Driven diffusive systems and growing stationary
configurations
T. Rafferty, P. Chleboun, S. Grosskinsky
Tom Rafferty
Paul Chleboun
Stefan Grosskinsky
University of Warwick
t.rafferty@warwick.ac.uk
September 24, 2013
( University
Growth
of Warwick
and Condensation
t.rafferty@warwick.ac.uk )
September 24, 2013
1 / 19
Overview
1
Introduction
2
Zero-range process
Stationary measures
3
Properties of a growth rule
4
The zero-range process - Constant rates
Results - Growth
5
Condensation
Results - Growth
6
Results
T. Rafferty, P. Chleboun, S. Grosskinsky
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September 24, 2013
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Introduction
Interacting particle system
Stationary product measures
Condensation transition
Relaxation times of order N 2
Fixed and finite lattice
The Aim
Construct a growth rule
CPU times scales linearly with N
Observe condensation
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The zero-range process - Definition
uHΗ3LpH3,4L
uHΗ6LpH5,6L
1
2
3
4
5
6
7
L
Definition
Lattice: Λ = {1, 2, . . . , L}
State space:
S = NΛ
Configurations: η = (ηx )x∈Λ where ηx ∈ N
Jump rates:
ux : N → [0, ∞)
ux (n) = 0 ⇔ n = 0
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The zero-range process - Stationary measures
T. Rafferty, P. Chleboun, S. Grosskinsky
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and Condensation
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September 24, 2013
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The zero-range process - Stationary measures
Site marginals (Grand-canonical ensemble)
νφx [ηx = n] =
T. Rafferty, P. Chleboun, S. Grosskinsky
wx (n)(φ)n
zx (φ)
and wx (n) =
n
Y
k=1
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1
ux (k)
September 24, 2013
5 / 19
The zero-range process - Stationary measures
Site marginals (Grand-canonical ensemble)
νφx [ηx = n] =
wx (n)(φ)n
zx (φ)
and wx (n) =
n
Y
k=1
1
ux (k)
P
n
Partition function zx (φ) = ∞
n=0 wx (n)(φ)
Fugacity φ ∈ [0, φc ) where φc ∈ [0, ∞] called critical fugacity
T. Rafferty, P. Chleboun, S. Grosskinsky
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Growth
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September 24, 2013
5 / 19
The zero-range process - Stationary measures
Site marginals (Grand-canonical ensemble)
νφx [ηx = n] =
wx (n)(φ)n
zx (φ)
and wx (n) =
n
Y
k=1
1
ux (k)
P
n
Partition function zx (φ) = ∞
n=0 wx (n)(φ)
Fugacity φ ∈ [0, φc ) where φc ∈ [0, ∞] called critical fugacity
P
k
Average density Rx (φ) = Eφ [ηx ] = zx 1(φ) ∞
k=1 kwx (k)(φ)
Critical density Rxc = limφ%φc ρx (φ) ∈ [0, ∞]
T. Rafferty, P. Chleboun, S. Grosskinsky
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Growth
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and Condensation
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September 24, 2013
5 / 19
The zero-range process - Stationary measures
Site marginals (Grand-canonical ensemble)
νφx [ηx = n] =
wx (n)(φ)n
zx (φ)
and wx (n) =
n
Y
k=1
1
ux (k)
P
n
Partition function zx (φ) = ∞
n=0 wx (n)(φ)
Fugacity φ ∈ [0, φc ) where φc ∈ [0, ∞] called critical fugacity
P
k
Average density Rx (φ) = Eφ [ηx ] = zx 1(φ) ∞
k=1 kwx (k)(φ)
Critical density Rxc = limφ%φc ρx (φ) ∈ [0, ∞]
Unique stationary measure (Canonical ensemble)
X
πL,N [η] = νφL ηx = n
ηx = N =
x∈Λ
Y
1
wx (ηx )
