Complex dynamics made simple: colloidal dynamics

Transcription

1 Complex dynamics made simple: colloidal dynamics Chemnitz, June 2010

2 Complex dynamics made simple: colloidal dynamics Chemnitz, June 2010

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4 Terminology Quench: sudden change of external parameter, e.g. temperature or density.

5 Terminology Quench: sudden change of external parameter, e.g. temperature or density. Aging: After a quench, complex glassy materials (glasses, polymers spin glasses) undergo a slow change of physical properties called aging 1.

6 Terminology Quench: sudden change of external parameter, e.g. temperature or density. Aging: After a quench, complex glassy materials (glasses, polymers spin glasses) undergo a slow change of physical properties called aging 1. Age: time elapsed from the initial quench

7 Terminology Quench: sudden change of external parameter, e.g. temperature or density. Aging: After a quench, complex glassy materials (glasses, polymers spin glasses) undergo a slow change of physical properties called aging 1. Age: time elapsed from the initial quench Time homogeneous dynamics: e.g. diffusion, no changes of mesoscopic rates with time.

8 Terminology Quench: sudden change of external parameter, e.g. temperature or density. Aging: After a quench, complex glassy materials (glasses, polymers spin glasses) undergo a slow change of physical properties called aging 1. Age: time elapsed from the initial quench Time homogeneous dynamics: e.g. diffusion, no changes of mesoscopic rates with time. Time inhomogeneous dynamics: e.g. aging: rate of significant events goes down with age.

9 Metastability and coarse-graining in energy landscapes Many basins of attraction. E.g. energy minima separated by energy barriers.

10 Metastability and coarse-graining in energy landscapes Many basins of attraction. E.g. energy minima separated by energy barriers. On long enough time scales neglect the details and look at the dynamics at the level of attractor changes.

11 Metastability and coarse-graining in energy landscapes Many basins of attraction. E.g. energy minima separated by energy barriers. On long enough time scales neglect the details and look at the dynamics at the level of attractor changes. The resulting dynamics may be simple: e.g. diffusion in sinusoidal potential

12 Metastability and coarse-graining in energy landscapes Many basins of attraction. E.g. energy minima separated by energy barriers. On long enough time scales neglect the details and look at the dynamics at the level of attractor changes. The resulting dynamics may be simple: e.g. diffusion in sinusoidal potential Or complex: e.g. spin glass dynamics, where averages have power-law or logarithmic dependences.

13 Metastability and coarse-graining in hard-sphere colloids Hard sphere colloidal systems-only short range interactions

14 Metastability and coarse-graining in hard-sphere colloids Hard sphere colloidal systems-only short range interactions Entropy driven dynamics.

15 Metastability and coarse-graining in hard-sphere colloids Hard sphere colloidal systems-only short range interactions Entropy driven dynamics. Clusters of highly correlated groups of particlesl (spatial coarse graining)

16 Metastability and coarse-graining in hard-sphere colloids Hard sphere colloidal systems-only short range interactions Entropy driven dynamics. Clusters of highly correlated groups of particlesl (spatial coarse graining) Dynamics described in terms of the probability per unit of time that a cluster survives the random forces from the rest of the system.

17 What does trivial or simple mean? Trivial dynamics def = time homogeneous (but perhaps not stationary)

18 What does trivial or simple mean? Trivial dynamics def = time homogeneous (but perhaps not stationary) For stationary processes:

19 What does trivial or simple mean? Trivial dynamics def = time homogeneous (but perhaps not stationary) For stationary processes: Correlation and response functions only depend on time differences

20 What does trivial or simple mean? Trivial dynamics def = time homogeneous (but perhaps not stationary) For stationary processes: Correlation and response functions only depend on time differences Can be expanded in eigenvalue expansions

21 The simplest stochastic process Add random and independent position increments

22 The simplest stochastic process Add random and independent position increments Use linearity of the variance

23 The simplest stochastic process Add random and independent position increments Use linearity of the variance MSD t : diffusion

24 Trivializing aging dynamics Record dynamics record sized fluctuations

25 Trivializing aging dynamics Record dynamics record sized fluctuations in the stationary white noise impinging on the system (e.g. thermal noise)

26 Trivializing aging dynamics Record dynamics record sized fluctuations in the stationary white noise impinging on the system (e.g. thermal noise) drive the aging dynamics in marginally stable (glassy) systems

