Large deviations of stochastic processes and lifetime of ... · Overview Stochastic processes...

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


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.


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.


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


Contact process: kinetic rules

Spontaneousrecovery

dt

Contamination

λdt

λdt

coordination number z


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?


Biology experiments

B. J. M. Bohannan & R. E.Lenski, Ecology 78 2303 (1997)

Kerr et al., Nature 418 171(2002)


Overview


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


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


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)


Metastable state, absorbing state

density of full sites

t

f x volume∼exp( )

system trapped in the metastable state

complete recovery


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


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.


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.


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


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


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


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







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







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







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







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


Towards a field theory (2)Main quantity: partition function / generating function

Z = 〈O|︸︷︷︸sum of all

states

exp

(∫dt W

)|P(0)〉︸︷︷︸initial

probabilitydistribution


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.


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


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

]


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〉.



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. . .






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. . .





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


Solutions of the mean field equations of motion

ψ 6= 0 → most probable improbable events

contact process rock-scissors-paper


Overview


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)


Results : contact process large deviation function(Top point shifted, quasistationary regime) curvature is exact


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



P(a, b, c , t) ∝ eNπ(a,b,c,t) ?

Computation → in mean field, π(a, b, c ; t = +∞) = 0.

Simulations:


-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


-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



P(a, b, c , t) ∝ eNπ(a,b,c,t) ?

Computation → in mean field, π(a, b, c ; t = +∞) = 0.Simulations:


-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


-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


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


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

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