Glassy  systems
Condensed  and  soft-­‐matter  systems,....,  computer  science  (optimization   problems),  economics  (agent  based  models)...
Super-­‐cooled  liquids  and  structural  glasses
Colloidal  liquids  and  colloidal  glasses

Slow  Dynamics  and  the  Glass  Transition
T g
/T
• Relaxation  time:
picoseconds  -­‐>  days
• It   takes   one   second   to   a   molecule   to   move   of   one
Angstrom!
exp D
T
0
T T
0

Slow  Dynamics  and  the  Glass  Transition
T g
/T
• Relaxation  time:
picoseconds  -­‐>  days
• It   takes   one   second   to   a   molecule   to   move   of   one
Angstrom!
exp D
T
0
T T
0

Slow  Dynamics  and  the  Glass  Transition
T g
/T
• Relaxation  time:
picoseconds  -­‐>  days
• It   takes   one   second   to   a   molecule   to   move   of   one
Angstrom!
exp D
T
0
T T
0

Disordered  structure
Liquid Glass
A  snapshot  of  a  glass  looks  like  the  one  of  a  liquid!
W.  Krauth

H =
X
U ( x i x i
) i<j
Z =
X exp( H )
C
Newtonian  Dynamics
Langevin    Dynamics
U(r) r
Super-­‐cooled  liquids Colloids

A  huge  number  of  minima,  saddles  and  maxima
38  LJ  particles
D.J.  Wales  et  al.

•Huge  number  of  competing  low  temperature  phases
•What  kind  of  long-­‐range  order:  amorphous  order?
•Properties  of  the  ideal  glass  state
•Dynamics  in  such  a  complex  landscape

•Huge  number  of  competing  low  temperature  phases
•What  kind  of  long-­‐range  order:  amorphous  order?
•Properties  of  the  ideal  glass  state
•Dynamics  in  such  a  complex  landscape

Mean-­‐Field  Theory  for  Rugged  Energy  Landscapes
• A  lot  of  metastable  states  below  a  characteristic  temperature
N ( f ) = exp( N s c
( f ))
• Competition  between  free-­‐energy  and  conQigurational  entropy
Z
Z =
Z
= df N ( f ) exp( N f ) df exp( N [ s c
( f ) f ]) s c
1
=
⇥ s c
( f )
⇥ f f
Ideal Glasses
A long story from the Random Energy Model to Hard Spheres in infinite dimensions

Mean-­‐Field  Theory  for  Rugged  Energy  Landscapes
• A  lot  of  metastable  states  below  a  characteristic  temperature
N ( f ) = exp( N s c
( f ))
• Competition  between  free-­‐energy  and  conQigurational  entropy
Z
Z =
Z
= df N ( f ) exp( N f ) df exp( N [ s c
( f ) f ]) s c
2
=
⇥ s c
( f )
⇥ f f
Ideal Glasses
A long story from the Random Energy Model to Hard Spheres in infinite dimensions

Mean-­‐Field  Theory  for  Rugged  Energy  Landscapes
• A  lot  of  metastable  states  below  a  characteristic  temperature
N ( f ) = exp( N s c
( f ))
• Competition  between  free-­‐energy  and  conQigurational  entropy
Z
Z =
Z
= df N ( f ) exp( N f ) df exp( N [ s c
( f ) f ]) s c
K
=
⇥ s c
( f )
⇥ f f
Ideal Glasses
A long story from the Random Energy Model to Hard Spheres in infinite dimensions

The  dynamical  (Mode-­‐Coupling)  transition
Q ( t ) :
1
N
1
X h s i
( t ) s i
(0) i
N
X h exp
✓ i
( x j
( t ) x 0 i
(0)) 2
2 a 2 i,j
◆ i
•Power  law  divergence  of  the  relaxation  time
•Discontinuous  Edwards  Anderson  Parameter
•Singular  behavior  in  time  before  the  transition q

Random  First  Order  Transition
Kirkpatrick,  Thirumalai  and  Wolynes,...
Franz-­‐Parisi
Potential
X exp ( H ( C ))
Z
C
[ Q ( C , C eq
) q ] = exp( N V ( q )) s c
•Mode   Coupling   Cross-­‐Over   analogous  to  a  spinodal
•Glass   Transition   analogous   to   a   phase  transition  of  Ising  in  a  Qield
Overlap                                                                                Magnetization
ConWigurational  entropy                  Magnetic  Wield

Pro  and  Cons  of  Mean-­‐Field  Theory
•Ideal  Glass  Transition  as  condensation  transition
•Qualitative  phase  diagrams  (structural  glasses,  colloids,...)
•Behavior  of  static  intensive  observables  (g(r),  speciQic  heat,...)
•No  stable  states  below   T
M CT
•Wrong  description  of  the  dynamics

Pro  and  Cons  of  Mean-­‐Field  Theory
•Ideal  Glass  Transition  as  condensation  transition
•Qualitative  phase  diagrams  (structural  glasses,  colloids,...)
•Behavior  of  static  intensive  observables  (g(r),  speciQic  heat,...)
•No  stable  states  below   T
M CT
•Wrong  description  of  the  dynamics

