On the stochastic transport in disordered systems

Фейгельман, М. В.; Винокур, В. М.

doi:10.1051/jphys:0198800490100173100

Cited by 44 publications

(35 citation statements)

References 20 publications

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“…This distribution actually corresponds to the distribution of barriers against the bias [23,29] in a Sinai potential. For µ ≤ 0, the appropriate rescaled variable reads…”

Section: A Saddle Point Methods In Each Samplementioning

confidence: 99%

Finite-size scaling properties of random transverse-field Ising chains: Comparison between canonical and microcanonical ensembles for the disorder

Monthus

2004

Phys. Rev. B

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The Random Transverse Field Ising Chain is the simplest disordered model presenting a quantum phase transition at T = 0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder (i) the canonical ensemble, where the disorder variables are independent (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation C(L) for a chain of length L. At criticality, each observable decays typically as e −w √ L in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular in their asymptotic behaviors. As a consequence, the dependence in L of averaged observables differ in the two ensembles. For instance, the correlation C(L) decays algebraically as 1/L in the canonical ensemble, but sub-exponentially as e −cL 1/3 in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent ν = 2 in both ensembles, but the following observables are governed by the exponentν = 1 in the microcanonical ensemble, instead of the exponent ν = 2 in the canonical ensemble (a) in the disordered phase : the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation (b) in the ordered phase : the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.

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“…This distribution actually corresponds to the distribution of barriers against the bias [23,29] in a Sinai potential. For µ ≤ 0, the appropriate rescaled variable reads…”

Section: A Saddle Point Methods In Each Samplementioning

confidence: 99%

Finite-size scaling properties of random transverse-field Ising chains: Comparison between canonical and microcanonical ensembles for the disorder

Monthus

2004

Phys. Rev. B

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“…It is known that this model without a bias exhibits non trivial ultraslow logarithmic behavior, as the walker typically moves as x ∼ (ln t) 2 , as well as several dynamical phases with anomalous diffusion as the bias is increased from zero. By contrast, there was until now no exact results about two time aging dynamics, despite several mostly qualitative and numerical studies [17,16] which found interesting aging behavior in this model. In addition, the Sinai model has interesting extensions to many interacting particles, and via domain walls, to the Glauber dynamics of 1D random field Ising ferromagnets and spin glasses in a magnetic field.…”

Section: Introductionmentioning

confidence: 99%

Random walkers in one-dimensional random environments: Exact renormalization group analysis

1999

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Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position x(t) of a particle, the probability of it not returning to the origin and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite but large sample. We compute the distribution of meeting time of two particles in the same environment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward. These quantities obey multifractal scaling, characterized by generalized persistence exponents θ(g) which we compute. In the presence of a small bias, the number of returns to the origin becomes finite, characterized by a universal scaling function which we obtain. The full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained, as well as detailed analytical information about correlations between directions and times of successive jumps. The two time distribution of the positions of a particle, x(t) and x(t ′ ) with t > t ′ , is also computed exactly. It is found to exhibit "aging" with several time regimes characterized by different behaviors. In the unbiased case, for t − t ′ ∼ t ′α with α > 1, it exhibits a ln t ln t ′ scaling, with a singularity at coinciding rescaled positions x(t) = x(t ′ ). This singularity is a novel feature, and corresponds to particles which remain in a renormalized valley. For closer times α < 1, the two time diffusion front exhibits a quasi-equilibrium regime with a ln(t − t ′ )/ ln t ′ behavior which we compute. The crossover to 1 a t/t ′ aging form in the presence of a small bias is also obtained analytically. Rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables, are also characterized in detail. Connections with the Green's function of a one-dimensional Schrödinger problem and quantum spin chains are discussed.

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“…It has been proposed for a long time [17,18,47] that the biased Sinai model should be asymptotically equivalent to a directed trap model defined by the master equation (17) in which the τ n are independent random variables distributed with the algebraic law…”

Section: Anomalous Diffusion Phasementioning

confidence: 99%

Random Walks and Polymers in the Presence of Quenched Disorder

Monthus

2006

Lett Math Phys

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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 Tc(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.

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On the stochastic transport in disordered systems

Cited by 44 publications

References 20 publications

Finite-size scaling properties of random transverse-field Ising chains: Comparison between canonical and microcanonical ensembles for the disorder

Finite-size scaling properties of random transverse-field Ising chains: Comparison between canonical and microcanonical ensembles for the disorder

Random walkers in one-dimensional random environments: Exact renormalization group analysis

Random Walks and Polymers in the Presence of Quenched Disorder

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