Dynamicsof

Complex Systems

M.Y. ChoiDepartment of Physics

Seoul National University

Seoul 151-747, Korea

May 2005 PITP Conference

Main CollaboratorsJ. Choi (KU), D.S. Koh (UW), B.J. Kim (AU), H. Hong (JNU), G.S. Jeon (PSU), J. Yi (PNU), M.-S. Choi, M. Lee (KU), H.J. Kim, Y. Shim (CMU), J.S. Lim, H. Kang, J. Jo (SNU)

Complex System

Many-particle system

many elements (constituents)

a large number of relations among elements interactions Nonlinearity (nonlinear relations) complicated behavior Open and dissipative structure environment essential Memory adaptation Aging properties Between order and disorder critical

Large variability ← frustration and randomness

Characteristic time-dependence → dynamic approach

information flow

Potpourri of Complex Systems Electron and superconducting systems: Josephson-junction arrays, Harper’

s equation, CDW Glass: glass, spin glass, charge glass, vortex glass, gauge glass Complex fluids: colloids, polymers, liquid crystals, powder, traffic flow, io

nic liquids Disordered systems: interface, growth, composites, fracture, coupled oscill

ators, fiber bundles Biological systems: protein, DNA, metabolism, regulatory and immune sy

stems, neural networks, population and growth, ecosystem and evolution Optimization problems: TSP, graph partitioning, coloring Complex networks: communication/traffic networks, social relations, dyna

mics on complex networks Socio-economic systems: prisoner’s dilemma, consumer referral, stock ma

rket , Zipf’s law

similarity out of diversity details irrelevant

Dynamics of Driven Systems

Relaxation and responses

Synchronization and stochastic resonance

Mode locking, dynamic transition, and resonance

Mesoscopic Systems

Quantum coherence and fluctuations

(Quantum) Josephson-junction arrays

Charge-density waves

Biological Systems

Insulin secretion and glucose regulation

Dynamics of failures

Information transfer and criticality

Other Systems

Complex networks

Consumer referral

Dynamics of Driven Systems

many-particle systemtime-dependent perturbation(external driving)

relaxation time τ0period τ ≡ 2π/Ω

• relaxation time τ0 ≠ 0 response not instantaneous

• competition between τ0 and τ rich dynamics

dynamic hysteresis, dynamic symmetry breaking, stochast

ic resonance, mode locking and melting

Ω

Ubiquitous but equilibrium concepts (free energy) inapplicable

No perturbation: equilibrium order parameter m

m ≠ 0 → broken symmetry

Time-dependent perturbation h(t):

dynamics ☜ Langevin equation, Fokker-Planck equation,

master equation, etc. equations of motion: symmetric in time

order parameter m(t): may not be symmetric in time

dynamic order parameter

→ dynamic symmetry breaking

1 Q dt m

0Q

ordered phase shrinks as ω→0 dynamic

divergence of the relaxation time and fluctuations

1D/2D Superconducting Arrays simple complex system

superconducting islandsweakly coupled by

Josephson junctions in magnetic fields driven by applied

currents

“Fancy” concepts: topological defects, symmetry and breaking, topological order, gauge field, fractional charge, frustration, randomness, gauge glass and algebraic glass order, chaos, Berry’s phase, topological quantization, mode locking and devil’s staircase, dynamic transition, stochastic resonance, anomalous relaxation, aging, complexity, quantum fluctuations and dissipation, quantum phase transition, charge-vortex duality, quantum vortex, QHE, AB/AC effects, persistent current and voltage, exciton

magnetic field/charge → frustration

Frustrated XY Model

Symmetry depends on f in a highly discontinuous fashion f = 0 (unfrustrated): U(1), BKT transition

T < Tc: critical, power-law decay of phase correlation

f = ½ (fully frustrated): U(1)Z2

ground state: doubly degenerate (discrete) → Z2 (Ising)

ji

ijjiJ AEH,

)cos(

→ double transitions (BKT + Ising?)two kinds of coupled degrees of freedom

phase (vortex excitation) chirality (domain-wall excitation)

fAdc

eA

Pij

j

iij 22,2

0 lA

current conservation → equations of motion

noise current I =Id: IV characteristics, current-induced unbinding, CR

I = Ia cos t: dynamics transition, SR

I = Id + Ia cos t: mode locking, melting and transition

L L SQ arrayuniform applied currents

extij ijijjiCijji IAIA

dt

d

eR

'

sin2

))('(2

)'()( jkiljlikklij ttR

kTtt

)( ,1, Lxxexti II

resistively shunted junction

Current-driven array of Josephson junctions

real dynamics (↔ kinetic Ising model)

