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Partially asymmetric exclusion processes with quenched disorder Ludger Santen 1, Robert Juhasz 1 and Ferenc Igloi 2 1 Universität des Saarlandes, Saarbrücken,

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Presentation on theme: "Partially asymmetric exclusion processes with quenched disorder Ludger Santen 1, Robert Juhasz 1 and Ferenc Igloi 2 1 Universität des Saarlandes, Saarbrücken,"— Presentation transcript:

1 Partially asymmetric exclusion processes with quenched disorder Ludger Santen 1, Robert Juhasz 1 and Ferenc Igloi 2 1 Universität des Saarlandes, Saarbrücken, Germany, 2 Szeged University, Szeged, Hungary

2 The ZRP with strong disorder: Definition ASEP with particlewise disorder Hopping rates: p i (forward) q i (backward) Direction of the bias is random L sites and N particles

3 Realisations of the disorder Control Parameter δ Asymmetry between forward and backward rates δ>0 (δ<0): Bias to the right (left) Uniform distribution (p 0 >0)Bimodal distribution

4 Stationary solution of the ZRP Stationary weights have factorised form: where: Ansatz & stationary master equation: 

5 Stationary solution of the ZRP Solution of stationarity condition: Conserved quantity: Choice (const=1):

6 Infinite Particle Limit Partition function Current: Occupation probability & density profile

7 Properties of the random variables Idenfication ↔ g Kesten variables Asymptotic behaviour (L→ ∞, δ>0): Example: Bimodal distribution Scaling of g L (inverse current):

8 Hopping rates and energy landscape Construction of the landscape: Size of the excursions: Probability of transversal excursions :

9 Strong disorder RG Effective rates: Decimation of a site i: Renormalized current:

10 J & remaining g‘s are invariant Elimination of the largest rate Ω is gradually decreasing Approximation (asymptotically exact): forward rate decimation: backward rate decimation Properties of the RG

11 SDRG: Results The unbiased case (δ=0): Relation between rate-scale and clustersize Accumulated distance : Current fluctuations :

12 SDRG: Results The biased case (δ>0): Existence of a limiting time scale τ~1/Ω ξ Ω > Ω ξ : elimination of forward and backward rates Ω < Ω ξ : TASEP with rates: Relation between rate-scale and clustersize

13 SDRG: Results The biased case (δ>0): Current distributions

14 :::: z→∞: Cumulated distance Stationary state: Transport properties z=0: Uniform bias  Situation similar to the TASEP 0

15 Distribution of particles Active particles: Single particle Transport (z>1): Finite number of active particles Accumulated distance: X~t 1/z Many particle Transport (z<1): L 1-z active particles Accumulated distance: X~t Inactive particles Particles in the „cloud“: The condensate is attractive; excursions of length ξ Subleading extrema of the energy landscape

16 Density profile (Griffith phase) Position of the condensate: i=M finite boundary layer of width:

17 Density profile at criticality (  Scaling form:

18 Renormalisation group analogous to real coarsening  Clustersize ~distance between occupied sites  Length scale  Distance l behaves as (z<∞) : Approaching the stationary state Critical point (z→∞)

19 Coarsening in the Griffith phase Size of the condensate (determines the variance of the occupation number):

20 Coarsening at criticality Critical point: Anomalous coarsening Scaling:

21 Particle & sitewise disorder

22 Summary TASEP with particle disorder: condensation of holes at low densities Strong disorder: Griffith phase Criticality: Site disorder:


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