large deviations of stochastic processes and lifetime of ... · overview stochastic processes...
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Overview Stochastic processes Mean-field approach Computation Results Conclusion
Large deviations of stochastic processes andlifetime of metastable states in high space
dimensions
Christophe Deroulers
Institute of Theoretical Physics, University of Cologne
MPIPKS Dresden, October 30th, 2006
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Overview
Stochastic processes
Mean-field approach
Computation
Results
Conclusion
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Overview
Stochastic processes
Mean-field approach
Computation
Results
Conclusion
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Framework
Graph with N sites (or nodes), e.g.
I D-dimensional hypercubic lattice
I Cayley tree (Bethe lattice)
I fully connected graph
I ...
where each site i is in one of a discrete set of states.
Evolution according to kinetic rules.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Framework
Graph with N sites (or nodes), e.g.
I D-dimensional hypercubic lattice
I Cayley tree (Bethe lattice)
I fully connected graph
I ...
where each site i is in one of a discrete set of states.
Evolution according to kinetic rules.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
1st example: Contact process
T. E. Harris, Ann. Prob. 2 969 (1974)
Inspired by epidemics, computer virus, social phenomena,...
States of the sites:sick/full/B or healthy/empty/A
Kinetic rules:
B1−→ A
B + Aλ−→ B + B
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Contact process: kinetic rules
Spontaneousrecovery
dt
Contamination
λdt
λdt
coordination number z
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Second example: rock-scissors-paper model
States: A, B, C.
Rules:
A + BkB−→ B + B
B + CkC−→ C + C
C + AkA−→ A + A
No detailed balance
Motivations: population biology; can species coexist? Populationoscillations?
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Biology experiments
B. J. M. Bohannan & R. E.Lenski, Ecology 78 2303 (1997)
Kerr et al., Nature 418 171(2002)
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Overview
Stochastic processes
Mean-field approach
Computation
Results
Conclusion
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Rate equations: contact processFor the contact process: B
1→ A, A + Bλ→ B + B
dρ(t)dt
= −ρ(t) + λρ(t)[1− ρ(t)]
where ρ(t) is the density of sick sites.
0 10 20 30time t
0
0.2
0.4
0.6
0.8
1
dens
ity ρ
(t) λ = 2
λ = 1
λ = 0.5
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Rate equations: contact processFor the contact process: B
1→ A, A + Bλ→ B + B
dρ(t)dt
= −ρ(t) + λρ(t)[1− ρ(t)]
where ρ(t) is the density of sick sites.
0 10 20 30time t
0
0.2
0.4
0.6
0.8
1
dens
ity ρ
(t) λ = 2
λ = 1
λ = 0.5
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Comparison to simulation
N = 100 sites
0 10 20 30date t
0
0.2
0.4
0.6
0.8
1de
nsity
ρ(t
) 0 2000 4000 60000
0.5
1
λ = 1.6
λ = 1
λ = 0.5
λ = 1.6 (continued)
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Metastable state, absorbing state
density of full sites
t
f x volume∼exp( )
system trapped in the metastable state
complete recovery
Overview Stochastic processes Mean-field approach Computation Results Conclusion
QuestionsWhat it the lifetime of the metastable state?
In the metastable phase, one may define a large deviations function(for N sites):
P(ρ, t) ∝ eNπ(ρ,t) for large N
Overview Stochastic processes Mean-field approach Computation Results Conclusion
QuestionsWhat it the lifetime of the metastable state?
In the metastable phase, one may define a large deviations function(for N sites):
P(ρ, t) ∝ eNπ(ρ,t) for large N
Overview Stochastic processes Mean-field approach Computation Results Conclusion
QuestionsWhat it the lifetime of the metastable state?
In the metastable phase, one may define a large deviations function(for N sites):
P(ρ, t) ∝ eNπ(ρ,t) for large N
π(ρ = 0, t = ∞) rules the lifetime :
lifetime = eN|π(ρ=0,t=∞)| × subdominant factors
→ How much is π(ρ, t)?
