Partially asymmetric exclusion processes with quenched disorder Ludger

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

The ZRP with strong disorder: Definition n n ASEP with particlewise disorder Hopping rates: pi (forward) qi (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): Scaling of g. L (inverse current): Example: Bimodal distribution

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

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

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

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/Ωξ Ω > Ωξ: elimination of forward and backward rates Ω < Ωξ: TASEP with rates: Relation between rate-scale and clustersize

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

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

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

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

Density profile at criticality (d=0) n Scaling form:

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

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

Coarsening at criticality n Critical point: n Scaling: Anomalous coarsening

Particle & sitewise disorder

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