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

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

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

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

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

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

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

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

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

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

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

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

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

:::: z→∞: Cumulated distance Stationary state: Transport properties z=0: Uniform bias  Situation similar to the TASEP 0<z<1 CurrentJ~L -z Active Particles N a ~L 1-z N a =O(1) Particle velocity v=O(1) v~L -z Griffith Phase:

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

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

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

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→∞)

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

Coarsening at criticality Critical point: Anomalous coarsening Scaling:

Particle & sitewise disorder

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