27 Trivializing aging dynamics Record dynamics record sized fluctuations in the stationary white noise impinging on the system (e.g. thermal noise) drive the aging dynamics in marginally stable (glassy) systems no. of records is a Poisson process in logarithmic time

28 Trivializing aging dynamics Record dynamics record sized fluctuations in the stationary white noise impinging on the system (e.g. thermal noise) drive the aging dynamics in marginally stable (glassy) systems no. of records is a Poisson process in logarithmic time the dynamics appears simple in logarithmic time

29 The rest of this talk Brief overview of results for thermally activated dynamics

30 The rest of this talk Brief overview of results for thermally activated dynamics Experimental 1 colloidal data re-analyzed.

31 The rest of this talk Brief overview of results for thermally activated dynamics Experimental 1 colloidal data re-analyzed. A cluster model for colloidal behavior numerically analyzed.

32 The rest of this talk Brief overview of results for thermally activated dynamics Experimental 1 colloidal data re-analyzed. A cluster model for colloidal behavior numerically analyzed. Trees, record dynamics and complexity

33 Aging set-up 6 5 Isothermal Aging quench temperature external field t The age t starts at the initial quench, and a field is (possibly) switched on at t = t w = 100. In a colloid, the system is centrifuged to a high density (quench). No fields are applied.

34 Edwards Anderson spin glass The Edwards Anderson spin-glass is an Ising spin model where interacting spins σ i = ±1 are placed on a lattice. The Hamiltonian is H = n.n. σ iσ j J ij.

35 Edwards Anderson spin glass The Edwards Anderson spin-glass is an Ising spin model where interacting spins σ i = ±1 are placed on a lattice. The Hamiltonian is H = n.n. σ iσ j J ij. For nearest neighbors J ij are Gaussian standard random variables, independent for i < j

36 Edwards Anderson spin glass The Edwards Anderson spin-glass is an Ising spin model where interacting spins σ i = ±1 are placed on a lattice. The Hamiltonian is H = n.n. σ iσ j J ij. For nearest neighbors J ij are Gaussian standard random variables, independent for i < j The thermal equilibrium properties of the model are complex

37 Edwards Anderson spin glass The Edwards Anderson spin-glass is an Ising spin model where interacting spins σ i = ±1 are placed on a lattice. The Hamiltonian is H = n.n. σ iσ j J ij. For nearest neighbors J ij are Gaussian standard random variables, independent for i < j The thermal equilibrium properties of the model are complex The time evolution, as given by a MC algorithm (Metropolis acceptance rule or equivalent) is complex and very similar to experimental data

38 p-spin glass model The p-spin model is an Ising spin model, The Hamiltonian is H = Plaq. σ iσ j σ k σ l

39 p-spin glass model The p-spin model is an Ising spin model, The Hamiltonian is H = Plaq. σ iσ j σ k σ l The thermal equilibrium properties of the model are trivial

40 p-spin glass model The p-spin model is an Ising spin model, The Hamiltonian is H = Plaq. σ iσ j σ k σ l The thermal equilibrium properties of the model are trivial The time evolution is nevertheless complex, featuring metastability and aging.

41 ROM model The Restricted Occupancy Model is a lattice model describing vortex creep in type II superconductors in terms of the number n i = of vortices on site i. the energy includes repulsive interactions between vortex lines on neighbor sites see L. P. Oliveira et al. Phys. Rev. B (2005) and references therein.

42 ROM model The Restricted Occupancy Model is a lattice model describing vortex creep in type II superconductors in terms of the number n i = of vortices on site i. the energy includes repulsive interactions between vortex lines on neighbor sites pinning to random sites see L. P. Oliveira et al. Phys. Rev. B (2005) and references therein.

43 ROM model The Restricted Occupancy Model is a lattice model describing vortex creep in type II superconductors in terms of the number n i = of vortices on site i. the energy includes repulsive interactions between vortex lines on neighbor sites pinning to random sites the configuration is updated with Metropolis dynamics see L. P. Oliveira et al. Phys. Rev. B (2005) and references therein.

44 ROM model intermittency N (t) t L. P. Oliveira et al. Phys. Rev. B (2005) The time variation of the total number of vortices N(t) on the system for a single realization of the pinning potential and the thermal noise in a lattice for T = 0.1.

