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Register a new userQuasi-Static Fractures in Disordered Media and Iterated Conformal Maps
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We study the geometrical characteristic of quasi-static fractures in disordered media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lame equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern; it crosses over from about 0.5 for small scales to about 0.75 for large scales, in excellent agreement with experiments. We propose that this cross-over reflects the increased ramification of the fracture pattern.
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We study Levy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites, in analogy with recent experiments. Our results are compared with numerical simulations, with excellent agreement.
Operators in ergodic spin-chains are found to grow according to hydrodynamical equations of motion. The study of such operator spreading has aided our understanding of many-body quantum chaos in spin-chains. Here we initiate the study of operator spreading in quantum maps on a torus, systems which do not have a tensor-product Hilbert space or a notion of spatial locality. Using the perturbed Arnold cat map as an example, we analytically compare and contrast the evolutions of functions on classical phase space and quantum operator evolutions, and identify distinct timescales that characterize the dynamics of operators in quantum chaotic maps. Until an Ehrenfest time, the quantum system exhibits classical chaos, i.e. it mimics the behavior of the corresponding classical system. After an operator scrambling time, the operator looks random in the initial basis, a characteristic feature of quantum chaos. These timescales can be related to the quasi-energy spectrum of the unitary via the spectral form factor. Furthermore, we show examples of emergent classicality in quantum problems far away from the classical limit. Finally, we study operator evolution in non-chaotic and mixed quantum maps using the Chirikov standard map as an example.
We investigate diffusion-limited aggregation (DLA) in a wedge geometry. Arneodo and collaborators have suggested that the ensemble average of DLA cluster density should be close to the noise-free selected Saffman-Taylor finger. We show that a different, but related, ensemble average, that of the conformal maps associated with random clusters, yields a non-trivial shape which is also not far from the Saffman-Taylor finger. However, we have previously demonstrated that the same average of DLA in a channel geometry is not the Saffman-Taylor finger. This casts doubt on the idea that the average of noisy diffusion-limited growth is governed by a simple transcription of noise-free results.
We make a review of the two principal models that allows to explain the imbibition of fluid in porous media. These models, that belong to the directed percolation depinning (DPD) universality class, where introduced simultaneously by the Tang and Leschhorn [Phys. Rev A 45, R8309 (1992)] and Buldyrev et al. [Phys. Rev. A 45, R8313 (1992)] and reviewed by Braunstein et al. [J. Phys. A 32, 1801 (1999); Phys. Rev. E 59, 4243 (1999)]. Even these models have been classified in the same universality class than the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889, (1986)] with quenched noise (QKPZ), the contributions to the growing mechanisms are quite different. The lateral contribution in the DPD models, leads to an increasing of the roughness near the criticality while in the QKPZ equation this contribution always flattens the roughness. These results suggest that the QKPZ equation does not describe properly the DPD models even when the exponents derived from this equation are similar to the one obtained from the simulations of these models. This fact is confirmed trough the deduced analytical equation for the Tang and Leschhorn model. This equation has the same symmetries than the QKPZ one but its coefficients depend on the balance between the driving force and the quenched noise.
We derive a systematic, multiple time-scale perturbation expansion for the work distribution in isothermal quasi-static Langevin processes. To first order we find a Gaussian distribution reproducing the result of Speck and Seifert [Phys. Rev. E 70, 066112 (2004)]. Scrutinizing the applicability of perturbation theory we then show that, irrespective of time-scale separation, the expansion breaks down when applied to untypical work values from the tails of the distribution. We thus reconcile the result of Speck and Seifert with apparently conflicting exact expressions for the asymptotics of work distributions in special systems and with an intuitive argument building on the central limit theorem.
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