Record dynamics in spin glassses, superconductors and biological evolution.

Transcription

1 Record dynamics in spin glassses, superconductors and biological evolution. Henrik Jeldtoft Jensen Institute of Mathematical Sciences and Department of Mathematics Collaborators: Paolo Sibani, Paul Anderson, Luis P Oliveira and Mario Nicodemi

2 The question: Is intermittent, logarithmically slow, dynamics, driven by record events, typical of complex systems?

3 List of content: Dynamics of complex systems Three models definition and dynamics Manifestation of record dynamics Consequences Conclusion/summary

4 Complex dynamics: Intermittent, non-stationary Jumping through collective adaptation space: quake driven Transitions f(t) Log(t) Motion within one quasistable epoch

5 The models: Tangled Nature Model of co-evolving biological species Restricted Occupancy Model of vortex dynamics in type II superconductors. Edward-Anderson Spin Glass nearest neighbour Gaussian couplings

6 The relaxation Tangled Nature model collective adaptation: configurations increasingly coupled together. ROM model magnetic pressure Spin Glass thermal quench

7 First Model: Tangled Nature

8 Tangled Nature model of evolution Definition: * Individuals, where and * Dynamics a time step: Annihilation: Choose indiv. at random, remove with probability L= 3

9 Reproduction: Choose indiv. at random Determine occupancy at the location

10 The coupling matrix J(S, S ) Either consider J(S, S ) to be uncorrelated or to vary smoothly through type space.

11 from reproduction probability 1

12 Asexual reproduction: Replace by two copies with probability

13 Mutations Mutations occur with probability, i.e.

14 Phenomenology Long time dynamics The evolved networks

15 Segregation in genotype space Initiation Non Correlated Only one genotype Jn term = 0 Total population N(t) adjusts Diversity Matt Hall

16 Intermittency at systems level: Non Correlated Type label # generations 1 generation = Matt Hall

17 Intermittency at systems level: Correlated Fig. 1 An occupation plot of a single run for a system with R = 10,000. For each timeslice a point appears where a phenotype is in existence but as the full space is in 16 dimensions a projection onto a single trait is used. Simon Laird

18 Time evolution of Distribution of active coupling strengths Θ=0.005 Non correlated Θ=0.25 Paul Anderson

19 Time evolution of Distribution of active coupling strengths Correlated Simon Laird

20 Increasing complexity? x=1 Random Simulation Θ=0.25 Note: Effect is significant for correlated type space

21 Time evolution of Species abundance distribution Non Correlated Paul Anderson Low connectivity High connectivity

22 The evolved degree distribution Correlated Simon Laird Exponential becomes 1/k in limit of vanishing mutation rate

23 Intermittent dynamics

24 Intermittency: q-ess = quasi-evolutionary Stable Strategy # of transitions in window Matt Hall 1 generation = # generations

25 Stability of the q-ess: Consider simple adiabatic approximation. Stability of genotype S assuming: Consider Stationary solution Fluctuation Fulfil i.e. stability

26 Transitions between q-ess caused by co-evolutionary collective fluctuations

27 Time dependence (averaged) Total population N(t) Diversity Time in Generations Matt Hall

28 Origin of drift? Effect of mutation Let convex

29 N(t) t Not the whole explanation: evolution not smooth.

30 Record dynamics

31 Record dynamics: the record stochastic signal Paolo Sibani and Peter Littlewood (1992): exponentially distributed

32 Record dynamics: exponentially distributed Poisson process in logarithmic time Mean and variance Rate of records constant as function of ln(t) Rate decreases

33 Tangled Nature model: Single realisation and record dynamics: Extracting records from the population size N(t) Time Paul Anderson

34 Record dynamics: Ratio r remains non-zero Cumulative Distribution Paul Anderson

35 Second Model: ROM

36 ROM Monte Carlo Kawasaki dynamics on stack of coarse grained superconducting planes x

37 ROM Hamiltonian Here

38 ROM: Temperature independent creep

39 Realisations of record dynamics

40 Manifestation of the decelerating activity.

41 Further evidence The cumulative distribution of the log waiting times. Comparison with exponential distribution.

42 Number of vortices in the bulk as function of time

43 Quake statistics and the total number vortices entering.

44 The temperature in-dependence of the quake rate.

45 The magnetic creep rate: comparison with experiment

46 Third Model: Spin Glass

47 Spin glass Microscopic magnetic moments or spins coupled together with random coupling constants. The Hamiltonian: H = 1 2 J ij S i S j where S i, S j = ±1 ij

48 Spin glass Quench from high temperature: time < 0: T = high time > 0: T = very low t 1 t 2 t 3 time

49 Spin glass: heat transfer Protocol: Quench from high temp. at time t= 0. Measure heat transfer, H, between spin glass and reservoir during time interval If If Gaussian p(h) exponential tail

50 Spin glass: heat transfer

51 Consequences of record dynamics. Statistics of quake times independent of underlying noise mechanism. Biology: same intermittent dynamics in micro as in macro evolution. Decreasing transition rate. Magnetic relaxation: temperature independent Spin glass: exponential tails creep rate

52 Conclusion/Summary Considered spin-glasses, superconductors and biological evolution as typical complex systems. Generic dynamics of complex systems: Non-stationary Intermittent record dynamics - quakes Rate of activity ~ 1/t Stationary as function of log(t) Collaborators: Paolo Sibani, Paul Anderson and Luis P Oliveira

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