Population dynamics method for rare events: systematic errors & feedback control

Transcription

1 Population dynamics method for rare events: systematic errors & feedback control Esteban Guevara (1), Takahiro Nemoto (2), Freddy Bouchet (3), Rob Jack (4), Vivien Lecomte (5) (1) IJM, Paris (2) ENS, Paris (3) ENS, Lyon (4) Bath University (5) LPMA, Paris & LIPhy, Grenoble IHP 24 April 2017 Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

2 Introduction Motivations Why studying rare events? 2003 heat wave, Europe [Terra MODIS] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

3 Introduction Motivations Why studying rare events? [Anomaly for 1-month average] 2003 heat wave, Europe [Terra MODIS] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

4 Introduction Motivations Why studying rare events? 2010 heat wave in Western Russia [Dole et al., 2011] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

5 Introduction Motivations Why studying rare events? t max=40 days { }}{ C 1 tmax dt T(t) > 2 C «Teleconnection patterns» [Bouchet et al.] t max 0 Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

6 Introduction Motivations Why studying rare events? Questions for physicists and mathematicians: Probability and dynamics of rare events? How to sample these in numerical modelisations? Numerical tools and methods to understand their formation? Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

7 Introduction Motivations Why studying rare events? Questions for physicists and mathematicians: Probability and dynamics of rare events? How to sample these in numerical modelisations? Numerical tools and methods to understand their formation? Evolution of the return time of the monthly averaged temperature 1 t max tmax 0 dt T(t) Due anthropogenic impact on climate? [Otto et al., 2012] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

8 Tools Large deviation functions Distribution of a time-extensive observable K 0.2 1/t Log Prob(K/t) Prob[K, t] e t φ(k/t) K /t Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

9 Tools Large deviation functions Distribution of a time-extensive observable K 0.2 1/t Log Prob(K/t) Prob[K, t] e t φ(k/t) K /t Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

10 Tools Changing ensemble s-modified dynamics Markov processes: t P(C, t) = { } W(C C)P(C, t) W(C C )P(C, t) }{{}}{{} C gain term loss term Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

11 Tools Changing ensemble s-modified dynamics K = activity = #events Markov processes: t P(C, t) = { } W(C C)P(C, t) W(C C )P(C, t) }{{}}{{} C gain term loss term More detailed dynamics for P(C, K, t): t P(C, K, t) = } {W(C C)P(C, K 1, t) W(C C )P(C, K, t) C Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

12 Tools Changing ensemble s-modified dynamics K = activity = #events Markov processes: t P(C, t) = { } W(C C)P(C, t) W(C C )P(C, t) }{{}}{{} C gain term loss term More detailed dynamics for P(C, K, t): t P(C, K, t) = } {W(C C)P(C, K 1, t) W(C C )P(C, K, t) C Canonical description: s conjugated to K ˆP(C, s, t) = K e sk P(C, K, t) Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

13 Tools Changing ensemble s-modified dynamics K = activity = #events Markov processes: t P(C, t) = { } W(C C)P(C, t) W(C C )P(C, t) }{{}}{{} C gain term loss term More detailed dynamics for P(C, K, t): t P(C, K, t) = } {W(C C)P(C, K 1, t) W(C C )P(C, K, t) C Canonical description: s conjugated to K ˆP(C, s, t) = e sk P(C, K, t) K s-modified dynamics [probability non-conserving] { } tˆp(c, s, t) = e s W(C C)ˆP(C, s, t) W(C C )ˆP(C, s, t) C Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

14 Tools Changing ensemble s-modified dynamics K = k C0 C 1 + k C1 C Markov processes: t P(C, t) = { } W(C C)P(C, t) W(C C )P(C, t) }{{}}{{} C gain term loss term More detailed dynamics for P(C, K, t): t P(C, K, t) = } {W(C C)P(C, K k C C, t) W(C C )P(C, K, t) C Canonical description: s conjugated to K ˆP(C, s, t) = e sk P(C, K, t) K s-modified dynamics [probability non-conserving] { } tˆp(c, s, t) = e sk C CW(C C)ˆP(C, s, t) W(C C )ˆP(C, s, t) C Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

