Condensation in Totally Asymmetric Inclusion Process
Jiarui Cao
Joint work with Paul Chleboun and Stefan Grosskinsky
January 10, 2013
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Outline
1. Totally Asymmetric Inclusion Process (TASIP)
2. Condensation in TASIP Model
3. Dynamics of Condensation
I
I
I
I
Stationary Regime
Saturation Regime
Coarsening Regime
Nucleation Regime
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Totally Asymmetric Inclusion Process
Lattice : ΛL = {1, 2, 3, ..., L} with periodic boundary condition
State space : X = {0, 1, 2, ...}ΛL
Configuration : η = (ηx )x∈ΛL . Conserved particles:
Generator : Lf (η) =
X
P
x∈ΛL
ηx = ρL L = N
p(x, y )ηx (dL + ηy )(f (η x,y ) − f (η))
x,y ∈Λ
(
where p(x, y ) =
1,
0,
if y = x + 1
(nearest neighbor jump).
otherwise
Stationary product measure [Grosskinsky, Redig, Vafayi. 2011]
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Condensation in TASIP model
Condensation : all particles accumulate on a single site.
In the limit dL → 0, a condensation phenomenon occurs in inclusion
process :
I
L and N are both fixed.
I
in the limit L → ∞, N → ∞ s.t.
N
L
I
in the limit L fixed, and N → ∞.
[Grosskinsky, Redig, Vafayi. 2012]
[Grosskinsky, Redig, Vafayi. 2011]
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→ ρ > 0.
[Chleboun. 2011]
Condensation in Totally Asymmetric Inclusion Process
Condensation in TASIP model
MOVIE [L = 64, ρL = 2, dL = L12 ]
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Dynamics of Condensation
120
120
100
100
80
80
60
60
Nucleation
−→
40
20
0
Coarsening
−→
40
20
0
10
20
30
40
50
0
60
0
120
120
100
100
80
80
60
20
30
40
50
60
60
Saturation
−→
40
20
0
10
Stationary
−→
40
20
0
10
20
30
40
50
0
60
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0
10
20
30
40
50
60
Condensation in Totally Asymmetric Inclusion Process
Condensation in TASIP model
Observable: σ 2 (t) = E[η 2 ] =
1
L
P
x∈ΛL
ηx2
σ 2 (0) = ρ2 + ρ − L1 ρ (converges to ρ(1 + ρ) as L → ∞).
t→∞
σ 2 (t) −→ ρ2 L
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Dynamics of Condensation
0
0
10
10
II
σ2/ρ2L
2
2
σ /ρ L
I
−1
10
III
IV
−1
10
L=64 ρ=2
L=64 ρ=4
L=128 ρ=2
L=128, ρ=4
Exp Fit
Power Law Fit
−2
10
0
5
10
t/τcoars x 10
−4
10
−3
10
−2
−1
10
10
0
10
coars
−6
t/τ
Simulation results with dL = 1/L2 , averaged with 200 realizations
I : Nucleation Regime. II : Coarsening Regime.
III : Saturation Regime. IV : Stationary Regime.
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Stationary Regime
I
A particle jumps by diffusion with rate dL ρL L.
I
Other particles on site x follow immediately by inclusion.
I
The single condensate moves ballistically with speed dL ρL L.
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Saturation Regime
Two condensates interaction
I
N1 , N2 are both in order L.
I
Large condensate will penetrate small one without macroscopic
change of number of particles.
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Saturation Regime
Exponential Approximation of σ 2 (t)
d
E [f (ηt )] = E[(LL f )(ηt )] , take f (η) = ηz2 , z ∈ ΛL .
dt
Assumption : ηz−1 = ηz+1 =
2
⇒ σ 2 (t) ' ρ2L L 1 − e − L t
ρL L−ηz
L−1
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Saturation Regime
0
10
0
10
−0.1
10
III
IV
−0.2
10
−0.3
10
2
2
σ /ρ L
σ2/ρ2L
II
−1
10
−0.4
10
L=64 ρ=2
L=64 ρ=4
L=128 ρ=2
L=128, ρ=4
Exp Fit
Power Law Fit
−4
10
−3
10
−2
−1
10
10
0
10
⇒
L=64 ρ=2
L=64 ρ=4
L=128 ρ=2
L=128, ρ=4
Exp Fit
−0.5
10
−0.6
10
−1
0
10
t/τcoars
10
coars
t/τ
Exponential fit of saturation regime
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Coarsening Regime
n(t) : number of piles. m(t) : size of a typical pile. (m(t)n(t) = ρL L)
v ∼ dL m(t) : speed of a pile. s ∼
L
n(t)
: average distance of two piles.