45 Spin-glass, heat flow intermittency P q (ε) PDF (t/t w ) n I /n G H(0.01 t ;0.5 t ;t ) w w w log(1+t/t ) w Heat transfer PDF for a spin glass model (PS & H J Jensen, Europhysics Lett. 2005) Heat ε transfer H over small time δt in the E-A spin glass model has a Gaussian part and an intermittent tail. Six different ages are considered with δt/t w =.01.

46 Heat flow rate, p-spin model r E T=0.05 T=0.75 T=1.00 T= T= age The average rate of energy flow is plotted versus the age for the temperatures shown. The full line has the form y = C(T)tw 1.

47 p-spin model, intermittent energy decay T=1.5 BSF Energy S. Christiansen & PS, New J. of Physics, to appear t

48 p-spin model, quakes in real space t w = 10 5 ; t obs = 10 6 ; T= S. Christiansen & PS, New J. of Physics

49 p-spin model, heat flow PDF 10 0 t w = 1000; t = 10000; δ t = 100 obs δ E The PDF of the heat exchanged between system and thermal bath over a time δt = 100. T = 1.5.

50 p-spin model, magnetic fluctuations H = 0 PDF t w = 1000; t obs = 10000; δ t = δ M The PDF of the spontaneous magnetic fluctuations over a time δt = 100 and T = 1.5.

51 p-spin model, magnetic fluctuations H = 0.3 PDF t w = 1000; t obs = 10000; δ t = 100 δ E > δ M PDF of the spontaneous magnetic fluctuations over a time δt = 100. T = 1.5. The magnetic response is SUBORDINATED to the quakes

52 Take-home message from thermally activated aging The evolution of aggregate variables (i.e. the energy and magnetization) is controlled by extremely rare and irreversible events quakes

53 Take-home message from thermally activated aging The evolution of aggregate variables (i.e. the energy and magnetization) is controlled by extremely rare and irreversible events quakes Reversible fluctuation of zero average with Gaussian PDF s describe the dynamics in quasi-equilibrium.

54 Experimental data Tracking data obtained by confocal microscopy Courtland and Weeks,J. Phys.: Condens. Matter 15, S359, (2003)

55 Experimental data Tracking data obtained by confocal microscopy particle radius 1.18µ Courtland and Weeks,J. Phys.: Condens. Matter 15, S359, (2003)

56 Experimental data Tracking data obtained by confocal microscopy particle radius 1.18µ trajectories [x(t), y(t), z(t)] available for thousands of tagged partices Courtland and Weeks,J. Phys.: Condens. Matter 15, S359, (2003)

57 Experimental data Tracking data obtained by confocal microscopy particle radius 1.18µ trajectories [x(t), y(t), z(t)] available for thousands of tagged partices dense colloids ρ > 0.62 sub-diffusive behavior, aging Courtland and Weeks,J. Phys.: Condens. Matter 15, S359, (2003)

58 Experimental data Tracking data obtained by confocal microscopy particle radius 1.18µ trajectories [x(t), y(t), z(t)] available for thousands of tagged partices dense colloids ρ > 0.62 sub-diffusive behavior, aging glass formers ρ < 0.62 diffusive behavior, time homogeneous Courtland and Weeks,J. Phys.: Condens. Matter 15, S359, (2003)

59 Experimental procedure Centrifuge the sample to desired density

60 Experimental procedure Centrifuge the sample to desired density Stir the sample

61 Experimental procedure Centrifuge the sample to desired density Stir the sample Wait (age t w calculated from the end of the stirring phase)

62 Experimental procedure Centrifuge the sample to desired density Stir the sample Wait (age t w calculated from the end of the stirring phase) Start tracking. Scanning through sample 20 s

63 MSD vs time. Glass-former (MSD/5) t (hrs)

64 MSD vs time. Dense MSD (µ 2 ) t/t w

65 Persistence analysis Partition the system volume in a set of subvolumes

66 Persistence analysis Partition the system volume in a set of subvolumes Pick for each sub-volume a pair of colloidal particle which are initially in touch

67 Persistence analysis Partition the system volume in a set of subvolumes Pick for each sub-volume a pair of colloidal particle which are initially in touch Identify the time at which each pair splits

68 Persistence analysis Partition the system volume in a set of subvolumes Pick for each sub-volume a pair of colloidal particle which are initially in touch Identify the time at which each pair splits Calculate the fraction of pairs which survive at time t.

