Transcript Susan Coppersmith - American Institute of Mathematics

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
xj-1/2
xj-1/2
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j
every j plotted
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every other j plotted
ordinate 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
(zj=xj-1/2)
j
Scale invariance quantitative prediction of
exponent values
-z2j = zj
zj
zj
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are needed to see this picture.
j
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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
only ab enters
-(1+ab) = 1  ab=-(1+1/)
2ab = -  -2(1+1/)= -  2-2-2=0
so, to this order: ≈(2+√5)/2≈2.12
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.
Can other nonequilibrium systems be
understood using this paradigm?
• Classic nonequilibrium system:
glass
– Technologically useful since
antiquity
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– Glass state is what many liquids
reach when cooled quickly enough
Is glass a phase, or is it a frozen liquid?
Very fast rise in
viscosity as
temperature
lowered toward
glass transition
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• Is glass transition
a phase transition,
or just a kinetic
freezing process?
Debenedetti & Stillinger, Nature (2001)
note: 1 year =
3107
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)
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?
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?
or
What 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:
H   JijSizS jz    Six
i, j 
i
• Spin glass models describe real physical systems (e.g.,
CuMn, LiYxHo1-xF4)
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 shortrange interactions is controversial (Fisher & Huse, Bray
and Moore, Newman and Stein)
Another point of view: glass transition is just
one manifestation of “jamming”.
Proposed jamming
phase diagram
Temperature
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.
Density
A.J. Liu and S.R. Nagel
In this view, glasses are fundamentally similar to granular materials.
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
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
QuickTime™ and a
TIFF (LZW) decompressor
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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
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
1) Occupy sites on a lattice with
probability p,
2) If an occupied site has fewer than
K occupied neighbors, empty it.
K-core or bootstrap percolation
K=3
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)
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 “easyhard” 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)
P is not a phase, but functions in P are either
in or close to non-generic phases
all Boolean functions
low order
polynomials
xi mod 3
majority
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