Differential equation
d
v
ρL
m(t) ∼ = ρL dL , initial condition : m(0) =
dt
s
r
r : average ratio of occupied sites after nucleation.
⇒
m(t) = C1 ρL dL t +
ρL
r
After time τLcoars , m(t) will grow to size L.
m(τLcoars ) ∼ L ⇒ τLcoars ∼
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L
dL
Condensation in Totally Asymmetric Inclusion Process
Coarsening Regime
Ratio of occupied sites:
n(t*dL)/L
n(t)
1
=
,
L
C1 dL t + 1/r
−1
10
L=64 ρ=2
L=64 ρ=4
L=128 ρ=2
L=128 ρ=4
L=256 ρ=2
L=256 ρ=4
L=512 ρ=2
CurveFit
−2
10
−1
10
0
10
1
t*dL
10
2
10
Simulation results with dL = 1/L2 , averaged with 200 realizations. Fitting constant C1 ≈ 0.8538
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Coarsening Regime
Second moment:
σL2 (t)
σ 2 (t)
1
∼ n(t)m2 (t) ⇒ L 2 = C̃
L
ρL
1
C1 dL t +
r
256
128
2
10
σ2/ρ2
64
L=64 ρ=2
L=64 ρ=4
L=128 ρ=2
L=128 ρ=4
L=256 ρ=2
L=256 ρ=4
L=512 ρ=2
Fit
1
10
−1
10
0
1
10
10
2
10
t*dL
Simulation results with dL = 1/L2 , averaged with 200 realizations. Fitting constant C1 ≈ 0.5513 and C̃ ≈ 3.3070
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Nucleation Regime [in progress]
120
120
100
100
Nucleation
80
−→
60
40
20
0
80
60
40
20
0
10
20
30
40
50
60
0
0
10
20
30
40
50
60
Time scale of this regime is much smaller compared with coarsening
regime.
I R: ratio of occupied sites when this regime ends. (appr. Normal
distributed with mean r ).
I T : duration of this regime. (appr. Gumbel distributed).
I
2
10
L=64
64 Gumbel
L=128
128 Gumbel
L=256
256 Gumbel
L=512
512 Gumbel
L=1024
1024 Gumbel
L=2048
2048 Gumbel
L=4096
4096 Gumbel
0.7
Density
0.6
0.5
0.4
1
10
L=64
64 Normal Fit
L=128
128 Normal Fit
L=256
256 Normal Fit
L=512
512 Normal Fit
Density
0.8
0.3
0
10
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
−1
10
Stop Time
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
Ratio : (# Non−empty sites)/L
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Nucleation Regime [in progress]
Toy model for R:
ηx = 1, ∀x ∈ ΛL . Waiting time Tx , i.i.d. r.v.
Particles merge if they stay on the same site.
...11111111... ⇒ ...11111101...
Absorbing state:
...1010001000100...
Constructed
by blocks ( e.g.0001 ) of size Xn , where 2 ≤ Xn ≤ L and
P
n Xn = L.
Block with ’1’ on fixed site x has k ’0’s ⇐⇒ Tx−k < Tx−k+1 < ... < Tx−1
Tx are i.i.d. =⇒ uniform permutation.
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Nucleation Regime [in progress]
Toy model for R:
L−1
X 1
1
⇒ E[Xn ] =
+ 1 → e , as L → ∞.
P[Xn − 1 ≥ k] =
k!
k!
k=1
Xn forms a renewal process :
(
N(L) = max n :
n
X
)
Xi ≤ L ,
i=1
where N(L) is number of particles in n blocks.
From renewal theorem:
N(L)
1
1
→
=
, as L → ∞ almost surely .
L
E[X1 ]
e
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
Review
0
0
10
10
II
σ2/ρ2L
2
2
σ /ρ L
I
−1
10
III
IV
−1
10
L=64 ρ=2
L=64 ρ=4
L=128 ρ=2
L=128, ρ=4
Exp Fit
Power Law Fit
−2
10
0
5
10
t/τcoars x 10
−4
10
−3
10
−2
−1
10
10
0
10
coars
−6
t/τ
Simulation results with dL = 1/L2 , averaged with 200 realizations
I : Nucleation Regime. II : Coarsening Regime.
III : Saturation Regime. IV : Stationary Regime.
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process
THANK YOU
Jiarui Cao
Condensation in Totally Asymmetric Inclusion Process