Transcript AIM, Palo Alto, August 2006 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. T.

AIM, Palo Alto, August 2006
QuickTime™ and a
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T. Witten, University of Chicago
Elements of a phase transition
0 0 1 0 1
• a set of N variables, e.g. "spins" {si} = 0 or 1:
any particular choice is a configuration
0 1 1 0 1
• a list or graph of connections between variables
e.g. square lattice with neighbors connected
1 0 1 1 0
0 1 0 1 0
• statistical weight for each configuration
eg "each connected pair of unlike spins
reduces weight by factor x"
0 0 1 1 0
1
• a rule for joining joining N systems by adding connections
eg. enlarge lattice.
Such a system may have a phase transition:
• averages change non-smoothly with weights
eg
d s
x = xc
dx
N

s
Phase 1
Phase 2
Non-smoothness requires N ––emergent
Variables are influenced by an infinite number of
others
0
1
x
Equilibrium phase transitions
–T. Witten, University of Chicago
• distinctions: two non-phase transitions
• varieties of phase transition
• nature of the transition state
Complex emergent structure with mysterious regularities
• theoretical methods
Mean field theory
Renormalization theory
Conformal symmetry
Stochastic Loewner evolution
• questions for workshop
Phase transition versus ?
• non abrupt: crossover
s
0
• non-emergent: abrupt without N
x
1
Ising model: paradigm phase transition
0 0 1 0 1
Probabilities P{s} are set by a "goal function" H{s}
0 1 1 0 1
Goal: all s's are the same as their connected neighbors
L
0 1 0 1 0

(sc1 - 1/2)(sc2 - 1/2)
connections
c=1
Specifically: H{s} = -J
Unlike neighbors  increased H  decreased probability P:
P{s} = (constant) 
connections
c
1 0 1 1 0
e-J/2 (= x)
1
if sc1 sc2
if sc1= sc2`
Has a phase transition with discontinuous derivitive d s
dx
… at x = xc N (and L)
0 0 1 1 0
1
Generalizing Ising model
Variables: Z2 (Ising), Zn (clock), O2 (superfluid helium) On (magnets) …
In general the variables are group elements and the goal function depends
on the group operation relating the connected elements
Graph: d-dimensional lattice, continuous space, Cayley tree …
connections beyond nearest-neighbor
connections among k > 2 variables
Goal function: seeks like values of variables / seeks different values
Satisfiable in finite fraction, vanishing fraction (Ising), or none (frustrated)
of the configurations
Definite / random:
eg, each connection has weight x or 1/x chosen randomly: spin glass
Probabilities: thermal equilibrium (Ising): each graph element contributes one
factor to the probability; factor depends on goal function on that element.
kinetic: configurations are generated by a sequential, stochastic
process whose probabilities are dictated by a goal function
Stochastic growth processes
"nearly empty lattice" generalizations
Self-avoiding random walk
Goal function allows only sites that form a linear sequence
from the origin.
Penalty factor x for each site
Random animals
Goal function allows only cluster of connected sites
Penalty x for each site
percolation
Place s variables on a lattice at random with a given s = x
Determine largest connected cluster
All have phase transitions: as x  xc , size of cluster grows to a nonzero
fraction of the lattice
Varieties of phase-transition behavior
How non-smooth?
Discontinuous: expectation values eg s jump at transition point eg boiling
aka first-order, subcritical
Continuous: only derivitives of s are discontinuous: eg Ising model, critical
opalescence
aka second-, third- …order, critical
The transition state
discontinuous
Uniform regions: eg liquid and vapor
State depends on history, boundary conditions
continuous
heterogeneous
System is uniform over length scales >
"correlation length" 


