Large deviations and population dynamics: finite-population effects · 2016. 6. 28. · Large deviations and population dynamics: finite-population effects Vivien Lecomte(1), Julien

Large deviations and population dynamics:finite-population effects

Vivien Lecomte(1), Julien Tailleur(2)

Freddy Bouchet(3), Rob Jack(4), Takahiro Nemoto(1)

Esteban Guevara(1,5)

(1)LPMA, Paris (2)MSC, Paris(3)ENS Lyon (4)Bath University (5)IJM, Paris

ENPC — June 28th, 2016

Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 1 / 28

Introduction Motivations

[Merrolle, Garrahan and Chandler, 2005]

Order parameter: additive observable K (= number of events)Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 2 / 28


0.0 0.5 1.0 1.5 2.0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

K /t

1/t

Log

Pro

b(K

/t)

Prob[K] ∼ etφ(K/t)



0.0 0.5 1.0 1.5 2.0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

K /t

1/t

Log

Pro

b(K

/t)

Prob[K] ∼ etφ(K/t)



0.0 0.5 1.0 1.5 2.0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

K /t

1/t

Log

Pro

b(K

/t)

Prob[K] ∼ etφ(K/t) Finite-time & -size scalings matter.Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 3 / 28

Introduction s-ensemble

s-modified dynamics

K =activity

Markov processes:

∂tP(C, t) =∑C′

{W(C′ → C)P(C′, t)︸︷︷︸

gain term

−W(C → C′)P(C, t)︸︷︷︸loss term

}

More detailed dynamics for P(C,K, t):

∂tP(C,K, t) =∑C′

{W(C′ → C)P(C′,K−1, t)−W(C → C′)P(C,K, t)

}Canonical description: s conjugated to K

P(C, s, t) =∑

Ke−sKP(C,K, t)

s-modified dynamics [probability non-conserving]

∂tP(C, s, t) =∑C′

{e−sW(C′ → C)P(C′, s, t)− W(C → C′)P(C, s, t)

}



s-modified dynamics K =activity

Markov processes:

∂tP(C, t) =∑C′

{W(C′ → C)P(C′, t)︸︷︷︸

gain term

−W(C → C′)P(C, t)︸︷︷︸loss term

}More detailed dynamics for P(C,K, t):

∂tP(C,K, t) =∑C′

{W(C′ → C)P(C′,K−1, t)−W(C → C′)P(C,K, t)

}Canonical description: s conjugated to K

P(C, s, t) =∑

Ke−sKP(C,K, t)

s-modified dynamics [probability non-conserving]

∂tP(C, s, t) =∑C′

{e−sW(C′ → C)P(C′, s, t)− W(C → C′)P(C, s, t)

}Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 4 / 28


Numerical method [with J. Tailleur]

Evaluation of large deviation functions [à la Monte-Carlo diffusion]

Z(s, t) =∑C

P(C, s, t) =⟨e−s K⟩ ∼ e−tψ(s)

discrete time: Giardinà, Kurchan, Peliti [PRL 96, 120603 (2006)]continuous time: Lecomte, Tailleur [JSTAT P03004 (2007)]

Cloning dynamics∂tP(C, s) =

∑C′

Ws(C′ → C)P(C′, s)− rs(C)P(C, s)︸︷︷︸modified dynamics

+ δrs(C)P(C, s)︸︷︷︸cloning term

Ws(C′ → C) = e−sW(C′ → C)rs(C) =

∑C′ Ws(C → C′)

δrs(C) = rs(C)− r(C)



Numerical method [with J. Tailleur]

Evaluation of large deviation functions [à la Monte-Carlo diffusion]

Z(s, t) =∑C

P(C, s, t) =⟨e−s K⟩ ∼ e−tψ(s)

discrete time: Giardinà, Kurchan, Peliti [PRL 96, 120603 (2006)]continuous time: Lecomte, Tailleur [JSTAT P03004 (2007)]

