Dynamics
of
Complex Systems
M.Y. Choi
Department of Physics
Seoul National University
Seoul 151-747, Korea
Main Collaborators
J. 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)
May 2005 PITP Conference
Complex System

Many-particle system
many elements (constituents)
a large number of relations among elements

Nonlinearity (nonlinear relations)

Open and dissipative structure

Memory

Aging properties

Between order and disorder
adaptation
interactions
complicated behavior
environment essential
information flow
critical
Large variability ← frustration and randomness
Characteristic time-dependence → dynamic approach
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,
ionic liquids
Disordered systems: interface, growth, composites, fracture, coupled
oscillators, fiber bundles
Biological systems: protein, DNA, metabolism, regulatory and immune
systems, neural networks, population and growth, ecosystem and evolution
Optimization problems: TSP, graph partitioning, coloring
Complex networks: communication/traffic networks, social relations,
dynamics on complex networks
Socio-economic systems: prisoner’s dilemma, consumer referral, stock
market , 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 system
time-dependent perturbation
(external driving)
Ω
period τ ≡ 2π/Ω
relaxation time τ0
•
relaxation time τ0 ≠ 0
response not instantaneous
•
competition between τ0 and τ
rich dynamics
dynamic hysteresis, dynamic symmetry breaking,
stochastic resonance, mode locking and melting
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

Q
dynamic order parameter
1

 dt m
Q  0 → dynamic symmetry breaking
ordered phase shrinks as ω→0
dynamic
divergence of the relaxation time and fluctuations
1D/2D Superconducting Arrays
simple complex system
superconducting islands
weakly coupled by
Josephson junctions in
magnetic fields driven by
applied currents
magnetic field/charge → frustration
“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
Frustrated XY Model
H   EJ  cos( i   j  Aij )
i, j

2e j
Aij   A  dl,
c i
A
ij
 2  0  2f
P
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)
→ double transitions (BKT + Ising?)
two kinds of coupled degrees of freedom
 phase (vortex excitation)
 chirality (domain-wall excitation)
Current-driven array of Josephson junctions
L  L SQ array
uniform applied currents
I iext  I ( x,1   x, L )
resistively shunted junction
current conservation → equations of motion
  d

ext








A

I
sin




A



I
 j  2eR dt i j ij C
i
j
ij
ij 
i

'
2kT
(t  t ' )(ik  jl  il  jk )
noise current
R
 I =Id: IV characteristics, current-induced unbinding, CR
ij (t )kl (t ' ) 


I = Ia cos t: dynamics transition, SR
I = Id + Ia cos t: mode locking, melting and transition
real dynamics (↔ kinetic Ising model)
Stochastic Resonance
ac driving I = Ia cos t
S 
SNR  10 log 10  
N 

signal S : power spectrum peak at 
N : background noise level
Ia = 0.8; /2 = 0.08: Q > 0 (no osc.) at T = 0
staggered magnetization
• SR phenomena
peak only at T >Tc
(
double peaks around Tc)
☜ τ → ∞ at T <Tc
Mode Locking
ac + dc driving I = Id + Ia cos t at T = 0
→ voltage quantization: giant Shapiro steps (GSS)
L 
IGSS
2e
n L 
f  r s: V 
FGSS
s 2e
f  0:
V n
(cf. devil’s staircase)
• mode locking ← topological invariance
• chaos
Dynamic phase diagram
melting of voltage steps
from the voltage step width w
V = 0(□), 1/2(O), 1(∆)
Inset:
V  1/ 4
Arnold tongue structure
dynamic transition ↔ melting of Shapiro steps
Biological Systems
Paradigm: complex systems
displaying life as cooperative phenomena
Physics: understanding by means of (simple) models
relevant and irrelevant elements
• fine-grained modeling: beta cells, protein dynamics
• coarse-grained modeling: synchronization, failure, evolution
Insulin Secretion and Glucose Regulation
β-cells in Islet of Langerhans
glucose → bursting behavior → insulin secretion
Pancreas
Islet of Langerhans
Action Potentials
Intact β-cells
V
Isolated β-cells
Kinard et al. (1999)
Synchronized bursting of β-cells
simultaneous
recording
of the electrical
activity
from two cells
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).
glucose
 ATP ↑
 K+ channel closed
 K+ ↓, depolarized
 Ca2+ channel open
 Ca2+ ↑
 insulin exocytosis
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)
optimal coupling (40 pS)
regular bursts
induced
strong coupling (100 pS)
Collective synchronization
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
N
i (t )  i (t )   J ij sin i (t )   j (t  )  Aij   i  Ii cos t  i (t )
j


natural frequency distribution
(e.g. Gaussian with variance σ2 ≡1)
phase order parameter

1
N
e
j
i j
 ei
1
g ()   ( j  )
N j
( 0 : synchronization)
Failures in biological systems
neurons (Alzheimer) , β cells (diabetes), T cells (AIDS) degenerative disease
Time course of HIV infection
HIV antibodies
CD4+ T cells
Virus
2-10 wks
Up to 10 yrs
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
1 s j
Vi  f  Vij
2
j
death of cell i depends on Vi and its tolerance gi:
(Vi  gi ) si  0 or
si ( Vij s j  hi )  0
j




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)
Time evolution of the average fraction of living cells
f  fc
Phase diagram
healthy state
Information transfer and evolution
Fossil record
evolution proceeds not at a steady pace but in an intermittent manner
 punctuated equilibrium
fossil data display power-law behavior  critical
number of taxa with n sub-taxa: M n ~ n 
2

lifetime distribution of genera: M t ~ t
number of extinction events of size s: M s ~ s 
power spectrum of mutation rate: P() ~ 
  1.5
Basic idea
molecular level: random mutation
natural selection
phenotypic level: power-law behavior
evolution dynamics: random mutation and natural selection
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
S ( F )  ln ( F )
( F )   j  dx j ( F   i f i ( x))
ecosystem directed to gather information from the environment
and to evolve continuously into a new configuration
information transfer dynamics
entropic sampling
environment
ecosystem x
S (F )
information
exchange
S0  ln 0
St ( F )  S  S0  St (0)  F

total entropy

probability for the ecosystem in state x
(  St F |0 )
P( x)  0  e S0  e St  S  eF ( x )  S ( F ( x ))
β → ∞: important sampling
β → 0: entropic sampling
(St = const., i.e., reversible info exchange)
 power-law behavior (γ ≈ τ ≈ 2)
Mutation Rate and Power Spectrum
P( )  1.5
critical, scale invariant
2D Ising model
power spectrum of magnetization and relaxation time
P( )  2

Scale-free behavior emerging from information transfer dynamics
L2.6
Other Systems
Complex Networks
•Regular networks (lattices)
highly clustered
characteristic path length:
O( N )
•Random networks
low clustering
characteristic path length:
O (log N )
•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

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

quantum system: 
classical system:
N2
N
 
 
N
log 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 methods
statistical mechanics
nonlinear dynamics
computational physics
 Physics of Complex Systems
biological physics, econophysics, sociophysics, …