Elastic string in a random potential

Dong, Mo; Marchetti, M. Cristina; Middleton, A. Alan; Vinokur, V. M.

doi:10.1103/physrevlett.70.662

Cited by 73 publications

(71 citation statements)

References 15 publications

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“…It is now well known that at very low driving, the response is highly nonlinear, leading to the so-called creep regime. Phenomenological arguments based on the Arrhenius activation of segments of the interface (thermal nuclei) showed that, instead of a linear response, one should expect a stretched exponential response [13][14][15][16][17], where the average velocity of the wall is exponentially small in a power law of the external force. This behavior was later confirmed by more microscopic derivations based on a functional renormalization group procedure (FRG) in d ¼ 4 − ϵ dimensions [18,19].…”

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confidence: 99%

“…The only available theoretical tools to address the dynamics of low dimensional interfaces, so far, are numerical simulations. In this respect, traditional molecular dynamics techniques have difficulties reaching the very long times which are necessary to deal with the ultraslow motion characterizing creep [17,20]. Thus, a well-controlled numerical technique that would not suffer from slowing down when the force is reduced would be highly suitable.…”

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confidence: 99%

“…The global velocity is determined using the Arrhenius formula, assuming that the energy barriers scale as the ground state fluctuations, i.e., as L θ opt with θ ¼ θ eq , resulting in the creep law [13][14][15][16][17][18][19] …”

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confidence: 99%

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Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics

et al. 2017

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Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics

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“…The direct numerical simulation of the dynamics, especially of long-range systems, in the depinning region is extremely tedious because the velocity of the manifold vanishes at the threshold, which is thus difficult to approach [18,19]. Many authors therefore preferred to treat the problem within the framework of cellular automaton models.…”

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confidence: 99%

Roughness at the depinning threshold for a long-range elastic string

2002

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In this paper, we compute the roughness exponent ζ of a long-range elastic string, at the depinning threshold, in a random medium with high precision, using a numerical method which exploits the analytic structure of the problem ('no-passing' theorem), but avoids direct simulation of the evolution equations. This roughness exponent has recently been studied by simulations, functional renormalization group calculations, and by experiments (fracture of solids, liquid meniscus in 4 He). Our result ζ = 0.390 ± 0.002 is significantly larger than what was stated in previous simulations, which were consistent with a one-loop renormalization group calculation. The data are furthermore incompatible with the experimental results for crack propagation in solids and for a 4 He contact line on a rough substrate. This implies that the experiments cannot be described by pure harmonic long-range elasticity in the quasi-static limit.The statics and dynamics of elastic manifolds in random media govern the physics of a variety of systems, ranging from vortices in type-II superconductors [1] and charge density waves [2] to interfaces in disordered magnets [3], contact lines of liquid menisci on a rough substrate [4] and to the propagation of cracks in solids [5].In most cases, the restoring elastic forces acting on a point on the manifold are local i.e. depend only on the deformation in its neighborhood. The corresponding short-range string has been the object of many theoretical and experimental studies. In the depinning limit, two different scenarios are possible: numerical simulations and analytical calculations [6,7] have established that a string with an elastic restoring force breaks at the depinning threshold, while percolation experiments and numerical studies on directed polymers in random media [8,9] agree that in those systems with stronger than harmonic restoring forces the roughness exponent is ζ = 0.63.It has also been shown [5,10] that for a contact line of a liquid meniscus or for crack propagation in a solid, the elastic force is long-range, rather than local. Nonlocal elasticity can be expected to modify the dynamic and static properties of these systems and to change the critical exponents. In this work, we compute one of these exponents, the roughness exponent ζ of a long-range elastic string at the depinning threshold f c .The theoretical approaches are up to now based on the assumption that the motion of the line at the threshold is quasi-static. This means that velocity-dependent terms in the equations of motion of the deformation field h(x, t) are taken to be irrelevant and can be derived from an energy function, which incorporates potential energy due to the driving force f and the disorder potential η(x, h), as well as an elastic energy. According to this hypothesis, the equation of motion of the deformation field at zero temperature is:The last term in this equation accounts for long-range restoring forces. Let us note that measurements of local velocities for a liquid 4 He contact line [4] have cast doub...

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“…dx dh ηη = 1, describes the movement of an elastic band over a rough surface [16] pulled by a transverse force acting at one end point only. Below it is shown that the λ-term disappears in the continuum, establishing the first rigorous identification of the Oslo model and the qEW equation.…”

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confidence: 99%

Oslo rice pile model is a quenched Edwards-Wilkinson equation

Pruessner

2003

Phys. Rev. E

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The Oslo rice pile model is a sandpile-like paradigmatic model of "Self-Organized Criticality" (SOC). In this paper it is shown that the Oslo model is in fact exactly a discrete realization of the much studied quenched Edwards-Wilkinson equation (qEW) [Nattermann et al., J. Phys. II France 2, 1483]. This is possible by choosing the correct dynamical variable and identifying its equation of motion. It establishes for the first time an exact link between SOC models and the field of interface growth with quenched disorder. This connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of SOC.PACS numbers: 64.60. Ht, 05.65.+b, 68.35.Fx, The Oslo rice pile model (Oslo model hereafter) was originally intended to model the relaxation processes in real rice piles [1]. Meanwhile, it has been subject to many investigations and publications in its own right. The model as defined below supposedly develops into a scale free state without the explicit tuning of external parameters, and is therefore regarded as an example of Self-Organized Criticality (SOC) [2]. In fact, contrary to many other "standard" models of SOC [3,4,5,6], it shows a reliable and consistent (simple) scaling behavior and is robust against certain changes in the details of the dynamics [7,8,9]. The most prominent observable in the model, the avalanche size s, is governed by a probability distribution P(s) which obeys simple scaling,

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Elastic string in a random potential

Cited by 73 publications

References 15 publications

Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics

Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics

Roughness at the depinning threshold for a long-range elastic string

Oslo rice pile model is a quenched Edwards-Wilkinson equation

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