Cloning dynamics∂tP(C, s) =

∑C′



Ws(C′ → C) = e−sW(C′ → C)rs(C) =

∑C′ Ws(C → C′)

δrs(C) = rs(C)− r(C)


Population dynamics Explicit construction

Explicit construction (1/3)

0

C0

t1

C1

t2

C2

. . . tK

CK

t

CK

jump probability:1

rs(C1)Ws(C1→C2)

Probability-preserving contribution

∂tP(C, t) =∑C′

{Ws(C′ → C)P(C′, t)︸︷︷︸

gain term

−Ws(C → C′)P(C, t)︸︷︷︸loss term

}




0

C0

t1

C1

t2

C2

. . . tK

CK

t

CK

jump probability:1

rs(C1)Ws(C1→C2)

Which configurations will be visited?Configurational part of the trajectory: C0 → . . .→ CK

Prob{hist of Ck’s} =

K−1∏n=0

Ws(Cn → Cn+1)

rs(Cn)

wherers(C) =

∑C′

Ws(C → C′)




0

C0

t1

C1

t2

C2

. . . tK

CK

t

CK

probability density:rs(C1)e−(t2−t1)rs(C1)

no jump probability:e−(t−tK)rs(CK)

When shall the system jump from one configuration to the next one?probability density for the time interval tn − tn−1

rs(Cn−1)e−(tn−tn−1)rs(Cn−1)

probability not to leave CK during the time interval t − tK

e−(t−tK)rs(CK)




0

C0

t1

C1

t2

C2

. . . tK

CK

t

CK

probability density:rs(C1)e−(t2−t1)rs(C1)

no jump probability:e−(t−tK)rs(CK)

When shall the system jump from one configuration to the next one?probability density for the time interval tn − tn−1

rs(Cn−1)e−(tn−tn−1)rs(Cn−1)

probability not to leave CK during the time interval t − tK

e−(t−tK)rs(CK)




∂tP(C, s) =∑C′



How to take into account loss/gain of probability?make evolve a large number of copies of the systemimplement a selection rule: on a time interval ∆ta copy in config C is replaced by e∆t δrs(C) copiesψ(s) = the rate of exponential growth/decay of the total populationoptionally: keep population constant by un-biased pruning/cloning




∂tP(C, s) =∑C′



How to take into account loss/gain of probability?make evolve a large number of copies of the systemimplement a selection rule: on a time interval ∆ta copy in config C is replaced by ⌊e∆t δrs(C) + ε⌋ copies, ϵ ∼ [0, 1]

ψ(s) = the rate of exponential growth/decay of the total populationoptionally: keep population constant by un-biased pruning/cloning




∂tP(C, s) =∑C′



How to take into account loss/gain of probability?make evolve a large number of copies of the systemimplement a selection rule: on a time interval ∆ta copy in config C is replaced by ⌊e∆t δrs(C) + ε⌋ copies, ϵ ∼ [0, 1]

ψ(s) = the rate of exponential growth/decay of the total populationoptionally: keep population constant by un-biased pruning/cloning



Non-constant population [with E Guevara]

0 20 40 60 80 100 120 1401

2

3

4

5

6

7

Time

Log−P

opulations



Questions

Source of errors when evaluating the CGF ψ(s):1 Diversity issue2 Finite-time and finite-population issues

CGF = (scaled) Cumulant Generating Function


Diversity issue Questions

How to perform averages? (1/2) [with R Jack, F Bouchet, T Nemoto]

∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞

etψ(s)|R⟩⟨L| ⟨L|Ws = ψ(s)⟨L|

[ ⟨L| = ⟨−| @ s = 0 ]

⋆ Final-time distribution: proportion of copies in C at t

⟨Nnc(t)⟩s

= ⟨−|etWs |Pi⟩N0 ∼t→∞

etψ(s)⟨L|Pi⟩N0

⟨Nnc(C, t)⟩s

= ⟨C|etWs |Pi⟩N0 ∼t→∞

etψ(s)⟨C|R⟩⟨L|Pi⟩N0

pend(C, t) =⟨Nnc(C, t)⟩s⟨Nnc(t)⟩s

∼t→∞

⟨C|R⟩ ≡ pend(C)

[Nnc = number in non-constant population dynamics]

Final-time distribution governed by right eigenvector of Ws.




