Restricted ASEP without particle Conservation flows to DP Urna - PowerPoint PPT Presentation

Restricted ASEP without particle Conservation flows to DP Urna Basu Joint work with P.K. Mohanty Theoretical Condensed matter Physics Division Saha Institute of Nuclear Physics Kolkata, India

Introduction Absorbing state Phase Transition (APT) occurs in certain C 1 C 2 C 4 non-equilibrium systems C 3 C 6 C 7 Absorbing configuration: C 5 can be reached but cannot be left Contact process, directed percolation, spreading etc.

DP conjecture Janssen Z Phys B 1981 Grassberger Z Phys B 1982 Continuous transitions from an active phase to an absorbing state governed by a fluctuating scalar order parameter belong to Directed Percolation (DP) universality …if the system has short range interaction no unconventional symmetry no additional conservation no quenched disorder

APT s not belonging to DP Branching annihilating Random Walk Parity Compact Directed Percolation Particle Hole Voter model Z 2 + Noise etc ...

Continued…… Manna, CLG, RASEP etc… & Sandpile models (Self organized) Fluctuating scalar order parameter No special symmetry Additional conserved field (density or height) Non-DP behaviour is due to Belief : coupling of order parameter to the conserved field.

Conservation is the cause ? Sandpile models + special perturbation DP Mohanty & Dhar PRL 2002 ( even in presence of conserved field ) ‘Conservation is the cause’ only if breaking of conservation leads to DP

Breaking Density Conservation May destroy the transition May destroy the structure of the absorbing configurations Need suitable non-conserving dynamics

Motivation Pick a simple, analytically tractable model : Restricted ASEP (RASEP) Find a suitable dynamics to break the density conservation Investigate the critical behaviour : does it flow to DP ? Very      different RASEP 1 1 0 DP 0.2764 1.09 0.2764

Restricted Asymmetric Simple Exclusion Process (RASEP) Restricted forward motion of hardcore particles on a periodic 1D lattice ( L sites ) Configuration   t h 1 if i site is occupied   s { , ... } s s s i  t h  0 if i site is empty 1 2 L A particle moves forward only when followed by atleast m particles o m=1 110 -> 101 ; o m=2 1110 - > 1101 etc… Particle conserving dynamics Isolated particles : absorbing configuration UB & Mohanty PRE 2009

Exact results : APT at a critical density Control parameter:  ρ Order parameter : a (density of active sites) m =1 <110 > m =2 <1110 > ... [ ρ-m(1- ρ)](1- ρ) ρ = a ρ-(m- 1)(1- ρ) m ρ = m +1 c Order parameter      vs density for m=1,2,3 ( ) a c  =1 Critical exponent

Spatial correlations Generic n-point correlations can be calculated exactly Correlation between two active sites separated by j sites for m=1 : j     1      2 (2 1)    j   spatial correlation for m=1      1, 1

Other exponents Numerical Exact Results estimates     z    || 1/2 2 RASEP 1 1 1 0 Jain PRE 2005 Lee & Lee PRE 2008 Da Silva & Oliveira J Phys A 2008 UB & Mohanty PRE 2009  Violate one scaling law  || z  

Breaking the density conservation Augment the dynamics with some particle addition/deletion moves w   Simplest one : 0 1  1-w   Fixes the density w Destroys all the absorbing states -> No Transition !

Keeping the absorbing states ... Need to keep the absorbing states intact One possible dynamics for m=1: w   - add & delete 110 111  1-w - original conserving hop 110 1 101    Absorbing configuration : ρ 1/2  same as Isolated 1s before  =<110 >  Activity a

Non-conserved dynamics w   110 111  1-w Works only on active configurations (some of the particles have neighbours) density increases with w.    1 Low w likely to be absorbed 2 likely to be active w 1 1  Expect an APT as w is decreased below some w c

Use Monte-Carlo simulation to study the critical behaviour

Decay of activity t    At w c activity decays as : ( ) ~ a t Starting from maximally active configuration 110110110…     0.1595 DP w  0.567(6) c L=10000 w = 0.565,0.567, 0.5677,0.569,0.571

Order parameter Density of active sites in the steady state    vanishes algebraically at w c : ( ) w w a c w  0.567(6) c Order parameter exponent     0.2764 DP L = 10000

Off-Critical Scaling t    At w= w c ( ) ~ a t       ( , ) ( | | ) || t w t F t w w For a c      ~ ( ) t w w  a c   with  || Curves with different w collapsed using     0.1595 DP     1.732 || || DP w= 0.50,0.52,0.54, 0.58,0.60,0.62

Finite size scaling  For a finite system at w c     z ( , ) ( / )  a t L L G t L Curves for different system sizes collapse using     DP 0.252     DP   1.5807 z z w=w c; L= 64, 128, 256 DP

Spreading Exponents At w c, starting from a single active seed - Number of activity grows : t  ( ) N t a - Survival probability decays : t   ( ) ~ P t sur    DP  0.313     0.1595 DP    Reminder: DP DP time reversal symmetry    RASEP RASEP

Propagation of activity below criticality w = 0.520 L = 1000

Density at critical point… Density  ( ) w  Well defined for w>w c,  Ill defined below w c (absorbing phase) Approaches as w-> w c  ( ) w c  =<10 > + <11 > 1- =< 00 > + < 01 >  ; at critical point : <11>=0 [no activity]  1 - < 00 >  ( ) w 2 c

continues … From numerical simulations ρ(w ) = 0.491 c Near the critical point     ) b ( ) ( ) ( w w w w c c b=0.277 (close to !)  DP L=10000

Density as the control ρ = ρ(w ) = 0.491 c c Critical exponent   vs w a vs w ρ = ρ-ρ )  * ( a c Reminder:   * 1 In RASEP     ( ) a c   a vs ρ-ρ = (w-w ) b c c  = =1  * b b =  ! DP

Other exponents    / z  / a L vs t L Do not change:  ( ) a t vs t       * *     * * z z Decay and spreading exponents  Correlation length exponents    *   DP change    * ||  || DP

More about density…    is an equivalent order parameter ( ) w c       ( ) ( ) w w w DP c c Non – conserved RASEP  DP as coupled to a DP transition in density ? w   110 111  1-w 110 101 1    No transition

Scenario for RASEP with m=2 Conserving hop 1110 -> 1101 Use similar dynamics to add & delete particles w   1110 1111  1-w Works ! w c = 0.7245 All critical exponents are same as DP

Symmetric RASEP Both forward and backward hopping Generic m : APT at same density Belongs to RASEP universality class m=1 :  110 -> 101 <- 011 1D CLG

Break density conservation Without conservation 1 1 110 101 011   w w   1 1 w w 111 Flows to DP

Conclusion : In RASEP and similar models (in 1D) + a suitable non-conserved dynamics leads to DP behaviour Exclusion processes On a ring + Restriction = APT + Non-conservation =DP Is it possible to get DP by breaking conservation in CLG, CTTP, Manna models ?