Some background on nonequilibrium and disordered systems

S.N. Coppersmith

• Equilibrium versus nonequilibrium systems — why is nonequilibrium so much harder?

• Concepts from non-random systems that have proven useful for understanding some nonequilibrium systems

– phase transitions

– scaling and universality

• Remarks on glasses

• Remarks on granular materials

• Remarks on usefulness of these concepts for problems in computational complexity

Thermal equilibrium versus the real world

• Thermal equilibrium is the state matter reaches when you wait long enough without disturbing it– If energy functional E({configuration}) known,

Probability(configuration) exp(-E/kBT)

• Many systems are not in thermal equilibrium– Disordered systems (equilibration times very long)– Strongly driven systems– Configuration observed typically depends on system preparation

• What concepts are useful for understanding systems out of thermal equilibrium?

Powerful concepts that apply to equilibrium systems

• Phases of matter– Liquid, solid, gas– Ferromagnet, paramagnet

…..

• Scale invariance near some phase transitions– Power laws– Scaling relations between exponents– Renormalization group (Kadanoff, Wilson, Fisher)– Universality

Concepts useful for equilibrium phase transitions have been show to apply to some

other nonequilibrium situations

• Phase transitions:– Depinning of driven elastic media with randomness (D. Fisher)– “Flocking” (Toner, Tu)– Oscillator synchronization (Kuramoto)

• Scale invariance:– Transition to chaos (Feigenbaum)

• Describes nonlinear dynamics of driven damped oscillators

• Scale invariance associated with phase transition

– Diffusion-limited aggregation (Witten-Sander)– “Self-organized criticality” (Bak, Tang, Wiesenfeld)

Concepts useful for equilibrium phase transitions have been show to apply to some

other nonequilibrium situations

• Phase transitions:– Depinning of driven elastic media with randomness (D. Fisher)– “Flocking” (Toner, Tu)– Oscillator synchronization (Kuramoto)

• Scale invariance:– Transition to chaos (Feigenbaum)

• Describes nonlinear dynamics of driven damped oscillators

• Scale invariance associated with phase transition

– Diffusion-limited aggregation (Witten-Sander)– “Self-organized criticality” (Bak, Tang, Wiesenfeld)

Scale invariance and renormalization group: the existence of scale invariance is enough

to find the exponents characterizing it

• Simplest example (Feigenbaum)

Consider “logistic equation” xj+1=xj(1-xj) with =3.57

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xj-1/2

j

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Resulting time series x1, x2, … has property that it looks the same except for a rescaling when every other point is plotted:

-z2j = zj (zj=xj-1/2)

every j plotted every other j plottedordinate upside down

xj-1/2

j

Scale invariance quantitative prediction of exponent values

-z2j = zj

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zjzj

j j

Scale invariance quantitative prediction of exponent values

-z2j = zj

-z2(j+1) = zj+1

Write zj+1=g(zj)

-g(g(z2j)) = g(zj)

-g(g(-zj/)) = g(zj)This nonlinear eigenvalue equation for g only has a solution (for g’s that can be expanded in Taylor series) when =2.5029….

Scale invariance only can occur with particular values of the scaling exponents.

Now show that scale invariance exponent determined

-g(g(-y/)) = g(y)Expand g in Taylor Series: g(y) = a - by2 + …

Now show that scale invariance exponent determined

-g(g(-y/)) = g(y)Expand g in Taylor Series: g(y) = a - by2 + …Calculate to order y2:

-a - b(a-b(y/)2)2 = a - by2

-a - b(a2-2ab(y/)2) = a - by2

Now show that scale invariance exponent determined

-g(g(-y/)) = g(y)Expand g in Taylor Series: g(y) = a - by2 + …Calculate to order y2:

-a - b(a-b(y/)2)2 = a - by2

-a - b(a2-2ab(y/)2) = a - by2

Equate coefficients of y0 and y2:-a-ba2 = a; 2ab2/ = -b

-(1+ab) = 1 ab=-(1+1/) 2ab = - -2(1+1/)= - 2-2-2=0

so, to this order: ≈(2+√5)/2≈2.12

only ab enters

Scale invariance is often associated with phase transitions

• Examples:

– Logistic map: scale invariance at value of at which the “transition to chaos” between periodic and chaotic time series occurs

– Ferromagnet: scale invariance at temperature at which there is a transition between ferromagnetic and paramagnetic phases

– Percolation: scale invariance when probability of site occupation is at the value at which a giant cluster of occupied sites first appears.

Can other nonequilibrium systems be understood using this paradigm?

• Classic nonequilibrium system: glass– Technologically useful since

antiquity

– Glass state is what many liquids reach when cooled quickly enough

Is glass a phase, or is it a frozen liquid?

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Very fast rise in viscosity as temperature

lowered toward glass transition

• Is glass transition a phase transition, or just a kinetic freezing process?

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Debenedetti & Stillinger, Nature (2001)note: 1 year = 3107 seconds

Kauzmann paradox (Kauzmann, 1948)

• “Entropy crisis” — extrapolation of entropies of crystal and glass would yield unphysical “negative entropy difference,” so something must happen

• Crossover or phase transition?

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Kauzmann Entropy “paradox” appears to occur

at nearly the same temperature

as the apparent divergence of the

viscosity

• Lubchenko and Wolynes (2006)

Cartoon of free energy surface

Glassy systems have rugged energy landscapes

Do energy barriers diverge as temperature is lowered towards glass transition?

