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 C2C4

C3C6 C7

C5Absorbing configuration:

can be reachedbut cannot be left

DP conjecture

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

ParticleHole

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 0

DP 0.2764 1.09 0.2764

Very different

Restricted Asymmetric Simple Exclusion Process (RASEP)

Restricted forward motion of hardcoreparticles on a periodic 1D lattice ( L sites )Configuration

A particle moves forward only when followed by atleast m particles

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

Particle conserving dynamicsIsolated particles : absorbing configuration

1 2{ , ... }Ls s s1

0

t

i h

h

t

if i site is occupieds

if i site is empty

UB & Mohanty PRE 2009

Exact results : APT at a critical density

Order parameter : (density of active sites)

...

m =1

m =2

aρ

a

[ρ-m(1- ρ)](1- ρ)=

ρ-(m-ρ

1)(1-ρ)

=1

cρm

=m +1

Critical exponent

Order parametervs 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

21)1

(2

j

j

spatial correlationfor m=1

Other exponents

Violate one scaling law

z

RASEP 1 1 1 0 1/2 2

||

||z

Lee & Lee PRE 2008

Exact ResultsNumericalestimates

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-w0 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-w110 111

1110 101

same asbefore

ρ 1/2

a =

Non-conserved dynamics

Works only on active configurations (some of the particles have neighbours)

w

1-w110 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

1w 1

Use Monte-Carlo simulation to studythe 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)cw L=10000 w = 0.565,0.567,

0.5677,0.569,0.571

Order parameter

Order parameter exponent

( )a cw w

0.567(6)cw

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 ct 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 ct w w

At w= wcFor

Finite size scaling

1.5807DPz z

For a finite system at wc

Curves for different

system sizes collapseusing

( , ) ( / )za t L L G t L

w=wc; L= 64, 128, 256

0.252DP

DP

Spreading Exponents

At wc, starting from a single active seed

- Number of activity grows :

- Survival probability decays :

( )aN t t

( ) ~surP t t

0.1595DP

0.313DP

Reminder: time reversal symmetry

DP DP

RASEP RASEP

Propagation of activity below criticality

w = 0.520L = 1000

Density at critical point…

Well defined for w>wc,

Ill defined below wc (absorbing phase)

Approaches as w-> wc

;

at critical point : =0 [no activity]

( )wDensity

( )cw

( )cw 1

- < 00 >2

= + 1- =< 00 > + < 01 >

continues …

From numerical simulations

Near the critical point

b=0.277 (close to !)DP

( ) ( ) ( )bc cw w w w

cρ(w ) =0.491

L=10000

Density as the control

Critical exponent

c cρ = ρ(w ) =0.491

*

( )a cρ = ρ-ρ

* 1

*

b

c c=ρ-ρ (w-w )

= =1b

!DPb =

( )a c

Reminder: In RASEP

a vs wvs w

a vs

Other exponents

Do not change:

Decay and spreading exponents

Correlation length exponents

change

*

*

||

||

DP

DP

* *

* *z z

( )a t vs t

/ / zaL 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-w110 111

1110 101

( ) ( ) DPc cw w w

( )cw

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-w1110 1111

Both forward and backward hopping

Generic m : APT at same density

Belongs to RASEP universality class

m=1 :

110 -> 101

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 behaviourExclusion processes

On a ring + Restriction = APT

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

+ Non-conservation =DP