Our wish: a reusable technique
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Rate equations: rock-scissors-paper model
A + BkB−→ B + B B + C
kC−→ C + C C + AkA−→ A + A
da(t)
dt= −kBa(t)b(t) + kAc(t)a(t)
db(t)
dt= −kCb(t)c(t) + kBa(t)b(t)
dc(t)
dt= −kAc(t)a(t) + kCb(t)c(t)
where a(t), b(t), c(t) are the densitiesof sites in states A, B, C.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Rate equations: rock-scissors-paper model
A + BkB−→ B + B B + C
kC−→ C + C C + AkA−→ A + A
da(t)
dt= −kBa(t)b(t) + kAc(t)a(t)
db(t)
dt= −kCb(t)c(t) + kBa(t)b(t)
dc(t)
dt= −kAc(t)a(t) + kCb(t)c(t)
where a(t), b(t), c(t) are the densitiesof sites in states A, B, C.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Comparison to simulationRock-scissors-paper model, N = 1600 sites
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100
popu
latio
n
t
population 0
Fully-connected graph(mean-field)
100
200
300
400
500
600
700
800
900
1000
0 10 20 30 40 50 60 70 80 90 100
popu
latio
n
t
population 0
2D square lattice
Transition: oscillating metastable state to absorbing stateRapid in mean-field (→ extinction of species), slow in finite D?Confirmed by biology experiments
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Comparison to simulationRock-scissors-paper model, N = 1600 sites
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100
popu
latio
n
t
population 0
Fully-connected graph(mean-field)
100
200
300
400
500
600
700
800
900
1000
0 10 20 30 40 50 60 70 80 90 100
popu
latio
n
t
population 0
2D square lattice
Transition: oscillating metastable state to absorbing stateRapid in mean-field (→ extinction of species), slow in finite D?Confirmed by biology experiments
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Overview
Stochastic processes
Mean-field approach
Computation
Results
Conclusion
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Route of the computation
Stochastic process↓
Master equation↓
“Quantum” formalism↓
Field theory: lnZ [{ρi (t), ψi (t)}]↓
ODE or PDE↓
Large deviation function
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (1)Master equation written with “quantum” operators:
Felderhof, Doi, Peliti, ...
ddt|P(t)〉 = W |P(t)〉
with, for the contact process,
W = Wrecov + λWcontam
Wrecov =∑
i
(1− a+i )ai
(1− a+)a |0〉 = 0
(1− a+)a |1〉 = |0〉 − |1〉
Wcontam =1
z
∑i
∑j∈i
[a+j (1 + aj)− 1]a+
i ai
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (1)Master equation written with “quantum” operators:
Felderhof, Doi, Peliti, ...
ddt|P(t)〉 = W |P(t)〉
with, for the contact process,
W = Wrecov + λWcontam
Wrecov =∑
i
(1− a+i )ai
(1− a+)a |0〉 = 0
(1− a+)a |1〉 = |0〉 − |1〉
Wcontam =1
z
∑i
∑j∈i
[a+j (1 + aj)− 1]a+
i ai
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (1)Master equation written with “quantum” operators:
Felderhof, Doi, Peliti, ...
ddt|P(t)〉 = W |P(t)〉
with, for the contact process,
W = Wrecov + λWcontam
Wrecov =∑
i
(1− a+i )ai
(1− a+)a |0〉 = 0
(1− a+)a |1〉 = |0〉 − |1〉
Wcontam =1
z
∑i
∑j∈i
[a+j (1 + aj)− 1]a+
i ai
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (1)Master equation written with “quantum” operators:
Felderhof, Doi, Peliti, ...
