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

Restricted ASEP without particle Conservation flows to DP

Urna Basu

Theoretical Condensed matter Physics Division

Saha Institute of Nuclear Physics Kolkata, India

Joint work with P.K. Mohanty

Introduction

Absorbing state Phase Transition (APT) occurs in certain non-equilibrium systems Contact process, directed percolation, spreading etc.

C1 C2 C4 C3 C6 C7 C5 Absorbing configuration: can be reached but cannot be left

DP conjecture

Continuous transitions from an active phase to an absorbing state governed by a fluctuating scalar

• rder parameter belong to Directed Percolation

(DP) universality

Janssen Z Phys B 1981 Grassberger Z Phys B 1982

short range interaction no unconventional symmetry no additional conservation no quenched disorder

…if the system has

APT s not belonging to DP

Branching annihilating Random Walk Compact Directed Percolation Voter model etc ...

Parity

Particle Hole Z2+ Noise

Continued……

Manna, CLG, RASEP etc…

& Sandpile models (Self organized) Fluctuating scalar order parameter No special symmetry Additional conserved field (density or height) Belief :

Non-DP behaviour is due to coupling of order parameter to the conserved field.

Conservation is the cause ?

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

Mohanty & Dhar PRL 2002

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 ?

RASEP 1 1 DP 0.2764 1.09 0.2764



 

 

Very different

Restricted Asymmetric Simple Exclusion Process (RASEP) Restricted forward motion of hardcore particles on a periodic 1D lattice ( L sites ) Configuration A particle moves forward only when followed by atleast m particles

• m=1 110 -> 101 ;
• m=2 1110 -> 1101 etc…

Particle conserving dynamics Isolated particles : absorbing configuration

1 2

{ , ... }

L

s s s

1

t i h h t

if i site is occupied s if i site is empty      

UB & Mohanty PRE 2009

Exact results : APT at a critical density

Order parameter :

(density of active sites)

... m =1 <110 > m =2 <1110 >

a

ρ

a

[ρ-m(1- ρ)](1- ρ) = ρ-(m- ρ 1)(1- ρ)

 =1

c

ρ m = m +1 Critical exponent Order parameter vs density for m=1,2,3 Control parameter: 

( )

a c 

   

Spatial correlations

Generic n-point correlations can be calculated exactly Correlation between two active sites separated by j sites for m=1 :

1, 1  

 



2

1) 1 (2

j j

             

spatial correlation for m=1

Other exponents

Violate one scaling law

z RASEP 1 1 1 1/2 2



 

 



||



||

z    

Lee & Lee PRE 2008 Exact Results Numerical estimates

Jain PRE 2005 Da Silva & Oliveira J Phys A 2008 UB & Mohanty PRE 2009

Breaking the density conservation

Augment the dynamics with some particle addition/deletion moves Simplest one : Fixes the density Destroys all the absorbing states -> No Transition !

w     

w 1-w

1

Need to keep the absorbing states intact

One possible dynamics for m=1:

• add & delete
• original conserving hop
• Absorbing configuration :

Isolated 1s

• Activity

Keeping the absorbing states ...

  

w 1-w

110 111

 

1

110 101

same as before

 ρ 1/2

a

 =<110 >

Non-conserved dynamics

Works only on active configurations (some

• f the particles have neighbours)

  

w 1-w

110 111

density increases with w. Low w likely to be absorbed likely to be active Expect an APT as w is decreased below some wc



  1 2

 1

w 1

Use Monte-Carlo simulation to study the critical behaviour

Decay of activity

( ) ~

a t

t  



At wc activity decays as :

0.1595

DP

   

Starting from maximally active configuration 110110110…

0.567(6)

c

w 

L=10000 w = 0.565,0.567, 0.5677,0.569,0.571

Order parameter

Order parameter exponent

( )

a c

w w



 

0.567(6)

c

w 

0.2764

DP

   

Density of active sites in the steady state vanishes algebraically at wc :

L = 10000

Off-Critical Scaling

Curves with different w collapsed using

( ) ~

a t

t  



||

( , ) ( | | )

a c

t w t F t w w

 





 

|| ||

0.1595 1.732

DP DP

       

w= 0.50,0.52,0.54, 0.58,0.60,0.62

||

   

with ~ ( )

a c

t w w



   

At w= wc For

Finite size scaling

1.5807

DP

z z  

For a finite system at wc Curves for different system sizes collapse using

( , ) ( / )

z a t L

L G t L

 









w=wc; L= 64, 128, 256

0.252

DP DP

   

 

 

Spreading Exponents

At wc, starting from a single active seed

• Number of activity grows :
• Survival probability decays :

( )

a

N t t ( ) ~

sur

P t t 



0.1595

DP

   

0.313

DP 

  

Reminder:

time reversal symmetry

DP DP

  

RASEP RASEP

  

Propagation of activity below criticality

w = 0.520 L = 1000

Density at critical point…

• Well defined for w>wc,
• Ill defined below wc (absorbing phase)

Approaches as w-> wc ; at critical point : <11>=0 [no activity]

( ) w  Density

( )

c

w 

( )

c

w   1 - < 00 > 2

 =<10 > + <11 >  1- =< 00 > + < 01 >

continues …

From numerical simulations Near the critical point

b=0.277 (close to !)

DP



( ) ( ) ( )b

c c

w w w w    

c

ρ(w ) = 0.491

L=10000

Density as the control

Critical exponent

c c

ρ = ρ(w ) = 0.491

*

( )

a c

ρ = ρ-ρ

*

1  

*

b





c c

= ρ-ρ (w-w ) = =1 b

!

DP

 b =

( )

a c

    Reminder: In RASEP

a vs w



vs w 

a vs

 

Other exponents

Do not change: Decay and spreading exponents Correlation length exponents change

* * || || DP DP

   

 

 

 

* * * *

z z          

( )

a t vs t



/

/

z aL

vs t L

 





More about density…

is an equivalent order parameter Non –conserved RASEP  DP as coupled to a DP transition in density ?  No transition

  

w 1-w

110 111

 

1

110 101

( ) ( )

DP

c c

w w w



     ( )

c

w   

Scenario for RASEP with m=2

Conserving hop 1110 -> 1101 Use similar dynamics to add & delete particles Works !

wc= 0.7245

All critical exponents are same as DP

  

w 1-w

1110 1111

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

Symmetric RASEP



Break density conservation

Without conservation Flows to DP

1 1 1 1 w w w w  

  110 101 011 111

Conclusion :

In RASEP and similar models (in 1D) + a suitable non-conserved dynamics leads to DP behaviour

Exclusion processes On a ring + Restriction = APT

Is it possible to get DP by breaking conservation in CLG, CTTP, Manna models ?

+ Non-conservation =DP