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TRANSCRIPT
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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
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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
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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
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APT s not belonging to DP
Branching annihilating Random Walk
Compact Directed Percolation
Voter model
etc ...
Parity
ParticleHole
Z2+ Noise
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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.
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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
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Breaking Density Conservation
May destroy the transition
May destroy the structure of the absorbing configurations
Need suitable non-conserving dynamics
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Propagation of activity below criticality
w = 0.520L = 1000
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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
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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
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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
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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
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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
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Both forward and backward hopping
Generic m : APT at same density
Belongs to RASEP universality class
m=1 :
110 -> 101
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Break density conservation
Without conservation
Flows to DP
1 1
1 1
w w
w w
110 101 011
111
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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