Stochastic Resonance

N

S10log10SNR signal S : power spectrum peak at

N : background noise level

staggered magnetization

• SR phenomena peak only at T >Tc

( double peaks around Tc)

☜ τ → ∞ at T <Tc

Ia = 0.8; /2 = 0.08: Q > 0 (no osc.) at T = 0

ac driving I = Ia cos t

ac + dc driving I = Id + Ia cos t at T = 0

→ voltage quantization: giant Shapiro steps (GSS)

FGSS2

:

IGSS2

:0

e

L

s

nVsrf

e

LnVf

(cf. devil’s staircase)

• mode locking ← topological invariance• chaos

Mode Locking

Dynamic phase diagram

from the voltage step width

w

melting of voltage steps

V = 0(□), 1/2(O), 1(∆)

Inset:

Arnold tongue

structure

1/ 4V

dynamic transition ↔ melting of Shapiro steps

Paradigm: complex systems

displaying life as cooperative phenomena

• fine-grained modeling: beta cells, protein dynamics

• coarse-grained modeling: synchronization, failure, evolution

Physics: understanding by means of (simple) models

relevant and irrelevant elements

Biological Systems

Insulin Secretion and Glucose Regulation

glucose → bursting behavior → insulin secretion

β-cells in Islet of Langerhans

Islet of Langerhans

Pancreas

Isolated β-cells

Intact β-cells

Kinard et al. (1999)

V

Action Potentials

Synchronized bursting of β-cells

simultaneous recording of the electrical activityfrom two cells

glucose

ATP ↑

K+ channel closed

K+ ↓, depolarized

Ca2+ channel open

Ca2+ ↑

insulin exocytosis

Bursting mechanism

Activation and inhibition of GLUT-1 and GLUT-2 transporters by secreted insulin are represented by the solid (+) and dashed (-) arrows. Thick arrows describe physical transport of materials (glucose and ions).

Coupled oscillator model

Current equation at each cell i, neighbors of which are linked by gap junctions

Noise (thermal fluctuation)

increase noise level

Noise (stochastic channel gating)

Multiplicative or colored noise induces more effectively several consecutive firings than white noise.

Coupling (Gap Junction)

weak coupling (10 pS)

strong coupling (100 pS)

optimal coupling (40 pS)

regular bursts

induced

coherent motion among many coupled cells Josephson junctions, CDW, laser, chemical reactions, pacemaker cells, neurons,

circadian rhythm, insulin secretion, Parkinson’s disease, epilepsy, flashing fireflies, swimming rhythms in fish, crickets in unison, menstrual periods, rhythms in applause

prototype model: set of N coupled oscillators each described by its phase φi and natural frequency ωi driven with amplitude Ii and frequency Ω

natural frequency distribution (e.g. Gaussian with variance σ2 ≡1)

phase order parameter

( ) ( ) sin ( ) ( ) cos ( )N

i i ij i j ij i i ij

t t J t t A I t t

1( ) ( )j

j

gN

1( 0 : synchronization)ji i

j

e eN

Collective synchronization

2-10 wks Up to 10 yrs

Pla

sma

leve

ls

CD4+ T cells

Virus

HIV antibodies

Time course of HIV infection

Failures in biological systems

neurons (Alzheimer) , β cells (diabetes), T cells (AIDS) degenerative disease

Simplest model: system of N cells under stress F = Nf

state of each cell: si = ±1 dead/alive

state of the system {s1, s2, …, sN } 2N states

If cell j becomes dead (sj = 1), stress Vij is transferred to cell i

total stress on cell i

death of cell i depends on Vi and its tolerance gi:

or

uncertainty due to random variations, environment probabilistic(noise effective temperature T)

time delay td in stress redistribution

cell regeneration in time t0 → healing parameter a ~ t0-1

a = 0: fiber bundle model rupture, destruction, earthquake, social failure dynamics ← master equation for probability P({si}, t; {si’}, t-td)