69 Persistence curves 10 0 Thr.=0.8 Pair Survival probability t (s) Thr.=0.6 Thr.= t/t w

70 Intermittency curves t = 360 t = 180 t = 18 MSD (µ 2 ) PDF t (s) Glass former r (µ)

71 Intermittency curves t = 100 t = 60 t = 20 MSD (µ 2 ) PDF t (s) Dense colloid r (µ)

72 Conclusion from colloid data analysis the MSD is linear in t (glass formers) and in ln(t) ln(t w ) (dense)

73 Conclusion from colloid data analysis the MSD is linear in t (glass formers) and in ln(t) ln(t w ) (dense) The persistence probability is exponential in t (glass formers) and ln(t) ln(t w ) (dense)

74 Conclusion from colloid data analysis the MSD is linear in t (glass formers) and in ln(t) ln(t w ) (dense) The persistence probability is exponential in t (glass formers) and ln(t) ln(t w ) (dense) Transformation t ln(t/t w ) trivializes the dynamics of dense colloids

75 Conclusion from colloid data analysis the MSD is linear in t (glass formers) and in ln(t) ln(t w ) (dense) The persistence probability is exponential in t (glass formers) and ln(t) ln(t w ) (dense) Transformation t ln(t/t w ) trivializes the dynamics of dense colloids Highly intermittent and correlated motion

76 Not much changes in a colloid Particle density is constant

77 Not much changes in a colloid Particle density is constant Energy density is constant

78 Not much changes in a colloid Particle density is constant Energy density is constant Particle displacement over observation time of the order of particle radius

79 Not much changes in a colloid Particle density is constant Energy density is constant Particle displacement over observation time of the order of particle radius Particle motion is jerky and spatially correlated

80 A cluster model particles in a cluster moves in fully correlated fashion on average zero CM displacement as long as the cluster persists

81 Particle motion when a cluster is destroyed

82 Particle motion when a cluster is destroyed the particles join neighboring clusters

83 Particle motion when a cluster is destroyed the particles join neighboring clusters and move in real space

84 Particle motion when a cluster is destroyed the particles join neighboring clusters and move in real space Probability (per MC query) that a cluster of size h is destroyed

85 Particle motion when a cluster is destroyed the particles join neighboring clusters and move in real space Probability (per MC query) that a cluster of size h is destroyed P k (h) = 1 k j=0 hj /j! ; P (h) = e h ; P 1 (h) = 1/(1+h)

86 Particle motion when a cluster is destroyed the particles join neighboring clusters and move in real space Probability (per MC query) that a cluster of size h is destroyed P k (h) = 1 k j=0 hj /j! ; P (h) = e h ; P 1 (h) = 1/(1+h) for all k: P k (h 0) for h

87 The algorithm Markov chain, L clusters

88 The algorithm Markov chain, L clusters pick a cluster at random

89 The algorithm Markov chain, L clusters pick a cluster at random destroy it with probability P k (h)

90 The algorithm Markov chain, L clusters pick a cluster at random destroy it with probability P k (h) or choose to another cluster

91 The algorithm Markov chain, L clusters pick a cluster at random destroy it with probability P k (h) or choose to another cluster iff cluster is destroyed: partition its particles randomly in two groups each sub-group joins neighboring cluster particles move one random unit step on the lattice

92 Diffusive behavior I P 1 (h) = 1 1+h glass former, diffusive <x 2 > L=512 L=256 L=128 L= 64 L= 32 L= 16 ~t P(h)~1/h Sweeps

93 Diffusive behavior II P (h) = e h <x 2 > L=512 L=256 L=128 L= 64 L= 32 L= 16 dense colloid, log diffusive P(h)~e -h Sweeps

94 Persistence I P 1 (h) = 1 1+h glass former, exponential decay Persistence P(h)~1/h L=512, θ= 4 L=512, θ= 8 L=512, θ=16 L=128, θ= 4 L=128, θ= 8 L=128, θ=16 L= 32, θ= 4 L= 32, θ= Sweeps

95 Persistence II P (h) = e h Int. Persistence L=512, θ=2 L=512, θ=4 L=512, θ=8 L=128, θ=2 L=128, θ=4 L=128, θ=8 L= 32, θ=2 L= 32, θ=4 L= 32, θ=8 dense colloid, exponential decay in logt P oo (h)~e -h Sweep time t