∂t|P⟩ = Ws|P⟩

Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞


[ ⟨L| = ⟨−| @ s = 0 ]


⟨Nnc(t)⟩s

= ⟨−|etWs |Pi⟩N0 ∼t→∞

etψ(s)⟨L|Pi⟩N0

⟨Nnc(C, t)⟩s




∼t→∞







∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞


[ ⟨L| = ⟨−| @ s = 0 ]


⟨Nnc(t)⟩s

= ⟨−|etWs |Pi⟩N0 ∼t→∞

etψ(s)⟨L|Pi⟩N0

⟨Nnc(C, t)⟩s




∼t→∞







∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞


[ ⟨L| = ⟨−| @ s = 0 ]


⟨Nnc(t)⟩s = ⟨−|etWs |Pi⟩N0 ∼t→∞

etψ(s)⟨L|Pi⟩N0

⟨Nnc(C, t)⟩s = ⟨C|etWs |Pi⟩N0 ∼t→∞



∼t→∞



Final-time distribution governed by right eigenvector of Ws.Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 11 / 28




How to perform averages? (2/2)

∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞


[ ⟨L| = ⟨−| @ s = 0 ]

⋆ Mid-time distribution: proportion of copies in C at t1 ≪ t

⟨Nnc(t)⟩s

= ⟨−|etWs |Pi⟩N0 ∼t→∞

etψ(s)⟨L|Pi⟩N0

⟨Nnc(t|C, t1)⟩s

= ⟨−|e(t−t1)Ws |C⟩⟨C|et1Ws |Pi⟩N0 ∼ etψ(s)⟨L|C⟩⟨C|R⟩⟨L|Pi⟩N0

p(t|C, t1) =⟨Nnc(t|C, t1)⟩s

⟨Nnc(t)⟩s

∼t→∞

⟨L|C⟩⟨C|R⟩ ≡ pave(C)

Mid-time distribution governed by left and right eigenvectors of Ws.




∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞


[ ⟨L| = ⟨−| @ s = 0 ]



etψ(s)⟨L|Pi⟩N0

⟨Nnc(t|C, t1)⟩s = ⟨−|e(t−t1)Ws |C⟩⟨C|et1Ws |Pi⟩N0 ∼ etψ(s)⟨L|C⟩⟨C|R⟩⟨L|Pi⟩N0


⟨Nnc(t)⟩s∼

t→∞⟨L|C⟩⟨C|R⟩ ≡ pave(C)





∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩

etWs ∼t→∞


[ ⟨L| = ⟨−| @ s = 0 ]



etψ(s)⟨L|Pi⟩N0

⟨Nnc(t|C, t1)⟩s = ⟨−|e(t−t1)Ws |C⟩⟨C|et1Ws |Pi⟩N0 ∼ etψ(s)⟨L|C⟩⟨C|R⟩⟨L|Pi⟩N0


⟨Nnc(t)⟩s∼

t→∞⟨L|C⟩⟨C|R⟩ ≡ pave(C)




Huge sampling issueVivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 14 / 28


Example distributions for a simple Langevin dynamics

3 2 1 0 1 2 3x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p(x)

3 2 1 0 1 2 3x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

p(x)

pend(x) pave(x)



0 5 10 15 20 25 30

t

0

20

40

60

80

100

N(t

)

N(t) = number of surviving trajectories on [t, tfinal]


Diversity issue Multicanonical Approach

How to make mid- and final-time distribution closer?