complicated
x  xc
slow: correlation time  diverges at xc
Jochen Voss: http://seehuhn.de/mathe/ising-0.219.jpg
Character of heterogeneity: dilation invariance
Dilated configurations are indistinguishable
Eg random walks with steps much smaller than resolution
4 of these are zoomed 3x relative to the other four. Can you tell which?
Dilation invariance means that statistical
averages are virtually unchanged by dilation:
s(0) s(r1)s(r2)…s(rk)
= -Ak s(0) s(r1)s(r2)…s(rk)
A is the "scaling dimension" of s
This invariance appears to hold generally for
continuous phase transitions
A governs response of length  to a change of x
 ~ (x - xc)-1/(d-A)
A set of "critical exponents" like A dictate response near
the transition point
Character of the transition state: universality
Discrete universality classes
Critical exponents like A are often invariant under continuous changes of the system
Eg Ising model on different lattices, different variables, different goal functions
Eg liquid vapor, ferromagnet, liquid demixing, ising
These changes are called "irrelevant variables"
"Relevant" features do change the exponents:
Symmetry of the system: Z2 (Ising), O2, … O3
Spatial dimension d
Theoretical understanding of phase transitions
•
•
•
•
•
•
Mean-field theory
Renormalization theory
Conformal symmetry
Stochastic Loewner evolution
(Defects (local departures from goal state) ….omitted)
(replica methods, omitted)
Mean-field approximation: neglect nonuniformity
In H(s), replace spins connected to sI by their average s
This converts system to a single degree of freedom sI
One can readily compute eg thermodynamic free energy F(x,s) for assumed s
The actual s is that which minimizes free energy.
Mean field theory accounts for phase transitions.
Average s  "magnetization" M takes the value that minimizes free energy
free energy at temperature = 1
-0.58
-0.6
p
p
-0.62
energy
entropy
-0.64
J = .45
J = .55
-0.66
-0.68
0.2
0.4
0.6
magnetization M
Inherently independent of space dimension d
0.8
1
Renormalization theory
To explain dilation symmetry requires a dilation-symmetric description
To describe large-scale structure, we may remove small-scale structure of {s} by
averaging over distance  >> graph connections to produce s(r)
If H[s] describes the system and s is dilation symmetric, then
H2 must have the same functional form as H
One infers the mapping H H2 from the weights P[s2]:
H P[s]
Local spatial averaging:
2
P[s2] H2
Eg for d-dimensional Ising model with d 4 at transition
H
 ( s)2 + g*(d) s4
––Wilson, Fisher 1972
From the transformation of H near transition point, one infers the effect of
dilation:
Many H's converge to same H:
"scaling dimension" A of s
explains universality
other critical exponents.
Conformal symmetry for 2-dimensional systems
configuration in z plane
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
g(z) shows deformed
configuration
Q
and a
Any analytic functionTIFaFre( UnneceodmupicreksTsimeed™
e d to s e ) d e c omp r e ss o
e th is p ic
r
tu re .
g(z)
Conformal symmetry: deformed configuration has same statistical weight as
original.
Critical exponents label a discrete set of representations of conformal group
Cf discrete angular momentum representations of rotation symmetry
Restricted to d=2
–Shankar, Friedan ~1980
Loewner: analytic map implements self-avoidance
Z
g(z) = a +
((z-a)2 + 4t)1/2
g
z0
t)
a
Any self-avoiding curve has a g(z), analytic in
a
Growing curve: tip z0 came from some point at on the real axis.
By varying at with time, we can make “arbitrary” self avoiding curves.
Note: if at is held fixed for t, z0 “diffuses upward” with diffusion constant 4
Loewner: affect of at on gt(z) is local in time:
Each point on the current curve gt(z) feels Coulomb repulsion from at.
Cf. John Cardy cond-mat/0503313
Schramm: Loewner growth with Brownian at
suppose at diffuses, with ( at)2  =   t
Curve is dilation symmetric (fractal) with fractal
dimension D = 1 - /8
Many known random lattice structures are Schramm curves, differing only in 
Self-avoiding walk:  = 8/3
Ising cluster perimeters:  = 3
Percolation cluster boundaries:  = 6
…
Brownian motion replaces field theory to explain many known universal dilationsymmetric structures.
… simple random walks can create a new range of “correlated” objects.
Generalizing the notion of phase transitions
Processes in this workshop (eg k-sat) act like phase transitions
Sharp change of behavior
Emergent structure
Scale invariance
complexity
Without the features thought fundamental in conventional phase transition
Statistical mechanics description
Spatial dimensionality.
Can we free phase transitions from these features to get a deeper
understanding?
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Correlation qualitatively alters structure
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Random walk: no correlation
Self-avoiding walk: correlation
Size ~ length1/2
Size ~ length3/4
 central limit theorem
Strong modifications needed to defeat
central limit theorem.
…interacting field theory
3/4 exponent was only known by
simulation
conjectures
Future hopes for Schramm-Loewner (SLE)
SLE shows that many known dilation-symmetric structures are simple kinetic,
random walks, viewed through the distorting lens of evolving maps
Can it be generalized to account for branched objects, eg. Random animals
Can it describe stochastic growth phenomena, eg. DLA?
Can it be generalized to higher dimensions?
Random structures like self avoiding walks are universal fractals in any
dimension
If they are Schramm-like in two-dimensions, why not in others?
What plays the role of gt(z)?
Displacement field of mapping a region into itself
Must preserve topology
Must preserve dilation symmetry.
Must have a singular point: growing tip.