Dynamics of Complex Systems

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DynamicsofComplex 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
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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
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Potpourri of Complex Systems

• Electron and superconducting systems
  Josephson-junction arrays, Harpers 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 prisoners dilemma,
  consumer referral, stock market , Zipfs law
• similarity out of
  diversity
• details
  irrelevant

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• 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

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Dynamics of Driven Systems
many-particle system
time-dependent perturbation (external driving)
O
period t 2p/O
relaxation time t0

• relaxation time t0 ? 0 response not
  instantaneous
• competition between t0 and t rich
  dynamics

dynamic hysteresis, dynamic symmetry breaking,
stochastic resonance, mode locking and melting
Ubiquitous but equilibrium concepts (free energy)
inapplicable
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• 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

ordered phase shrinks as ??0
dynamic divergence of the relaxation time and
fluctuations
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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, Berrys
phase, topological quantization, mode locking and
devils 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
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Frustrated XY Model

• Symmetry depends on f in a highly discontinuous
  fashion
• f 0 (unfrustrated) U(1), BKT transition T
  lt 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)

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Current-driven array of Josephson junctions
L ? L SQ array uniform applied currents
resistively shunted junction

• 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

real dynamics (? kinetic Ising model)
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Stochastic Resonance
ac driving I Ia cos ?t

• Ia 0.8 ?/2? 0.08 Q gt 0 (no osc.) at T 0

• staggered magnetization
• SR phenomena
• peak only at T gtTc
• ( double peaks around Tc)
• ? t ? 8 at T ltTc

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Mode Locking

• ac dc driving I Id Ia cos ?t at T
  0
• ? voltage quantization giant Shapiro
  steps (GSS)

• mode locking ? topological invariance
• chaos

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Dynamic phase diagram
melting of voltage steps
from the voltage step width w
V 0(?), 1/2(O), 1(?)
Inset Arnold tongue structure
dynamic transition ? melting of Shapiro steps
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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

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Insulin Secretion and Glucose Regulation
ß-cells in Islet of Langerhans
glucose ? bursting behavior ? insulin secretion
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Pancreas
Islet of Langerhans
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Action Potentials
Intact ß-cells
Kinard et al. (1999)
Isolated ß-cells
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Synchronized bursting of ß-cells
simultaneous recording of the electrical
activity from two cells
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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

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Coupled oscillator model
Current equation at each cell i, neighbors of
which are linked by gap junctions
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Noise (thermal fluctuation)
increase noise level
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Noise (stochastic channel gating)
Multiplicative or colored noise induces more
effectively several consecutive firings than
white noise.
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Coupling (Gap Junction)
regular bursts induced
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Collective synchronization

• coherent motion among many coupled cells
• Josephson junctions, CDW, laser, chemical
  reactions, pacemaker cells, neurons, circadian
  rhythm, insulin secretion, Parkinsons 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 fi
  and natural frequency ?i driven with
  amplitude Ii and frequency O
• natural frequency distribution (e.g. Gaussian
  with variance s2 1)
• phase order parameter

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Failures in biological systems
neurons (Alzheimer) , ß cells (diabetes), T cells
(AIDS) degenerative disease
Time course of HIV infection
HIV antibodies
Plasma levels
CD4 T cells
Virus
2-10 wks
Up to 10 yrs
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• 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)

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Time evolution of the average fraction of living
cells
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Phase diagram
healthy state
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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
• 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

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Evolution dynamics

• ecosystem consisting of N interacting species
• configuration xxi (i 1,2,,N)
• fitness of each species fi(x)
• total fitness F(x) ?i fi(x) ( - energy)
• entropy

ecosystem directed to gather information from the
environment and to evolve continuously into a new
configuration
entropic sampling
information transfer dynamics
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• total entropy
• probability for the ecosystem in state x
  ß ? 8 important sampling
  ß ? 0 entropic sampling
• (St const.,
  i.e., reversible info exchange) ?
  power-law behavior (? t 2)

informationexchange
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Mutation Rate and Power Spectrum
critical, scale invariant
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Scale-free behavior emerging from information
transfer dynamics
2D Ising model power spectrum of
magnetization and relaxation time
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Complex Networks
Other Systems

• 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

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• 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

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Coauthorships in network research
MEJ Newman M Girvan
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• Dynamics on small-world networks
• Phase transition, Synchronization, Resonance
• spin (Ising, XY) models and coupled oscillators
• mean-field behavior for p gt 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
fast world
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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 d. Referral fee r is
paid if (s)he convinced a linked consumer to buy
one. The procedure is continued.
3 4 5 6
? ? ? ? ? ?
? ?
? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ? ?
1 2
7 8
N
branched chain with branching probability P
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Maximum profit (per consumer) vs N
P 0 maximum profit per consumer 1/N (? 0 as
N ? 8)P? 0 maximum profit per consumer
saturates (? finite value as N ? 8)
small-world transition
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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,