Driven/auxiliary dynamics: [Maes, Jack&Sollich, Touchette&Chetrite]Probability preservingNo mismatch between pave and pend

Constructed as: [L = diagonal matrix of elements L]

Wauxs = LWsL−1 − ψ(s)1

Issue: determining L is difficultSolution: evaluate L as Ltest on the fly and simulate

Wtests = LtestWsL−1

test

Whichever Ltest, the simulation is still correct.Similar in spirit to multi-canonical (e.g. Wang-Landau) approach instatic thermodynamics









test

Whichever Ltest, the simulation is still correct.Similar in spirit to multi-canonical (e.g. Wang-Landau) approach instatic thermodynamics









test

Whichever Ltest, the simulation is still correct.

Similar in spirit to multi-canonical (e.g. Wang-Landau) approach instatic thermodynamics









test

Whichever Ltest, the simulation is still correct.

Similar in spirit to multi-canonical (e.g. Wang-Landau) approach instatic thermodynamics









test

Whichever Ltest, the simulation is still correct.Similar in spirit to multi-canonical (e.g. Wang-Landau) approach instatic thermodynamicsVivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 17 / 28


Improvement of the depletion-of-ancestors problem:

0 5 10 15 20 25 30

t

0

20

40

60

80

100

N(t

)

Red: without multicanonical sampling.Green: with multicanonical sampling (Ltest evaluated on the fly)Dashed: lower temperature



Improvement of the small-noise crisis:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Normal (Nc= 20)Normal (Nc= 100)Normal (Nc= 200)Feedback (Nc= 20)

noise intensity

ψ(s)

[s is fixed]


Finite-time and finite-population effects Observations

Numerical observations (1/3) [with T. Nemoto & E. Guevara]

Large-t asymptotics at fixed population size N0.

100 200 300 400 500 600 700 800 900 10000.089

0.09

0.091

0.092

0.093

0.094

analytical fit

⟨Ψ

(N0)s (t)

⟩︸︷︷︸numericalestimator

= Ψ(N0)s + 1

t∆Ψ(N0)s




Large-N0 asymptotics at fixed final time T.

100 200 300 400 500 600 700 800 900 10000.0932

0.0934

0.0936

0.0938

0.094

0.0942

0.0944

fitanalytical

⟨Ψ

(N0)s (T )

⟩︸︷︷︸numericalestimator

= Ψ(∞)s + 1

N0∆Ψ

(∞)s




Distribution of large-deviation estimators.

0.05 0.06 0.07 0.08 0.09 0.1 0.11

100

200

300

400

500

600(b)

0.075 0.08 0.085 0.09 0.095

200

400

600

800

1000

1200

1400

1600

1800

2000(c)

N0 = 100 N0 = 1000Stochastic errors and systematic errors at finite N0.




Large deviations of large-deviations.

0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1

1

2

3

4

5

6

7x 10−3

P(Ψ

(N0)s

)∼ e−t I(Ψ(N0)s )


Finite-time and finite-population effects Death-birth representation

Analytical approach (1/3) [with T. Nemoto & E. Guevara]

(Fixed N0) death-birth process to represent the population dynamics:Preserves probability.LDF estimator is (1t×) an additive observable of the trajectory.

Ψ(N0)s =

1

t log∏

events

N0 +

∈{−1,0,1,...}︷︸︸︷number ofoffspringsN0

Large-deviation principle applies to the distribution of Ψ(N0)s :

P(Ψ

(N0)s

)∼ e−t I(Ψ(N0)s )

Aim:⟨Ψ(N0)

s ⟩︸︷︷︸death-birth

process

→ ψ(s)︸︷︷︸original

CGF




(Fixed N0) death-birth process to represent the population dynamics:Preserves probability.LDF estimator is (1t×) an additive observable of the trajectory.