1

2j

i ijj

sV f V

( ) 0i i iV g s ( ) 0i ij j ij

s V s h

Time evolution of the average fraction of living cells

cf f

Phase diagram

healthy state

Information transfer and evolution

Fossil record

evolution proceeds not at a steady pace but in an intermittent manner punctuated equilibriumfossil data display power-law behavior critical

number of taxa with n sub-taxa:

lifetime distribution of genera:

number of extinction events of size s:

power spectrum of mutation rate:

Basic idea

molecular level: random mutation

natural selection

phenotypic level: power-law behavior

evolution dynamics: random mutation and natural selection

~nM n

~tM t

~sM s

2

( ) ~P 1.5

Evolution dynamics

ecosystem consisting of N interacting species

configuration x≡{xi} (i = 1,2,…,N)

fitness of each species fi(x)

total fitness F(x) ≡ ∑i fi(x) (≡ − energy)

entropy( ) ln ( )

( ) ( ( ))j iij

S F F

F dx F f x

ecosystem directed to gather information from the environment and to evolve continuously into a new configuration

entropic sampling

information transfer dynamics

total entropy probability for the ecosystem in state x

β → ∞: important sampling β → 0: entropic sampling

(St = const., i.e., reversible info exchange)

power-law behavior (γ ≈ τ ≈ 2)

environment

0 0lnS ecosystem x

( )S F informationexchange

0 0( ) (0) ( | )t t tS F S S S F S F

0 ( ) ( ( ))0( ) tS S S F x S F xP x e e e

Mutation Rate and Power Spectrum

1.5( )P critical, scale invariant

Scale-free behavior emerging from information transfer dynamics

2D Ising model

power spectrum of magnetization and relaxation time

2( )P

2.6L

Complex Networks

•Regular networks (lattices)

highly clustered characteristic path length:

•Random networks

low clustering characteristic path length:

•Networks in nature: in between regular and random → complex

– Biological networks: neural networks, metabolic reactions, protein networks, food webs– Communication/Transportation networks: WWW, Internet, air route, subway and bus route– Social networks: citations, collaborations, actors, sexual partners

( )O N

(log )O N

Other Systems

Small-world networks Start from regular networks with N sites

connected to 2k nearest neighbors Rewire each link (or add a link) to a randomly

chosen site with probability p Highly clustered ≈ regular network (p = 0) Average distance between pairs increase

slowly with size N ≈ random network (p = 1)

Scale-free networks preferential linking hub structure power-law distribution of degrees

Coauthorships in network research

MEJ Newman & M Girvan

Dynamics on small-world networks

Phase transition, Synchronization, Resonance:spin (Ising, XY) models and coupled oscillators mean-field behavior for p > pc ( = 0 ?) fast propagation of information for p ≥ 0.5 lower SR peak enhanced system size resonance

→ cost effective

Vibrations: Netons excitation gap → rigidity against low energy deformation

Diffusion

classical system:

quantum system:

2N N

logN N fast world

Economic Systems: Consumer referral on a network

A monopolist having a link with only one out of and N consumers

Each consumer considers his/her valuation distributed according to f(v), and decides whether to purchase one at price p. If yes, (s)he decides whether to refer other(s) linked at referral cost δ. Referral fee r is paid if (s)he convinced a linked consumer to buy one. The procedure is continued.

● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ●

● ● ● ● ●

● ● ● ●

1 2

3 4 5 6

7 8

N

branched chain with branching probability P

Maximum profit (per consumer) vs N

P = 0: maximum profit per consumer ~ 1/N (→ 0 as N → ∞)P≠ 0: maximum profit per consumer saturates (→ finite value as N → ∞)

small-world transition

Concluding Remarks Physics pursuits universal knowledge (“theory”) “theoretical science”

how to understand phenomena and how to interpret nature Physics in 20th century: fundamental principles

Reductionism and determinism Simple phenomena (limited, exceptional) Particles and fields

Physics in 21st century: interpretation of nature Emergentism, holism, and unpredictability complementary Complex phenomena (diverse, generic) Information

Appropriate methodsstatistical mechanicsnonlinear dynamicscomputational physics

Physics of Complex Systems biological physics, econophysics, sociophysics, …