ddt|P(t)〉 = W |P(t)〉
with, for the contact process,
W = Wrecov + λWcontam
Wrecov =∑
i
(1− a+i )ai
(1− a+)a |0〉 = 0
(1− a+)a |1〉 = |0〉 − |1〉
Wcontam =1
z
∑i
∑j∈i
[a+j (1 + aj)− 1]a+
i ai
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (1)Master equation written with “quantum” operators:
Felderhof, Doi, Peliti, ...
ddt|P(t)〉 = W |P(t)〉
with, for the contact process,
W = Wrecov + λWcontam
Wrecov =∑
i
(1− a+i )ai
(1− a+)a |0〉 = 0
(1− a+)a |1〉 = |0〉 − |1〉
Wcontam =1
z
∑i
∑j∈i
[a+j (1 + aj)− 1]a+
i ai
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (2)Main quantity: partition function / generating function
Z = 〈O|︸︷︷︸sum of all
states
exp
(∫dt W
)|P(0)〉︸ ︷︷ ︸initial
probabilitydistribution
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Towards a field theory (2)Main quantity: partition function / generating function
Z = 〈O| exp(∫
dt W
)|P(0)〉
Idea of the computation :
Gaunt et Baker (1970), Georges et Yedidia (1990)
I We look for the probability of each trajectory ρ(t).
I It is proportional (if N is large) to Z where we constrain,thanks to Lagrange multipliers,
〈a+i ai 〉(t) = ρi (t) 〈ai 〉 = ρi (t) eψi (t).
I We then restrict to dominant trajectories (instantons)→ P(ρ, t) by integration of ODE/PDE.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Remarks on the field theory
I We can compute Z (only) when sites are decoupled (λ = 0)and W matrix is triangular.
I ⇒ Perturbative expansion in powers of λ and of anotherparameter.
I λ ∝ 1z = 1
2D thus the expansion is in powers of 1/D :
λWcontam =1.23
z
∑i
∑j∈i
[a+j (1 + aj)− 1]︸ ︷︷ ︸
χj
a+i ai︸︷︷︸φi
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Action
lnZ = boundary terms +
N
∫ T
0dt
(−ρ(t)dψ(t)
dt+ WMF(ρ(t), ψ(t))
)+
N
D
∫ T
0dt
∫ t
0dt ′F
{ρ(t), ψ(t), ρ(t ′), ψ(t ′)
}+
N
D2. . .
WMF(ρ, ψ) = ρ[exp(ψ)− 1
]+ λ ρ (1− ρ)
[exp(−ψ)− 1
]
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Memory kernelsMotion equations:
dψdt
(t) = +∂ρWMF[ρ(t), ψ(t)] +1
D
∫ t
0dt ′ . . .+ . . .
dρdt
(t) = −∂ψWMF[ρ(t), ψ(t)] +1
D
∫ t
0dt ′ . . .+ . . .
∂π
∂t= WMF
[ρ(t),
∂π
∂ρ
]+
1
D. . .
Mean-field properties of the process given by WMF
To orders 1/D, 1/D2, ..., memory kernels appear, even if theprocess is Markovian.
I ⇒ decorrelation when |t ′ − t| → ∞I Consequence of the projection of 2N states on the subset |ρ〉.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Memory kernelsMotion equations:
dψdt
(t) = +∂ρWMF[ρ(t), ψ(t)] +1
D
∫ t
0dt ′ . . .+ . . .
dρdt
(t) = −∂ψWMF[ρ(t), ψ(t)] +1
D
∫ t
0dt ′ . . .+ . . .
∂π
∂t= WMF
[ρ(t),
∂π
∂ρ
]+
1
D. . .
Mean-field properties of the process given by WMF
To orders 1/D, 1/D2, ..., memory kernels appear, even if theprocess is Markovian.
I ⇒ decorrelation when |t ′ − t| → ∞I Consequence of the projection of 2N states on the subset |ρ〉.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Memory kernelsMotion equations:
dψdt
(t) = +∂ρWMF[ρ(t), ψ(t)] +1
D
∫ t
0dt ′ . . .+ . . .
dρdt
(t) = −∂ψWMF[ρ(t), ψ(t)] +1
D
∫ t
0dt ′ . . .+ . . .