Ψ(N0)s =

1

t log∏

events

N0 +

∈{−1,0,1,...}︷︸︸︷number ofoffspringsN0

Large-deviation principle applies to the distribution of Ψ(N0)s :

P(Ψ

(N0)s

)∼ e−t I(Ψ(N0)s )

Aim:⟨Ψ(N0)

s ⟩︸︷︷︸death-birth

process

→ ψ(s)︸︷︷︸original

CGF




Route to follow:Determine the rates of the death-birth process (at fixed N0).Write the steady-state equation for the death-birth process.Project on the occupation number.Show that, in the steady-state the fractions

pC ≡⟨number of copies

in configuration CN0

⟩obey the original s-modified master equation.Hypothesis: at large N0, independence⟨number of copies

in configuration CN0

number of copiesin configuration C′

N0

⟩∝ δCC′




Infinite-N0 limit and large-N0 asymptotics:

fluctuating xC ≡number of copiesin configuration C

N0

(⇒ pC = ⟨xC⟩

)Large N0 expansion (à la van Kampen)At infinite N0: the xC obey deterministic equations.

At large N0: noise of amplitude√

N0.Large-deviation principle

P(Ψ

(N0)s

)∼ e−tN0 J(Ψ(N0)s )

i.e.lim

N0→∞lim

t→∞

1

N0tlogP

(Ψ

(N0)s

)= J(Ψ(N0)

s )




Infinite-N0 limit and large-N0 asymptotics:

fluctuating xC ≡number of copiesin configuration C

N0

(⇒ pC = ⟨xC⟩

)Large N0 expansion (à la van Kampen)At infinite N0: the xC obey deterministic equations.At large N0: noise of amplitude

√N0.

Large-deviation principle

P(Ψ

(N0)s

)∼ e−tN0 J(Ψ(N0)s )

i.e.lim

N0→∞lim

t→∞

1

N0tlogP

(Ψ

(N0)s

)= J(Ψ(N0)

s )



AveragingExercise/riddle for you:⟨

log∏

events

N0 +number ofoffspringsN0︸︷︷︸

this is Ψ(N0)s

⟩vs. log

⟨ ∏events

N0 +number ofoffspringsN0

⟩

Which is the “best” estimator of the CGF ψ(s)?


Conclusion

Summary and questions

(1) Multicanonical approach [with F Bouchet, R Jack, T Nemoto]Sampling problem (depletion of ancestors)On-the-fly evaluated auxiliary dynamicsSolution to the small-noise crisisSystems with chaos/large number of degrees of freedom?

(2) Finite-time t and finite-population N0 [with E Guevara, T Nemoto]LDF estimator converges at (i) t → ∞ (ii) N0 → ∞Order of the correction understoodGives an interpolation procedure for the numericsPhase transitions? Absence of steady-state?Combine (1) and (2)?


Conclusion

Summary and questions

(1) Multicanonical approach [with F Bouchet, R Jack, T Nemoto]Sampling problem (depletion of ancestors)On-the-fly evaluated auxiliary dynamicsSolution to the small-noise crisisSystems with chaos/large number of degrees of freedom?

(2) Finite-time t and finite-population N0 [with E Guevara, T Nemoto]LDF estimator converges at (i) t → ∞ (ii) N0 → ∞Order of the correction understoodGives an interpolation procedure for the numericsPhase transitions? Absence of steady-state?Combine (1) and (2)?


Discreteness of effects at non-constant populations

Initial transient regime [with E. Guevara]

Non-constant population dynamics

0 20 40 60 80 100 120 1401

2

3

4

5

6

7

Time

Log−P

opulations



Time-delay

−10 0 10 20 30 40 50 601

2

3

4

5

6

7Lo

g−P

opul

atio

ns

Time

Student Version of MATLAB



Improvement of the numerical evaluation of ψ(s)

2 2.5 3 3.5 4 4.5 5

0.044

0.0442

0.0444

0.0446

0.0448

0.045

[ ]



Distribution of time delays

−50 −40 −30 −20 −10 0 10 20 30

100

200

300

400

500

600

700

800

900

s = −0.1s = −0.2

s = −0.3


Documents

Large deviations and population dynamics: finite-population effects · 2016. 6. 28. · Large deviations and population dynamics: finite-population effects Vivien Lecomte(1), Julien