Z (L, N)
x∈Λ
Reference: [Spitzer, 1970] [Andjel, 1982]
T. Rafferty, P. Chleboun, S. Grosskinsky
( University
Growth
of Warwick
and Condensation
t.rafferty@warwick.ac.uk )
September 24, 2013
5 / 19
The zero-range process - Stationary measures
Site marginals (Grand-canonical ensemble)
νφx [ηx = n] =
wx (n)(φ)n
zx (φ)
and wx (n) =
n
Y
k=1
1
ux (k)
P
n
Partition function zx (φ) = ∞
n=0 wx (n)(φ)
Fugacity φ ∈ [0, φc ) where φc ∈ [0, ∞] called critical fugacity
P
k
Average density Rx (φ) = Eφ [ηx ] = zx 1(φ) ∞
k=1 kwx (k)(φ)
Critical density Rxc = limφ%φc ρx (φ) ∈ [0, ∞]
Unique stationary measure (Canonical ensemble)
X
πL,N [η] = νφL ηx = n
ηx = N =
x∈Λ
Y
1
wx (ηx )
Z (L, N)
x∈Λ
Reference: [Spitzer, 1970] [Andjel, 1982]
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and Condensation
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September 24, 2013
5 / 19
Growth - Properties
Α3=4
Α0=1
Grow L chains independently
Product measures
Continuous time birth
process
1
2
3
Correct marginals
X 1 HtL
Condition on N particles to
regain canonical stationary
measure
Birth Chains
Time vs Fugacity
T. Rafferty, P. Chleboun, S. Grosskinsky
4
Α4 =5
Α2 =3
X 2 HtL
Α5 =6
X 3 HtL
Α0 =1
X 4 HtL
0
1
2
3
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4
5
September 24, 2013
6 / 19
The zero-range process - Constant rates
ux (k) = α > 0 ∀k ≥ 1 =⇒ wx (n) =
1
αn
Stationary measures (Grand-canonical ensemble)
νφx [ηx
n
φ
φ
= n] = 1 −
α
α
(Geometric RV)
Stationary measures (Canonical ensemble)
πL,N [η] =
1
Z (L, N)
Defined for all φ ∈ [0, α)
Average density: ρx (φ) =
φ
α−φ
Critical density: ∞
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September 24, 2013
7 / 19
Results - Growth
Α2 =3
0
1
2
Αi =i+1
3
4
5
Master Equation
d
P(Xt = n) = nP(Xt = n − 1) − (n + 1)P(Xt = n)
dt
d
P(Xt = 0) = −P(Xt = 0)
dt
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September 24, 2013
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Results - Pure-birth process - Chain distribution
Use generating function: F (s, t) =
Solve the PDE:
P∞
k=0 s
k P(X
t
= k)
∂
∂t F (s, t)
(
F (0, t) = 0 ∀t ≥ 0
Boundary conditions
F (s, 0) = 1 ∀s ∈ [0, 1]
Use the identity:
T. Rafferty, P. Chleboun, S. Grosskinsky
1 ∂k
k! ∂s k F (s, t)
= P(Xt = k)
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September 24, 2013
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Results - Pure-birth process - Chain distribution
Use generating function: F (s, t) =
Solve the PDE:
P∞
k=0 s
k P(X
t
= k)
∂
∂t F (s, t)
(
F (0, t) = 0 ∀t ≥ 0
Boundary conditions
F (s, 0) = 1 ∀s ∈ [0, 1]
Use the identity:
1 ∂k
k! ∂s k F (s, t)
= P(Xt = k)
Chain distribution
n
P(Xt = n) = e −t 1 − e −t
n
φ
φ
x
νφ [ηx = n] = 1 −
α
α
φ
α
(Geometric RV)
(Geometric RV)
= 1 − e −t
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T. Rafferty, P. Chleboun, S. Grosskinsky
Section 5
Condensation
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September 24, 2013
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The zero-range process - Condensation
20
Rd(φ)
R(φ)
R(φ)
10
ρc
0
0.5
φ
1
φc=0.8
Conditions for condensation [Angel et al., 2004] [Ferrari et al., 2007]
∀x ∈ Λ \ {d}
Constant jump rates
ux (k) = 1
Single-site defect
ud (k) = r < 1
T. Rafferty, P. Chleboun, S. Grosskinsky
∀k ≥ 1
∀k ≥ 1
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Condensation
