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Spin Glasses and Spin Glasses and Complexity: Lecture Complexity: Lecture 2 2 Brief review of yesterday’s Brief review of yesterday’s lecture lecture Spin glass energy and broken Spin glass energy and broken symmetry symmetry Applications Applications - Combinatorial optimization and traveling - Combinatorial optimization and traveling salesman salesman - Simulated annealing - Simulated annealing - Hopfield-Tank neural network computation - Hopfield-Tank neural network computation - Protein conformational dynamics and - Protein conformational dynamics and folding folding Geometry of interactions and the Geometry of interactions and the infinite-range model infinite-range model

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Page 1: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Spin Glasses and Complexity: Spin Glasses and Complexity: Lecture 2Lecture 2

• Brief review of yesterday’s lectureBrief review of yesterday’s lecture

• Spin glass energy and broken symmetrySpin glass energy and broken symmetry

• ApplicationsApplications

- Combinatorial optimization and traveling salesman- Combinatorial optimization and traveling salesman

- Simulated annealing- Simulated annealing

- Hopfield-Tank neural network computation- Hopfield-Tank neural network computation

- Protein conformational dynamics and folding- Protein conformational dynamics and folding

• Geometry of interactions and the infinite-range modelGeometry of interactions and the infinite-range model

Page 2: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Homogeneous systems possess symmetries that greatly simplify Homogeneous systems possess symmetries that greatly simplify mathematical analysis and physical understanding--- Bloch’s mathematical analysis and physical understanding--- Bloch’s

theorem, broken symmetry, order parameter, Goldstone theorem, broken symmetry, order parameter, Goldstone modes, …modes, …

But for glasses, spin glasses, and other systems with But for glasses, spin glasses, and other systems with quenched disorder; many new ideas and concepts have quenched disorder; many new ideas and concepts have

been proposed, but so far no universal onesbeen proposed, but so far no universal ones

Examples: crystals, ferromagnets, superconductors and superfluids, liquid Examples: crystals, ferromagnets, superconductors and superfluids, liquid crystals, ferroelectrics, …crystals, ferroelectrics, …

Page 3: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Spin Glasses – a prototype disordered system?Spin Glasses – a prototype disordered system?

Dilute magnetic alloy, e.g., CuMnDilute magnetic alloy, e.g., CuMnFrustrationFrustration

Page 4: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Ground StatesGround States

CrystalCrystal GlassGlass

FerromagnetFerromagnet Spin GlassSpin Glass

x

xNM 1

Page 5: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

The Edwards-Anderson (EA) Ising Model

Site in Zd

xy

yxxyJ J

H

Site in Site in ZZdd

Nearest neighbor spins onlyNearest neighbor spins only

model) (Ising 1Coupling realizationCoupling realization

The couplings are i.i.d. random variables:The couplings are i.i.d. random variables:

]2/exp[2

1)( 2

xyxy JJP

S.F. Edwards and P.W. Anderson, J. Phys. F S.F. Edwards and P.W. Anderson, J. Phys. F 55, 965 (1975)., 965 (1975).

x

xxh

Page 6: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

EA ’75: A low-temperature spin glass phase should be described by presence of EA ’75: A low-temperature spin glass phase should be described by presence of temporal order (freezing) along with absence of spatial disorder.temporal order (freezing) along with absence of spatial disorder.

Broken symmetry in the spin glassBroken symmetry in the spin glass

01

lim

01

lim

1

2

1

N

ii

N

N

ii

N

SN

q

SN

M

EA

But there are some surprises in store …But there are some surprises in store …

Page 7: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

The most fundamental questions remain unanswered:The most fundamental questions remain unanswered:

• Is there a phase transition? Is there a phase transition?

• What is the nature of low-temperature phase (broken symmetry, order parameter)?What is the nature of low-temperature phase (broken symmetry, order parameter)?

• How does one account for the anomalous dynamical behavior (slow relaxation, How does one account for the anomalous dynamical behavior (slow relaxation, memory, aging…)?memory, aging…)?

Important not only for physics, but may lend important Important not only for physics, but may lend important concepts to other areas …concepts to other areas …

V. Cannella and J.A. Mydosh, V. Cannella and J.A. Mydosh, PhysPhys. . Rev. Rev. BB 66, 4220 (1972). , 4220 (1972).

L.E. Wenger and P.H. Keesom, L.E. Wenger and P.H. Keesom, Phys. Phys. Rev. BRev. B 1313, 4053 (1976), 4053 (1976)

Page 8: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

• Quenched disorderQuenched disorder

• FrustrationFrustration

• Combinatorially huge possible number of Combinatorially huge possible number of configurations, or states, or outcomesconfigurations, or states, or outcomes

• Many statistically equivalent `ground’ states Many statistically equivalent `ground’ states (more or less equally good optimal solutions)?(more or less equally good optimal solutions)?

• Slow equilibrationSlow equilibration

• Memory, aging …Memory, aging …

Applications to combinatorial optimization (graph theory) Applications to combinatorial optimization (graph theory) problems, neural networks, biological evolution, protein problems, neural networks, biological evolution, protein

dynamics and folding, …dynamics and folding, …

Example – the traveling salesman problemExample – the traveling salesman problem

• N=5 N=5 12 tours 12 tours

• N=10 N=10 181,440 tours 181,440 tours

• N=50 N=50 Forget it. Forget it.