Analytical  Frameworks
•Replica  Field  Theory  for  the  statics
•Dynamical  Field  Theory
•Replica  Qield  Theory  for  the  dynamics

Replica  Field  Theory
R q
↵ , R
( x ) |
B
=1
D q
↵ , exp ( L ( q
↵ ,
( x ))) O ( q
1 ,m
) q
↵ ,
( x ) |
B
=1
L ( q
↵ ,
( x )) =
Z
D q
↵ , exp ( L ( q
↵ ,
( x ))) d d x
2
4
X
↵ <
( r q
↵ ,
( x )) 2
2
= 1 , . . . , m !
1
3
+ H ( q
↵ ,
( x ))
5
Mean-­‐Field  Theory: q
↵ ,
( x ) = q &  saddle  point  method
H ( q
↵ ,
( x )) = ( m 1) V ( q )

Dynamical  Field  Theory
Similar  procedure  but
Mean-­‐Field  Theory:                                                                                                  and  saddle  point  method
Mode-­‐Coupling  Theory  equations

Replica  Field  Theory  for  the  dynamics quasi-­‐equilibrium  inside  a  “state” perturbation  around  the  secondary  minimum

Static  length-­‐scale
Liquid Glass

Dynamical  length-­‐scales
Hedges  et  al,  3D  KA  liquid
Local  overlap q ( x ; t, 0)
Dynamical  length-­‐scales  grow  approaching  the  glass  transition
L.  Berthier,   G.B.  J.-­‐P.   Bouchaud,   L.  Cipelletti,  W.   van   Saarloos,   Dynamical  heterogeneity   in   glasses,  colloids  and  granular  media ,  Oxford  University  Press  2011

Measurements  of  Point-­‐to-­‐Set  length
GB,  JP  Bouchaud,  A.  Cavagna,  T.S.  Grigera,  P.  Verrocchio,  Nature  Physics,  2007   and  several  other  studies  later  on.
•PS  length-­‐scale  grows  (but  mildly)
•The  relationship  with  the  time-­‐scale  is,  for  the  moment,  unknown

Measurements  of  dynamic  length
Berthier,  GB,  Bouchaud  et  al.  Science  ’05;  PRE  ’07 Flenner,  Zhang,  Szamel  (Hard  Spheres  Mixture)    PRE  ’11
•Qualitative  behavior  is  in  agreement  with  theory
•“Critical”  exponents  are  not  in  agreement  with  MF  theory  (                                                                    )

Qualitative  facts  not  accounted  for
/ exp( ` 3 F/k
B
T )
1 + exp( ` 3 F/k
B
T )
Where  is  the  sharp  change?
Candelier,  GB,  Dauchot  et  al.   Relaxation  close  to  MCT:  dynamical  facilitation  and  avalanches,   quite  different  from  spinodal  Qluctuations

Gradenigo,  GB  (ongoing)
FLUCTUATIONS V(q) N = 70
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1
sample A sample B sample C
0.2
0.3
0.4
q
0.5
0.6
0.7
0.8

/ exp( ` 3 F/k
B
T )
1 + exp( ` 3 F/k
B
T )
Smooth  change  since            and              Qluctuate
GB,  JP  Bouchaud,  A.  Cavagna,  T.S.  Grigera,  P.  Verrocchio,  Nature  Physics,  2007

Fytas,  Malakis  ’08
Para
Strong  disorder  destroys  the   transition  in  the  RFIM
Ferro
How  strong  is  the  self-­‐induced  disorder  ?
Too  strong  in  several  disordered  lattice  models:  no  transition  and  different   physics   Cammarota,  GB,  Tarjus,  Tarzia  ’13,...
Not  known  yet  in  glass-­‐forming  liquids:  crucial  and  within  reach!

Avalanches  &  dynamical  facilitation
This  is  expected  for  a  spinodal  in  presence  of  disorder,  cf.  RFIM
Taking   into  account   the  effect   of   disorder   is   crucial   to   understand   the  critical   d = 8
Nandi,  GB,  Tarjus,  Parisi  (to  appear)
Rizzo  ’13,’14

Replica  Field  Theory
X e
C eq
H ( C eq
)
P
Z e
C
H ( C ) ⇧ x 2 B
[ e H ( C ) ⇧ x 2 B q (
[
C q
,
(
C
C , eq
C
, x eq
)
, x )
1] O
1]
( q )

Replica  Field  Theory
X e
C eq
X e
C eq
H ( C eq
)
P
Z e
C
H ( C ) ⇧ x 2 B
[ e H ( C ) ⇧ x 2 B q (
[
C q
,
(
C
C , eq
C
, x eq
)
, x )
1] O
1]
( q )
H ( C eq
)
Z
P
P
C
C e H ( C ) ⇧ x 2 B e H ( C ) ⇧ x 2 B
[ q ( C , C eq
, x ) 1] O ( q )
1 n
[ q ( C , C eq
, x ) 1]