15 Numerical method Population dynamics Numerical method [JB Anderson; D Aldous; P Grassberger; P Del Moral; ] Evaluation of large deviation functions [à la Diffusion Monte-Carlo ] ˆP(C, s, t) = e s K e t ψ(s) C discrete time: Giardinà, Kurchan, Peliti [PRL 96, (2006)] continuous time: VL, Tailleur [JSTAT P03004 (2007)] Cloning dynamics tˆp(c, s) = W s (C C)ˆP(C, s) r s (C)ˆP(C, s) C } {{} modified dynamics W s (C C) = e s W(C C) r s (C) = C W s(c C ) δr s (C) = r s (C) r(c) + δr s (C)ˆP(C, s) } {{ } cloning term Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

16 Numerical method Population dynamics Numerical method [JB Anderson; D Aldous; P Grassberger; P Del Moral; ] Evaluation of large deviation functions [à la Diffusion Monte-Carlo ] ˆP(C, s, t) = e s K e t ψ(s) C discrete time: Giardinà, Kurchan, Peliti [PRL 96, (2006)] continuous time: VL, Tailleur [JSTAT P03004 (2007)] Cloning dynamics tˆp(c, s) = W s (C C)ˆP(C, s) r s (C)ˆP(C, s) C } {{} modified dynamics W s (C C) = e s W(C C) r s (C) = C W s(c C ) δr s (C) = r s (C) r(c) + δr s (C)ˆP(C, s) } {{ } cloning term Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

17 Population dynamics Explicit construction tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a copy in config C is replaced by e t δrs(c) copies ψ(s) = the rate of exponential growth/decay of the total population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

18 Population dynamics Explicit construction tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a 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 population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

19 Population dynamics Explicit construction tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a 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 population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

20 Population dynamics Explicit construction tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a 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 population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

21 Population dynamics Explicit construction tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term Biological interpretation copy in configuration C organism of genome C dynamics of rates W s mutations cloning at rates δr s selection rendering atypical histories typical Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

22 Population dynamics Example An example: 4 copies, 1 degree of freedom C = x R Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

23 Population dynamics Questions How to perform averages? (i) [with R Jack, F Bouchet, T Nemoto] Final-time distribution: proportion of copies in C at t N nc (t) s N nc (C, t) s p end (C, t) = N nc(c, t) s N nc (t) s [N nc = number in non-constant population dynamics] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

24 Population dynamics Questions How to perform averages? (i) [with R Jack, F Bouchet, T Nemoto] t ˆP = W s ˆP Final-time distribution: proportion of copies in C at t N nc (t) s N nc (C, t) s p end (C, t) = N nc(c, t) s N nc (t) s [N nc = number in non-constant population dynamics] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

25 Population dynamics Questions How to perform averages? (i) [with R Jack, F Bouchet, T Nemoto] t ˆP = W s ˆP W s R = ψ(s) R L W s = ψ(s) L [ L s = 0 ] Final-time distribution: proportion of copies in C at t N nc (t) s N nc (C, t) s p end (C, t) = N nc(c, t) s N nc (t) s [N nc = number in non-constant population dynamics] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

26 Population dynamics Questions How to perform averages? (i) [with R Jack, F Bouchet, T Nemoto] t ˆP = W s ˆP W s R = ψ(s) R e tws e tψ(s) R L t L W s = ψ(s) L [ L s = 0 ] Final-time distribution: proportion of copies in C at t N nc (t) s N nc (C, t) s p end (C, t) = N nc(c, t) s N nc (t) s [N nc = number in non-constant population dynamics] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

27 Population dynamics Questions How to perform averages? (i) [with R Jack, F Bouchet, T Nemoto] t ˆP = W s ˆP W s R = ψ(s) R e tws e tψ(s) R L t L W s = ψ(s) L [ L s = 0 ] Final-time distribution: proportion of copies in C at t N nc (t) s = e tws P i N 0 t e tψ(s) L P i N 0 N nc (C, t) s = C e tws P i N 0 t e tψ(s) C R L P i N 0 p end (C, t) = N nc(c, t) s N nc (t) s t C R p end (C) [N nc = number in non-constant population dynamics] Final-time distribution governed by right eigenvector. Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

28 Population dynamics Questions An example: 4 copies, 1 degree of freedom C = x R Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

29 Population dynamics Questions How to perform averages? (ii) Intermediate times t ˆP = W s ˆP W s R = ψ(s) R e tws e tψ(s) R L t L W s = ψ(s) L [ L s = 0 ] Mid-time distribution: proportion of copies in C at t 1 t N nc (t) s N nc (t C, t 1 ) s p(t C, t 1 ) = N nc(t C, t 1 ) s N nc (t) s Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

