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Dynamics of Complex Systems

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Dynamics of Complex Systems

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  1. 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

  2. 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←frustrationandrandomness Characteristic time-dependence→ dynamic approach information flow

  3. 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

  4. 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

  5. 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

  6. 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 ω→0 dynamic divergence of the relaxation time and fluctuations

  7. 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

  8. Frustrated XY Model • Symmetry depends on f in a highly discontinuous fashion • f = 0(unfrustrated): U(1), BKT transitionT < Tc: critical, power-law decay of phase correlation • f = ½(fully frustrated): U(1)Z2ground state: doubly degenerate (discrete)→ Z2 (Ising) → double transitions(BKT + Ising?)two kinds of coupled degrees of freedom • phase (vortex excitation) • chirality (domain-wall excitation)

  9. Current-driven array of Josephson junctions L  L SQ array uniform applied currents 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 resistively shunted junction realdynamics(↔ kinetic Ising model)

  10. signal S : power spectrum peak at  N : background noise level Stochastic Resonance ac drivingI = Ia cos t • 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

  11. (cf. devil’s staircase) Mode Locking ac + dc drivingI = Id + Ia cos tat T = 0 →voltage quantization:giant Shapiro steps (GSS) • mode locking ← topological invariance • chaos

  12. 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

  13. 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

  14. Insulin Secretion and Glucose Regulation β-cells in Islet of Langerhans glucose→ bursting behavior→ insulin secretion

  15. Pancreas Islet of Langerhans

  16. V Action Potentials Intact β-cells Kinard et al. (1999) Isolated β-cells

  17. Synchronized bursting of β-cells simultaneous recording of the electrical activity from two cells

  18. 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

  19. Coupled oscillator model Current equation at each cell i, neighbors of which are linked by gap junctions

  20. Noise (thermal fluctuation) increase noise level

  21. Noise (stochastic channel gating) Multiplicative or colored noise induces more effectively several consecutive firings than white noise.

  22. weak coupling (10 pS) optimal coupling (40 pS) strong coupling (100 pS) Coupling (Gap Junction) regular bursts induced

  23. 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 oscillatorseach 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

  24. Failures in biological systems neurons (Alzheimer) , βcells (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

  25. 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)

  26. Time evolution of the average fraction of living cells

  27. Phase diagram healthy state

  28. 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

  29. 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) ≡∑ifi(x)(≡ − energy) • entropy ecosystem directed to gather information from the environment and to evolve continuously into a new configuration information transfer dynamics entropic sampling

  30. ecosystem x environment • total entropy • probability for the ecosystem in state xβ → ∞: important samplingβ → 0: entropic sampling (St = const., i.e., reversible info exchange) power-law behavior (γ≈τ≈ 2) informationexchange

  31. Mutation Rate and Power Spectrum critical, scale invariant

  32. Scale-free behavior emerging frominformation transfer dynamics 2D Ising model power spectrum of magnetization and relaxation time

  33. Other Systems 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

  34. 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

  35. Coauthorships in network research MEJ Newman & M Girvan

  36. 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: fast world

  37. 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. 3 4 5 6 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 2 7 8 N branched chain with branching probability P

  38. 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

  39. 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, …