∂π
∂t= WMF
[ρ(t),
∂π
∂ρ
]+
1
D. . .
Mean-field properties of the process given by WMF
To orders 1/D, 1/D2, ..., memory kernels appear, even if theprocess is Markovian.
I ⇒ decorrelation when |t ′ − t| → ∞I Consequence of the projection of 2N states on the subset |ρ〉.
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Solutions of the mean field equations of motion
ψ = 0 → rate equations
0 10 20 30time t
0
0.2
0.4
0.6
0.8
1
dens
ity ρ
(t) λ = 2
λ = 1
λ = 0.5
contact processrock-scissors-paper
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Solutions of the mean field equations of motion
ψ 6= 0 → most probable improbable events
contact process rock-scissors-paper
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Overview
Stochastic processes
Mean-field approach
Computation
Results
Conclusion
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Results : contact process transition threshold
λc = 1+1
2D+
7
3(2D)2+O(1/D3) λc = 1 +
1
z+
4
3z2+ O(1/z3)
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Results : contact process large deviation function(Top point shifted, quasistationary regime) curvature is exact
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Results : rock-scissors-paper large deviation function
P(a, b, c , t) ∝ eNπ(a,b,c,t) ?
Computation → in mean field, π(a, b, c ; t = +∞) = 0.Simulations:
Fully connected graph
-2
-1
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ln(P
)
a.b.c
N=100N=200N=400N=625N=900
2D square lattice: π 6= 0
-0.025
-0.02
-0.015
-0.01
-0.005
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-ln(P
)/N
a.b.c
N=20x20N=25x25N=30x30
Work in progress
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Results : rock-scissors-paper large deviation function
P(a, b, c , t) ∝ eNπ(a,b,c,t) ?
Computation → in mean field, π(a, b, c ; t = +∞) = 0.
Simulations:
Fully connected graph
-2
-1
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ln(P
)
a.b.c
N=100N=200N=400N=625N=900
2D square lattice: π 6= 0
-0.025
-0.02
-0.015
-0.01
-0.005
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-ln(P
)/N
a.b.c
N=20x20N=25x25N=30x30
Work in progress
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Results : rock-scissors-paper large deviation function
P(a, b, c , t) ∝ eNπ(a,b,c,t) ?
Computation → in mean field, π(a, b, c ; t = +∞) = 0.Simulations:
Fully connected graph
-2
-1
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ln(P
)
a.b.c
N=100N=200N=400N=625N=900
2D square lattice: π 6= 0
-0.025
-0.02
-0.015
-0.01
-0.005
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-ln(P
)/N
a.b.c
N=20x20N=25x25N=30x30
Work in progress
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Overview
Stochastic processes
Mean-field approach
Computation
Results
Conclusion
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Conclusion
I The lifetime of metastable states of stochastic processes inhigh dimension D can be computed sytematically in powers of1/D thanks to this formalism.
I The idea of a “quantum” representation of the masterequation is not new, but was mainly used for the computationof universal quantities (Renormalization Group).
I Systematic recipe for many stochastic processes.
I Already in mean-field it allows simple calculations thanks toODE and/or PDE.
I Effects of finite dimension can be very important, mean-fieldapproximation can be qualitatively wrong.
Thanks to Remi Monasson
Overview Stochastic processes Mean-field approach Computation Results Conclusion
Conclusion
I The lifetime of metastable states of stochastic processes inhigh dimension D can be computed sytematically in powers of1/D thanks to this formalism.
I The idea of a “quantum” representation of the masterequation is not new, but was mainly used for the computationof universal quantities (Renormalization Group).
I Systematic recipe for many stochastic processes.
I Already in mean-field it allows simple calculations thanks toODE and/or PDE.
I Effects of finite dimension can be very important, mean-fieldapproximation can be qualitatively wrong.
Thanks to Remi Monasson