30 Population dynamics Questions How to perform averages? (ii) Intermediate times t ˆP = W s ˆP W s R = ψ(s) R e tws e tψ(s) R L t L W s = ψ(s) L [ L s = 0 ] Mid-time distribution: proportion of copies in C at t 1 t N nc (t) s = e tws P i N 0 t e tψ(s) L P i N 0 N nc (t C, t 1 ) s = e (t t1)ws C C e t1ws P i N 0 e tψ(s) L C C R L P i N 0 p(t C, t 1 ) = N nc(t C, t 1 ) s N nc (t) s t L C C R p ave (C) Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

31 Population dynamics Questions How to perform averages? (ii) Intermediate times t ˆP = W s ˆP W s R = ψ(s) R e tws e tψ(s) R L t L W s = ψ(s) L [ L s = 0 ] Mid-time distribution: proportion of copies in C at t 1 t N nc (t) s = e tws P i N 0 t e tψ(s) L P i N 0 N nc (t C, t 1 ) s = e (t t1)ws C C e t1ws P i N 0 e tψ(s) L C C R L P i N 0 p(t C, t 1 ) = N nc(t C, t 1 ) s N nc (t) s t L C C R p ave (C) Mid-time distribution governed by left and right eigenvectors. Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

32 Population dynamics Questions An example: 4 copies, 1 degree of freedom C = x R Huge sampling issue Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

33 Population dynamics Questions How to perform averages? Mid-time ancestor distribution: fraction of copies (at time t 1 ) which were in configuration C, knowing that there are in configuration C f at final time t f : p anc (C, t 1 ; C f, t f ) = N nc(c f, t f C, t 1 ) s C N nc(c f, t f C, t 1 ) s t f,1 L C C R = p ave(c) The ancestor statistics of a configuration C f is thus independent (far enough in the past) of the configuration C f. Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

34 Population dynamics Questions Example distributions for a simple Langevin dynamics final-time: p end (x) intermediate-time: p ave (x) Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

35 Population dynamics Questions The small-noise crisis: systematic errors grow as ϵ 0 Cause: as ϵ 0, p ave (x) & p end (x) sharply peaked at different points i.e. the clones do not attack sample correctly the phase space Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

36 Multicanonical Approach Description How to make mid- and final-time distribution closer? Driven/auxiliary dynamics: Probability preserving [Maes, Jack&Sollich, Touchette&Chetrite] No mismatch between p ave and p end Constructed as W aux s = LW s L 1 ψ(s)1 Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

37 Multicanonical Approach Description How to make mid- and final-time distribution closer? Driven/auxiliary dynamics: Probability preserving [Maes, Jack&Sollich, Touchette&Chetrite] No mismatch between p ave and p end Constructed as W aux s = LW s L 1 ψ(s)1 Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

38 Multicanonical Approach Description How to make mid- and final-time distribution closer? Driven/auxiliary dynamics: Probability preserving [Maes, Jack&Sollich, Touchette&Chetrite] No mismatch between p ave and p end Constructed as W aux s = LW s L 1 ψ(s)1 Issue: determining L is difficult Solution: evaluate L as L test on the fly and simulate W test s = L test W s L 1 test Whichever L test, the simulation is still correct. Iterate Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

39 Multicanonical Approach Description How to make mid- and final-time distribution closer? Driven/auxiliary dynamics: Probability preserving [Maes, Jack&Sollich, Touchette&Chetrite] No mismatch between p ave and p end Constructed as W aux s = LW s L 1 ψ(s)1 Issue: determining L is difficult Solution: evaluate L as L test on the fly and simulate W test s = L test W s L 1 test Whichever L test, the simulation is still correct. Iterate Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

40 Multicanonical Approach Description How to make mid- and final-time distribution closer? Driven/auxiliary dynamics: Probability preserving [Maes, Jack&Sollich, Touchette&Chetrite] No mismatch between p ave and p end Constructed as W aux s = LW s L 1 ψ(s)1 Issue: determining L is difficult Solution: evaluate L as L test on the fly and simulate W test s = L test W s L 1 test Whichever L test, the simulation is still correct. Iterate Similar in spirit to multi-canonical (e.g. Wang-Landau) approach in static thermodynamics Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

41 Multicanonical Approach Results Improvement of the small-noise crisis (i.i) Physical insight: probability loss transformed into effective forces Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

42 Multicanonical Approach Results Improvement of the small-noise crisis (i.ii) Much more efficient evaluation of the biased distribution. Even for a very crude (polynomial) approximation of the effective force. Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