η
100
ρ < ρc
50
0
0
256
512
Λ
768
η
100
1024
ρ > ρc
50
0
0
T. Rafferty, P. Chleboun, S. Grosskinsky
256
512
Λ
768
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1024
September 24, 2013
12 / 19
Results - Pure-birth process - Time inhomogeneous
αi = i + 1 −→ αi (t) = h(t)(i + 1)
Geometric random variable
P(Xt = n) = e −H(t) 1 − e −H(t)
Rt
H(t) = 0 h(s)ds
n
φ
νφd [ηd = n] = 1 − φr
r
T. Rafferty, P. Chleboun, S. Grosskinsky
n
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September 24, 2013
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Results - Pure-birth process - Time inhomogeneous
αi = i + 1 −→ αi (t) = h(t)(i + 1)
Geometric random variable
P(Xt = n) = e −H(t) 1 − e −H(t)
Rt
H(t) = 0 h(s)ds
n
φ
νφd [ηd = n] = 1 − φr
r
n
Intensity function
Solve e −H(t) = 1 − φ/r
1 − e −t
H(t) = − log 1 −
r
?
T = − log(1 − r )
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Results - Pure-birth process - Simulation
Scaled Events (T)
2
T*
H−1(t)
1
0
0
1
2
Events (E)
3
4
Blue - Time inhomogeneous process
Red - Time homogeneous process
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Results - Pure-birth process - Simulation
80
70
Growth Process
Zero Range Simulation
Defect Site
Condensate
60
ηx
50
40
30
20
10
0
0
100
T. Rafferty, P. Chleboun, S. Grosskinsky
200
300
400
500
Lattice Site (x)
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Results - Pure-birth process - CPU time
−6
6
x 10
CPU time / L
5
4
L=2
L=256
L=512
ρ =4
c
3
2
1
0
0
1
T. Rafferty, P. Chleboun, S. Grosskinsky
2
3
4
Density (ρ)
5
6
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7
8
September 24, 2013
16 / 19
Conclusion
Discussion
Sampled from the stationary product measure with CPU times scaling
linearly with N
Used a time inhomogeneous pure-birth process
The intensity function exhibited finite time blow up
Future work
Use coupling techniques to sample form the stationary measure of the
attractive ZRP with general rates
Construct a coupling of general product measure to sample from
general driven diffusive systems
T. Rafferty, P. Chleboun, S. Grosskinsky
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Growth
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and Condensation
t.rafferty@warwick.ac.uk )
September 24, 2013
17 / 19
Acknowledgements
Supervisors
Paul Chleboun
Stefan Grosskinsky
Centre for Complexity Science
EPSRC - for the money
Any questions???
T. Rafferty, P. Chleboun, S. Grosskinsky
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Growth
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and Condensation
t.rafferty@warwick.ac.uk )
September 24, 2013
18 / 19
References
Andjel, E. D. (1982).
Invariant Measures for the Zero Range Process.
Ann. Probab., 10(3):525–547.
Angel, A. G., Evans, M. R., and Mukamel, D. (2004).
Condensation transitions in a one-dimensional zero-range process with
a single defect site.
Journal of Statistical Mechanics: Theory and Experiment,
2004(04):P04001.
Ferrari, P. A., Landim, C., and Sisko, V. (2007).
Condensation for a Fixed Number of Independent Random Variables.
J. Stat. Phys., 128(5):1153–1158.
Spitzer, F. (1970).
Interaction of Markov processes.
Adv. Math., 5:246–290.
T. Rafferty, P. Chleboun, S. Grosskinsky
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September 24, 2013
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