d where L^d is the system volume and (ii) a "thermodynamic" regime, v^2 ~ N. In regime (i) all the equivalent replica symmetry breaking (RSB) saddle points within the Gaussian variational approximation contribute, while in regime (ii) the effect of RSB enters only through a single anomaly. When the RSB is continuous (e.g., for short-range disorder, in dimension 2 <= d <= 4), we prove that regime (ii) yields the large-N FRG function obtained previously. In that case, the disorder correlator exhibits a cusp in both regimes, though with different amplitudes and of different physical origin. When the RSB solution is 1-step and non- marginal (e.g., d < 2 for SR disorder), the correlator R(v) in regime (ii) is considerably reduced, and exhibits no cusp. Solutions of the FRG flow corresponding to non-equilibrium states are discussed as well. In all cases the regime (i) exhibits a cusp non-analyticity at T=0, whose form and thermal rounding at finite T is obtained exactly and interpreted in terms of shocks. The results are compared with previous work, and consequences for manifolds at finite N, as well as extensions to spin glasses and related models are discussed.

cond-mat.dis-nn↗

arXiv2007

Depinning in a two-layer model of plastic flow

We study a model of two layers, each consisting of a d-dimensional elastic object driven over a random substrate, and mutually interacting through a viscous coupling. For this model, the mean-field theory (i.e. a fully connected model) predicts a transition from elastic depinning to hysteretic plastic depinning as disorder or viscous coupling is increased. A functional RG analysis shows that any small inter-layer viscous coupling destablizes the standard (decoupled) elastic depinning FRG fixed point for d <= 4, while for d > 4 most aspects of the mean-field theory are recovered. A one-loop study at non-zero velocity indicates, for d<4, coexistence of a moving state and a pinned state below the elastic depinning threshold, with hysteretic plastic depinning for periodic and non-periodic driven layers. A 2-loop analysis of quasi-statics unveils the possibility of more subtle effects, including a new universality class for non-periodic objects. We also study the model in d=0, i.e. two coupled particles, and show that hysteresis does not always exist as the periodic steady state with coupled layers can be dynamically unstable. It is also proved that stable pinned configurations remain dynamically stable in presence of a viscous coupling in any dimension d. Moreover, the layer model for periodic objects is stable to an infinitesimal commensurate density coupling. Our work shows that a careful study of attractors in phase space and their basin of attraction is necessary to obtain a firm conclusion for dimensions d=1,2,3.

cond-mat.dis-nn↗

arXiv2007

How to measure the effective action for disordered systems

In contrast to standard critical phenomena, disordered systems need to be treated via the Functional Renormalization Group. The latter leads to a coarse grained disorder landscape, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We review recent progress on how the non-analytic effective action can be measured both in simulations and experiments, and confront theory with numerical work.

cond-mat.dis-nn↗

arXiv2007

A growth model for RNA secondary structures

A hierarchical model for the growth of planar arch structures for RNA secondary structures is presented, and shown to be equivalent to a tree-growth model. Both models can be solved analytically, giving access to scaling functions for large molecules, and corrections to scaling, checked by numerical simulations of up to 6500 bases. The equivalence of both models should be helpful in understanding more general tree-growth processes.

q-bio.BM↗

arXiv2007

Random RNA under tension

The Laessig-Wiese (LW) field theory for the freezing transition of random RNA secondary structures is generalized to the situation of an external force. We find a second-order phase transition at a critical applied force f = f_c. For f < f_c forces are irrelevant. For f > f_c, the extension L as a function of pulling force f scales as (f-f_c)^(1/gamma-1). The exponent gamma is calculated in an epsilon-expansion: At 1-loop order gamma = epsilon/2 = 1/2, equivalent to the disorder-free case. 2-loop results yielding gamma = 0.6 are briefly mentioned. Using a locking argument, we speculate that this result extends to the strong-disorder phase.

q-bio.BM↗

arXiv2007

Statics and dynamics of elastic manifolds in media with long-range correlated disorder

We study the statics and dynamics of an elastic manifold in a disordered medium with quenched defects correlated as r^{-a} for large separation r. We derive the functional renormalization-group equations to one-loop order, which allow us to describe the universal properties of the system in equilibrium and at the depinning transition. Using a double epsilon=4-d and delta=4-a expansion, we compute the fixed points characterizing different universality classes and analyze their regions of stability. The long-range disorder-correlator remains analytic but generates short-range disorder whose correlator exhibits the usual cusp. The critical exponents and universal amplitudes are computed to first order in epsilon and delta at the fixed points. At depinning, a velocity-versus-force exponent beta larger than unity can occur. We discuss possible realizations using extended defects.

cond-mat.dis-nn↗

arXiv2006

Functional Renormalization for Disordered Systems, Basic Recipes and Gourmet Dishes

We give a pedagogical introduction into the functional renormalization group treatment of disordered systems. After a review of its phenomenology, we show why in the context of disordered systems a functional renormalization group treatment is necessary, contrary to pure systems, where renormalization of a single coupling constant is sufficient. This leads to a disorder distribution, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how the non-analyticity can be measured in a simulation or experiment. We then construct a renormalizable field theory beyond leading order. We discuss an elastic manifold embedded in N dimensions, and give the exact solution for N to infinity. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. We further consider random field magnets, and supersymmetry. We finally discuss depinning, both isotropic and anisotropic, and universal scaling function.

cond-mat.dis-nn↗

arXiv2006

Numerical Calculation of the Functional renormalization group fixed-point functions at the depinning transition

We compute numerically the sequence of successive pinned configurations of an elastic line pulled quasi-statically by a spring in a random bond (RB) and random field (RF) potential. Measuring the fluctuations of the center of mass of the line allows to obtain the functional renormalization group (FRG) functions at the depinning transition. The universal form of the second cumulant Delta(u) is found to have a linear cusp at the origin, to be identical for RB and RF, different from the statics, and in good agreement with 2-loop FRG. The cusp is due to avalanches, which we visualize. Avalanches also produce a cusp in the third cumulant, whose universal form is obtained, as predicted by FRG.

cond-mat.dis-nn↗