43 Multicanonical Approach Results Improvement of the small-noise crisis (ii) Interacting system in 1D. Effective force: 1-, 2-, 3- body interactions only [also crude approx.]. Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

44 Conclusion Summary and questions (1) Multicanonical approach [with F Bouchet, R Jack, T Nemoto] Sampling problem (depletion of ancestors) On-the-fly evaluated auxiliary dynamics Solution to the small-noise crisis Systems with large number of degrees of freedom Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

45 Conclusion Summary and questions (1) Multicanonical approach [with F Bouchet, R Jack, T Nemoto] Sampling problem (depletion of ancestors) On-the-fly evaluated auxiliary dynamics Solution to the small-noise crisis Systems with large number of degrees of freedom Finite-population effects [with E Guevara, T Nemoto] Quantitative finite-n clones scaling interpolation method Initial transient regime due to small population Analogy with biology: many small islands vs. few large islands? Question: effective forces selection? Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

46 Conclusion Questions (2): why is it working? Improvement of the depletion-of-ancestors problem: Dashed line: lower noise Continuous line: higher noise Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

47 Conclusion Thanks for your attention! Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

48 Conclusion Supplementary material Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

49 Conclusion 0.2 1/t Log Prob(K/t) Prob[K] e t φ(k/t) K /t Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

50 Conclusion 0.2 1/t Log Prob(K/t) Prob[K] e t φ(k/t) K /t Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

51 Conclusion 0.2 1/t Log Prob(K/t) Prob[K] e t φ(k/t) K /t Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

52 Conclusion 0.2 1/t Log Prob(K/t) K /t Prob[K] e t φ(k/t) Finite-time & -size scalings matter. Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

53 Conclusion [Merrolle, Garrahan and Chandler, 2005] Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

54 Conclusion Exponential divergence of the susceptibility Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

55 Conclusion Explicit construction Explicit construction (1/3) 0 t 1 t 2... t K t C 0 C 1 C 2 C K C K Probability-preserving contribution { } tˆp(c, t) = W s (C C)ˆP(C, t) W }{{} s (C C )ˆP(C, t) }{{} C gain term loss term Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

56 Conclusion Explicit construction Explicit construction (1/3) 0 jump probability: 1 rs(c 1 ) Ws(C 1 C 2 ) t 1 t 2... t K t C 0 C 1 C 2 C K C K Which configurations will be visited? Configurational part of the trajectory: C 0... C K Prob{hist} = K 1 n=0 W s (C n C n+1 ) r s (C n ) where r s (C) = C W s (C C ) Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

57 Conclusion Explicit construction Explicit construction (2/3) 0 probability density: r s(c 1 )e (t 2 t 1 )rs(c 1 ) t 1 t 2... t K t C 0 C 1 C 2 C K C K When shall the system jump from one configuration to the next one? probability density for the time interval t n t n 1 r s (C n 1 )e (tn t n 1)r s(c n 1 ) Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

58 Conclusion Explicit construction Explicit construction (2/3) 0 probability density: r s(c 1 )e (t 2 t 1 )rs(c 1 ) t 1 t 2... t K no jump probability: e (t t K )rs(c K ) t C 0 C 1 C 2 C K C K When shall the system jump from one configuration to the next one? probability density for the time interval t n t n 1 r s (C n 1 )e (tn t n 1)r s(c n 1 ) probability not to leave C K during the time interval t t K e (t t K)r s(c K ) Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

59 Conclusion Explicit construction Explicit construction (3/3) tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a copy in config C is replaced by e t δrs(c) copies ψ(s) = the rate of exponential growth/decay of the total population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

60 Conclusion Explicit construction Explicit construction (3/3) tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a 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 population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

61 Conclusion Explicit construction Explicit construction (3/3) tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a 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 population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

62 Conclusion Explicit construction Explicit construction (3/3) tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term How to take into account loss/gain of probability? handle a large number of copies of the system implement a selection rule: on a time interval t a 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 population optionally: keep population constant by non-biased pruning/cloning Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22

63 Conclusion Explicit construction Explicit construction (3/3) tˆp(c, s) = C W s (C C)ˆP(C, s) r s (C)ˆP(C, s) }{{} modified dynamics + δr s (C)ˆP(C, s) } {{ } cloning term Biological interpretation copy in configuration C organism of genome C dynamics of rates W s mutations cloning at rates δr s selection rendering atypical histories typical Vivien Lecomte (LPMA & LIPhy) Population dynamics method 24/04/ / 22