(NP-complete)(NP-complete)

2/)!1( Nntours

Page 9: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Simulated annealingSimulated annealing

S. Kirkpatrick, C.D. Gelatt, Jr., and M.P. Vecchi, S. Kirkpatrick, C.D. Gelatt, Jr., and M.P. Vecchi, Science Science 220220, 671 (1983), 671 (1983)

• Cost function (plays role of energy function)Cost function (plays role of energy function)• Quenched disorderQuenched disorder

• FrustrationFrustration

• Combinatorially huge possible number of Combinatorially huge possible number of configurations, or states, or outcomesconfigurations, or states, or outcomes

• Many statistically equivalent `ground’ states Many statistically equivalent `ground’ states (more or less equally good optimal solutions)(more or less equally good optimal solutions)

- TSP: length of a tour- TSP: length of a tour

- ``Placement’’ in computer design- ``Placement’’ in computer design

- - k-SATk-SAT

Many of these resemble spin glass Hamiltonian!Many of these resemble spin glass Hamiltonian!

• Add a ``temperature’’, and treat problem like a statistical Add a ``temperature’’, and treat problem like a statistical mechanical problemmechanical problem

)/exp(1

)( TEZ

EP

Metropolis algorithmMetropolis algorithm

M. MM. Méézard, G. Parisi, and R. Zecchina, zard, G. Parisi, and R. Zecchina, Science Science 297, 297, 812 (2002)812 (2002)

Page 10: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

• Construct a ``cooling schedule’’Construct a ``cooling schedule’’

Page 11: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin
Page 12: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Neural circuit computationNeural circuit computation

J.J. Hopfield and D.W. Tank, Science J.J. Hopfield and D.W. Tank, Science 233233, 625 (1986), 625 (1986)

• Circuit element (``neuron’’) can be in one of two states (on/off: 0/1, Circuit element (``neuron’’) can be in one of two states (on/off: 0/1, spin up/spin down)spin up/spin down)

W.S. McCullough and W.H. Pitts, Bull. Math. Biophys. W.S. McCullough and W.H. Pitts, Bull. Math. Biophys. 55, 115 (1943), 115 (1943)

• Dynamics of ``neurons’’ given by Dynamics of ``neurons’’ given by

),,1()(1

, NiIR

uufT

dt

duC i

i

ijj

N

jji

ii

where where iu is the potential of neuronis the potential of neuron ii..

Page 13: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

• Dynamics corresponds to an energy functionDynamics corresponds to an energy function

jj

jjiji

ij VIVVTE ,2

1

Page 14: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

D.L. Stein, ed., D.L. Stein, ed., Spin Glasses and BiologySpin Glasses and Biology (World Scientific, Singapore, 1992) (World Scientific, Singapore, 1992)

Protein Conformational Dynamics Protein Conformational Dynamics

MyoglobinMyoglobin

Page 15: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

• Fluctuations important for biological processes (e.g., ligand diffusion)Fluctuations important for biological processes (e.g., ligand diffusion)

• Recombination experiments imply many conformational substatesRecombination experiments imply many conformational substates

A. Osterman A. Osterman et alet al., ., NatureNature 404404, 205 (2000), 205 (2000)

Page 16: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

D.L. Stein, Proc. Natl. Acad. Sci. USA D.L. Stein, Proc. Natl. Acad. Sci. USA 8282, 3670 (1985), 3670 (1985)

Spin Glass Model of Protein Conformational SubstatesSpin Glass Model of Protein Conformational Substates

Page 17: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Protein FoldingProtein Folding

• Levinthal paradoxLevinthal paradox

• ``Principle of minimal frustration’’``Principle of minimal frustration’’

J.D. Bryngelson and P.G. Wolynes, J.D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. USAProc. Natl. Acad. Sci. USA 8484, 7524 (1987), 7524 (1987)

Page 18: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Folding landscapes as a ``rough funnel’’Folding landscapes as a ``rough funnel’’

C.L. Brooks III, J.N. Onuchic, and D.J. Wales, C.L. Brooks III, J.N. Onuchic, and D.J. Wales, ScienceScience 293293, 612 (2001), 612 (2001)

Used to develop algorithms for structure prediction (J. Pillardy Used to develop algorithms for structure prediction (J. Pillardy et alet al., ., PNAS 98, 2329 2001); designing ``knowledge-based potentials for PNAS 98, 2329 2001); designing ``knowledge-based potentials for

fold recognition; etc.fold recognition; etc.

Page 19: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

Back to spin glasses proper …Back to spin glasses proper …

By now, it’s (hopefully) clear that understanding the behavior of these By now, it’s (hopefully) clear that understanding the behavior of these systems is important not only for condensed matter physics and systems is important not only for condensed matter physics and

statistical mechanics, but for many other fields as well…statistical mechanics, but for many other fields as well…

… … so we will now turn to examine what we know about them.so we will now turn to examine what we know about them.

Unfortunately, understanding their nature has been very difficult --- Unfortunately, understanding their nature has been very difficult --- theoretically, experimentally, and numerically!theoretically, experimentally, and numerically!

Page 20: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

The Geometry of ``Information Propagation’’The Geometry of ``Information Propagation’’

xy

yxxyJ J

H x

xxh

Page 21: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin

The Sherrington-Kirkpatrick (SK) Model The Sherrington-Kirkpatrick (SK) Model

i

iiji

ji hJ ijNN

1

h,J,

H

Phase transition with TPhase transition with Tcc=1. What is the thermodynamic =1. What is the thermodynamic

structure of the low-temperature phase?structure of the low-temperature phase?

``Infinite-range’’ model – no geometry left!``Infinite-range’’ model – no geometry left!

D. Sherrington and S. Kirkpatrick, D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett.Phys. Rev. Lett. 3535, 1792 (1975)., 1792 (1975).

``Mean-field’’ model; infinite-dimensional model.``Mean-field’’ model; infinite-dimensional model.

Broken replica symmetry --- one of the biggest surprises of all. Broken replica symmetry --- one of the biggest surprises of all.

Stay tuned …Stay tuned …

Page 22: Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin