THE EFFICIENT USE OF SUPPLEMENTARY INFORMATION IN FINITE POPULATION SAMPLING
Large deviations and population dynamics: finite-population effects · 2016. 6. 28. · Large...
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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
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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
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Introduction Motivations
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)
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Introduction Motivations
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)
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 3 / 28
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Introduction Motivations
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
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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)
}
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 4 / 28
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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)
}Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 4 / 28
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Introduction s-ensemble
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)
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Introduction s-ensemble
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)
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 5 / 28
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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
}
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 6 / 28
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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)
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′)
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Population dynamics Explicit construction
Explicit construction (2/3)
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)
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Population dynamics Explicit construction
Explicit construction (2/3)
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)
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 7 / 28
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Population dynamics Explicit construction
Explicit construction (3/3)
∂tP(C, s) =∑C′
Ws(C′ → C)P(C′, s)− rs(C)P(C, s)︸ ︷︷ ︸modified dynamics
+ δrs(C)P(C, s)︸ ︷︷ ︸cloning term
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
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 8 / 28
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Population dynamics Explicit construction
Explicit construction (3/3)
∂tP(C, s) =∑C′
Ws(C′ → C)P(C′, s)− rs(C)P(C, s)︸ ︷︷ ︸modified dynamics
+ δrs(C)P(C, s)︸ ︷︷ ︸cloning term
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
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 8 / 28
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Population dynamics Explicit construction
Explicit construction (3/3)
∂tP(C, s) =∑C′
Ws(C′ → C)P(C′, s)− rs(C)P(C, s)︸ ︷︷ ︸modified dynamics
+ δrs(C)P(C, s)︸ ︷︷ ︸cloning term
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
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 8 / 28
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Population dynamics Explicit construction
Non-constant population [with E Guevara]
0 20 40 60 80 100 120 1401
2
3
4
5
6
7
Time
Log−P
opulations
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Population dynamics Explicit construction
Questions
Source of errors when evaluating the CGF ψ(s):1 Diversity issue2 Finite-time and finite-population issues
CGF = (scaled) Cumulant Generating Function
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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.
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 11 / 28
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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.
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 11 / 28
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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.
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 11 / 28
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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.Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 11 / 28
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Diversity issue Questions
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 12 / 28
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Diversity issue Questions
How to perform averages? (2/2)
∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩
etWs ∼t→∞
etψ(s)|R⟩⟨L| ⟨L|Ws = ψ(s)⟨L|
[ ⟨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.
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Diversity issue Questions
How to perform averages? (2/2)
∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩
etWs ∼t→∞
etψ(s)|R⟩⟨L| ⟨L|Ws = ψ(s)⟨L|
[ ⟨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.
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 13 / 28
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Diversity issue Questions
How to perform averages? (2/2)
∂t|P⟩ = Ws|P⟩ Ws|R⟩ = ψ(s)|R⟩
etWs ∼t→∞
etψ(s)|R⟩⟨L| ⟨L|Ws = ψ(s)⟨L|
[ ⟨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.
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Diversity issue Questions
Huge sampling issueVivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 14 / 28
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Diversity issue Questions
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)
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Diversity issue Questions
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]
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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
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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
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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
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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
Vivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 17 / 28
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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 thermodynamicsVivien Lecomte (LPMA - Paris) LDFs and population dynamics 28/06/2016 17 / 28
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Diversity issue Multicanonical Approach
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
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Diversity issue Multicanonical Approach
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]
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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
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Finite-time and finite-population effects Observations
Numerical observations (2/3) [with T. Nemoto & E. Guevara]
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
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Finite-time and finite-population effects Observations
Numerical observations (3/3) [with T. Nemoto & E. Guevara]
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.
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Finite-time and finite-population effects Observations
Numerical observations (3/3) [with T. Nemoto & E. Guevara]
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 )
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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
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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
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Finite-time and finite-population effects Death-birth representation
Analytical approach (2/3) [with T. Nemoto & E. Guevara]
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′
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Finite-time and finite-population effects Death-birth representation
Analytical approach (3/3) [with T. Nemoto & E. Guevara]
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 )
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Finite-time and finite-population effects Death-birth representation
Analytical approach (3/3) [with T. Nemoto & E. Guevara]
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 )
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Finite-time and finite-population effects Death-birth representation
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)?
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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)?
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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)?
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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
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Discreteness of effects at non-constant populations
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
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Discreteness of effects at non-constant populations
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
[ ]
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Discreteness of effects at non-constant populations
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
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