arXiv subjects

B. Derrida

Publications and source records attributed to B. Derrida.

39 records · Page 3Linked to original sources

arXiv1997

Shock Profiles for the Asymmetric Simple Exclusion Process in One Dimension

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at rates $p$ and $1-p$ (here $p>1/2$) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers' equation; the latter has shock solutions with a discontinuous jump from left density $ρ_-$ to right density $ρ_+$, $ρ_-<ρ_+$, which travel with velocity $(2p-1)(1-ρ_+-ρ_-)$. In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice site $n$, measured from this particle, approaches $ρ_{\pm}$ at an exponential rate as $n\to\pm\infty$, with a characteristic length which becomes independent of $p$ when $p/(1-p)>\sqrt{ρ_+(1-ρ_-)/ρ_-(1-ρ_+)}$. For a special value of the asymmetry, given by $p/(1-p)=ρ_+(1-ρ_-)/ρ_-(1-ρ_+)$, the measure is Bernoulli, with density $ρ_-$ on the left and $ρ_+$ on the right. In the weakly asymmetric limit, $2p-1\to0$, the microscopic width of the shock diverges as $(2p-1)^{-1}$. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.

cond-mat↗

arXiv1994

Exact Exponent $λ$ of the Autocorrelation Function for a Soluble Model of Coarsening

The exponent $λ$ that describes the decay of the autocorrelation function $A(t)$ in a phase ordering system, $A(t) \sim L^{-(d-λ)}$, where $d$ is the dimension and $L$ the characteristic length scale at time $t$, is calculated exactly for the time-dependent Ginzburg-Landau equation in $d=1$. We find $λ= 0.399\,383\,5\ldots$. We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as $M(t) \sim L^μ$ and prove that for the $1d$ Ginzburg-Landau equation, $μ=λ$, exemplifying a general result.

cond-mat↗

arXiv1993

Weak Disorder Expansion for the Anderson Model on a Tree

We show how certain properties of the Anderson model on a tree are related to the solutions of a non-linear integral equation. Whether the wave function is extended or localized, for example, corresponds to whether or not the equation has a complex solution. We show how the equation can be solved in a weak disorder expansion. We find that, for small disorder strength $λ$, there is an energy $E_c(λ)$ above which the density of states and the conducting properties vanish to all orders in perturbation theory. We compute perturbatively the position of the line $E_c(λ)$ which begins, in the limit of zero disorder, at the band edge of the pure system. Inside the band of the pure system the density of states and conducting properties can be computed perturbatively. This expansion breaks down near $E_c(λ)$ because of small denominators. We show how it can be resummed by choosing the appropriate scaling of the energy. For energies greater than $E_c(λ)$ we show that non-perturbative effects contribute to the density of states but have been unable tell whether they also contribute to the conducting properties.

cond-mat↗