Or, is the apparent transition just a smooth increase in barrier height plus an exponential dependence of relaxation rate on temperature?

Whether or not a glass transition exists is controversial

• Yes: – Coincidence of Kauzmann temperature and extrapolated

temperature where viscosity diverges– Nagel scaling (Dixon et al., Menon et al.)– Superexponential growth of relaxation times limits range of

experimental data

• No: – 2-d systems have lots of configurations that interpolate smoothly

between “glassy” and “crystalline” (Santen & Krauth, Donev, Stillinger, Torquato)

• “The deepest and most interesting unsolved problem in solid state theory is probably the nature of glass and the glass transition. This could be the next breakthrough in the coming decade.” P.W. Anderson, Science 267, 1615 (1995)

What is crucial physics underlying the behavior of glasses?

orWhat other, simpler models can give insight

into structural glasses?

• Finite temperature important models of spins with random couplings, at finite temperature

“spin glass” (Edwards & Anderson)

• Key physics is not thermal but geometric “jamming”Consider models at zero temperature with geometrical constraints (Liu & Nagel, Biroli et al.)

Trying to simplify the glass problem — Spin glasses (Edwards & Anderson, 1975)

• In structural glasses, disorder is not intrinsic (crystal typically has lower energy). Assume some “slow” degrees of freedom cause others to “see” random environment.

• So consider model with quenched disorder and random couplings:

• Spin glass models describe real physical systems (e.g., CuMn, LiYxHo1-xF4)

∑Γ+∑=>< i

ixjzj,i

izij SSSJH

Edwards-Anderson spin glass

• Ising spins with couplings of random sign

ferromagnetic bond

antiferromagnetic bond

(Ising spins at each vertex)

Three-dimensional spin glasses undergo a phase transition. Exact nature is still controversial.

Studies of spin glasses yield interesting results, possibly relevant to structural

glasses• Infinite range spin glass model (“mean field”) -- novel

broken-symmetry phase “replica symmetry-breaking”

• Multi-spin couplings yield phenomenology similar to structural glasses (Kirkpatrick et al., Mezard and Parisi)

– Dynamical phase transition at temperature above thermodynamic phase transition

• Relevance of mean-field results to models with short-range interactions is controversial (Fisher & Huse, Bray and Moore, Newman and Stein)

Another point of view: glass transition is just one manifestation of “jamming”.

A.J. Liu and S.R. Nagel

Temperature

Density

Stress

Liu and Nagel propose that glass transition (reached by lowering temperature) is fundamentally similar to “jamming” transition of large particles at zero temperature as density is increased.

Proposed jamming phase diagram

In this view, glasses are fundamentally similar to granular materials.

Intro to granular materials

• Practical importance– industrial (e.g. construction, roads, etc.)– agricultural (e.g. grain silos)

• Fundamental questions– system has many degrees of freedom, is far from

thermal equilibrium

“complex system”

+ amenable to controlled experiments

material has both solid- and liquid-like aspects

Definition: Collection of classical particles interacting only via contact forces [negligible particle deformations]

Why study granular materials?

Interesting aspects of granular materials

• Importance of dilatancy in determining response to external stresses

• Nonlinear dynamics and pattern formation• Can “effective temperature” be used to describe

effects of driving + collisions?• Is a given configuration a random sample from

an ensemble of configurations? (Edwards)• Statistics of stress propagation in stationary

systems• Jamming — as density is increased, how does

the material begin to support stress?

Unjammed versus jammed configurations

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Lattice models of jamming

• K-core or bootstrap percolation (Schwarz, Liu, Chayes)

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

Lattice models of jamming

• K-core or bootstrap percolation (Schwarz, Liu, Chayes; Toninelli et al.)

1) Occupy sites on a lattice with probability p,

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

K-core or bootstrap percolation

1) Occupy sites on a lattice with probability p,

2) If an occupied site has fewer than K occupied neighbors, empty it.

K=3

Particles remain only if they have enough neighbors

“Coordination number” is discontinuous at transition (similar phenomenology to number of contacts at jamming transition)

Questions

• How related are the dynamical phase transition in p-spin spin glasses and the K-core percolation transition?

• Do these models contain the essential physics underlying the behavior of structural glasses and/or granular materials?

Applications of ideas from phase transitions to problems in computational complexity

• SAT-unSAT transition for problems chosen from a random ensemble exhibits a phase transition that obeys scaling (Selman & Kirkpatrick)

• Cavity method from spin glasses can be used to characterize SAT-unSAT transition (Biroli, Mézard, Monasson, Parisi)– phase transition within SAT region in which solution

space breaks up into disconnected clusters (replica symmetry-breaking)

Is the renormalization group useful for studying problems in computational complexity?

• Renormalization group gives insight into “easy-hard” transition in satisfiability problems

• Renormalization approach to P versus NP question

Given Boolean function f(x1,x2,…,xN)

f(x1,x2,…,xN) f(0,x2,…,xN) f(1,x2,…,xN)

transforms function of N variables into one of N-1 variables

Renormalization group approach to characterizing P (problems that can be

solved in polynomial time)

f(x1,x2,…,xN) f(0,x2,…,xN) f(1,x2,…,xN)

all Boolean functions

low order polynomials

majorityxi mod 3

P is not a phase, but functions in P are either in or close to non-generic phases

functions in P that are close to low order polynomials

Summary

• Phase transitions and scale invariance have proven to be useful concepts for nonequilibrium systems, but general theoretical understanding is lacking

• Glasses and granular materials may have deep similarities, but general theoretical understanding is lacking