96 Spatial heterogeneity I P 1 (h) = 1 1+h glass former, approaches stationary state P(h)~1/h 5 Deviation σ L=512 L=256 L=128 L= 64 L= 32 L= Sweeps

97 Spatial heterogeneity II P (h) = e h dense colloid, equilibrium is out of reach P(h)~e -h 6 Deviation σ 4 2 L=512 L=256 L=128 L= 64 L= 32 L= Sweeps

98 Quakes in colloidal model Collapses of clusters of different sizes are widely separated in time All clusters of size < σ(t) only provide background of mobile particles A quake on a time scale t corresponds to the collapse of the smallest cluster > σ

99 Quakes in thermally activated aging Energy fluctuations which are negative

100 Quakes in thermally activated aging Energy fluctuations which are negative And not reversible on the time scale t at which they occur

101 Quakes in thermally activated aging Energy fluctuations which are negative And not reversible on the time scale t at which they occur Size δe/t >> logt

102 Temporal statistics of quakes In the aging regime, each cluster collapse is called a quake Rate of Quakes/L L=512 L=256 L=128 L= 64 L= 32 L= 16 ~1/t P(h)~e -h Sweeps

103 Temporal statistics of quakes II τ k = log(t k ) log(t k 1 ): waiting log-times P(h)~e -h L= 32 L=128 L=512 P(τ) τ

104 P(log(t k /t k 1 ) >= x) x The distribution of the logarithmic differences log(t k ) log(t k 1 ). is approximately exponential.

105 The log Poisson distribution P(n,t 1,t 2 ) probability that n quakes occur in [t 1,t 2 ).

106 The log Poisson distribution P(n,t 1,t 2 ) probability that n quakes occur in [t 1,t 2 ). P(n,t 1,t 2 ) = µn n! exp( µ) µ(t 1,t 2 ) = αlog(t 2 /t 1 ) (1)

107 The log Poisson distribution P(n,t 1,t 2 ) probability that n quakes occur in [t 1,t 2 ). P(n,t 1,t 2 ) = µn n! exp( µ) µ(t 1,t 2 ) = αlog(t 2 /t 1 ) (1) The statistics applies to e.g. spin-glasses, colloids, evolutionary dynamics etc.

108 The log Poisson distribution P(n,t 1,t 2 ) probability that n quakes occur in [t 1,t 2 ). P(n,t 1,t 2 ) = µn n! exp( µ) µ(t 1,t 2 ) = αlog(t 2 /t 1 ) (1) The statistics applies to e.g. spin-glasses, colloids, evolutionary dynamics etc. Where can it possibly come from?

109 Marginal increase in stability In colloids: The dynamics decelerates once a smallest cluster marginally larger than its predecessor is formed This cluster is toppled by a record sized fluctuation on a time scale t 2 > t 1... Sequence of magnitude records in the probability of toppling the smallest cluster marks the evolution of the dynamics.

110 Relation to hierarchies and trees? Two different ways to describe the same generic situation ( Fischer Hoffmann & Sibani, PRE 2008 )

111 Relation to hierarchies and trees? Two different ways to describe the same generic situation Record dynamics needs an underlining hierarchical structure ( Fischer Hoffmann & Sibani, PRE 2008 )

112 Relation to hierarchies and trees? Two different ways to describe the same generic situation Record dynamics needs an underlining hierarchical structure work with Andreas Fischer and KH elaborates on this point ( Fischer Hoffmann & Sibani, PRE 2008 )

113 Acknowledgments Colloid analysis in collaboration with Stefan Boettcher, Emory University, GA, USA. Thamks to Eric Weeks Emory University, for kindly providing the colloidal data.

114 Acknowledgments Colloid analysis in collaboration with Stefan Boettcher, Emory University, GA, USA. Thamks to Eric Weeks Emory University, for kindly providing the colloidal data. The p-model data shown are part of a (former) student, Simon Christiansen s master thesis

115 Acknowledgments Colloid analysis in collaboration with Stefan Boettcher, Emory University, GA, USA. Thamks to Eric Weeks Emory University, for kindly providing the colloidal data. The p-model data shown are part of a (former) student, Simon Christiansen s master thesis Thanks to Jesper Dall, Christian Schön Karl Heinz Hoffmann & Henrik J. Jensen for inputs and discussions over the years.