Some background on nonequilibrium and disordered systems S.N. Coppersmith

1
Some background on nonequilibrium and disordered
systemsS.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

2
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?

3
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

4
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)

5
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)

6
Scale invariance and renormalization group the
existence of scale invariance is enough to find
the exponents characterizing it

• Simplest example (Feigenbaum)
• Consider logistic equation xj1?xj(1-xj)
  with ?3.57

xj-1/2
xj-1/2
j
j
every j plotted
every other j plottedordinate upside down
Resulting time series x1, x2, has property that
it looks the same except for a rescaling when
every other point is plotted -?z2j zj
(zjxj-1/2)
7
Scale invariance ?quantitative prediction of
exponent values

• -?z2j zj

zj
zj
j
j
8
Scale invariance ?quantitative prediction of
exponent values

• -?z2j zj
• -?z2(j1) zj1
• Write zj1g(zj)
• -?g(g(z2j)) g(zj)
• -?g(g(-zj/?)) g(zj)

This nonlinear eigenvalue equation for g only has
a solution (for gs that can be expanded in
Taylor series) when ?2.5029.
? Scale invariance only can occur with particular
values of the scaling exponents.
9
Now show that scale invariance ? exponent
determined

• -?g(g(-y/?)) g(y)
• Expand g in Taylor Series g(y) a - by2

10
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

11
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
• -?(1ab) 1 ? ab-(11/?)
• 2ab -? ? -2(11/?) -? ? ?2-2?-20

only ab enters
so, to this order ?(2v5)/22.12
12
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.

13
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?
14
Very fast rise in viscosity as temperature
lowered toward glass transition

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

Debenedetti Stillinger, Nature (2001)
note 1 year 3?107 seconds
15
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?

16
Kauzmann Entropy paradox appears to occur at
nearly the same temperature as the apparent
divergence of the viscosity

• Lubchenko and Wolynes (2006)

17
Glassy systems have rugged energy landscapes
Cartoon of free energy surface
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?
18
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)

19
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 ?
  jammingConsider models at zero temperature
  with geometrical constraints (Liu Nagel, Biroli
  et al.)

20
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)

21
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.
22
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)

23
Another point of view glass transition is just
one manifestation of jamming.
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
Temperature
Stress
Density
A.J. Liu and S.R. Nagel
In this view, glasses are fundamentally similar
to granular materials.
24
Intro to granular materials
Definition Collection of classical particles
interacting only via contact forces negligible
particle deformations
Why study 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

25
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?

26
Unjammed versus jammed configurations
27
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.
28
Lattice models of jamming

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

1) Occupy sites on a lattice with probability p,
29
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
30
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
31
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
32
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
33
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
34
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
35
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
36
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
37
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
38
K-core or bootstrap percolation
K3
1) Occupy sites on a lattice with probability
p, 2) If an occupied site has fewer than K
occupied neighbors, empty it.
Particles remain only if they have enough
neighbors Coordination number is discontinuous
at transition (similar phenomenology to number of
contacts at jamming transition)
39
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?

40
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)

41
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
42
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)

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
43
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