Replica  Field  Theory
X e
C eq
X e
C eq
H ( C eq
)
P
Z e
C
H ( C ) ⇧ x 2 B
[ e H ( C ) ⇧ x 2 B q (
[
C q
,
(
C
C , eq
C
, x eq
)
, x )
1] O
1]
( q )
H ( C eq
)
Z
P
P
C
C e H ( C ) ⇧ x 2 B e H ( C ) ⇧ x 2 B
[ q ( C , C eq
, x ) 1] O ( q )
1 n
[ q ( C , C eq
, x ) 1]
X e
C eq
H ( C eq
) X e
Z
C
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1] O ( q )
!
X e
C
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1]
!
n 1

Replica  Field  Theory
X e
C eq
H ( C eq
) X e
Z
C
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1] O ( q )
!
X e
C
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1]
!
n 1

Replica  Field  Theory
X e
C eq
H ( C eq
) X e
Z
C
X
C eq e
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1] O ( q )
!
X e
C
H ( C eq
)
0
@
Z
X
C
1
,..., C n e
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1]
!
n 1
1
H ( C
1
) ··· H ( C n
)
⇧ a,x 2 B
[ q ( C a
, C eq
, x ) 1] O ( q
1 , eq
)
A

Replica  Field  Theory
X e
C eq
H ( C eq
) X e
Z
C
X
C eq e
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1] O ( q )
!
X e
C
H ( C eq
)
0
@
Z
X
C
1
,..., C n e
H ( C )
⇧ x 2 B
[ q ( C , C eq
, x ) 1]
!
n 1
1
H ( C
1
) ··· H ( C n
)
⇧ a,x 2 B
[ q ( C a
, C eq
, x ) 1] O ( q
1 , eq
)
A
P
C
1
,..., C n
, C eq e H ( C
1
) ··· H ( C n
) H ( C eq
) ⇧ a,x 2 B
Z
[ q ( C a
, C eq
, x ) 1] O ( q
1 ,eq
)

Replica  Field  Theory
P
C
1
,..., C n
, C eq e H ( C
1
) ··· H ( C n
) H ( C eq
) ⇧ a,x 2 B
Z
[ q ( C a
, C eq
, x ) 1] O ( q
1 ,eq
)

Replica  Field  Theory
P
C
1
,..., C n
, C eq e H ( C
1
) ··· H ( C n
) H ( C eq
) ⇧ a,x 2 B
Z
[ q ( C a
, C eq
, x ) 1] O ( q
1 ,eq
)
P
C
1
,..., C m e H ( C
1
) ··· H ( C m
)
Z
⌦
( B ) q
↵ ,
=1
O ( q
1 ,m
) m !
1
↵ = 1 , . . . , m

Replica  Field  Theory
P
C
1
,..., C n
, C eq e H ( C
1
) ··· H ( C n
) H ( C eq
) ⇧ a,x 2 B
Z
[ q ( C a
, C eq
, x ) 1] O ( q
1 ,eq
)
P
C
1
,..., C m e H ( C
1
) ··· H ( C m
)
Z
⌦
( B ) q
↵ ,
=1
O ( q
1 ,m
) m !
1
↵ = 1 , . . . , m
P
C
1
,..., C m
P e
C
1
,..., C m
H ( C
1
) ··· H ( C m
) ⌦
( B ) q
↵ , e H ( C
1
) ··· H ( C m
) ⌦
=1 q
↵ ,
O ( q
1 ,m
)
( B )
=1

Replica  Field  Theory
P
C
1
,..., C n
, C eq e H ( C
1
) ··· H ( C n
) H ( C eq
) ⇧ a,x 2 B
Z
[ q ( C a
, C eq
, x ) 1] O ( q
1 ,eq
)
P
C
1
,..., C m e H ( C
1
) ··· H ( C m
)
Z
⌦
( B ) q
↵ ,
=1
O ( q
1 ,m
) m !
1
↵ = 1 , . . . , m
P
C
1
,..., C m
P e
C
1
,..., C m
H ( C
1
) ··· H ( C m
) ⌦
( B ) q
↵ , e H ( C
1
) ··· H ( C m
) ⌦
=1 q
↵ ,
O ( q
1 ,m
)
( B )
=1 m !
1 replicas  that  coincide  on  the  boundary

Replica  Field  Theory
P
C
1
,..., C m
P e
C
1
,..., C m
H ( C
1
) ··· H ( C m
) ⌦
( B ) q
↵ , e H ( C
1
) ··· H ( C m
) ⌦
=1 q
↵ ,
O ( q
1 ,m
)
( B )
=1

Replica  Field  Theory
P
C
1
,..., C m
P e
C
1
,..., C m
H ( C
1
) ··· H ( C m
) ⌦
( B ) q
↵ , e H ( C
1
) ··· H ( C m
) ⌦
=1 q
↵ ,
O ( q
1 ,m
)
( B )
=1
0
@
X
C
1
,..., C m e
1
H ( C
1
) ··· H ( C m
)
[ q
↵ ,
( x ) q ( C
↵
, C , x ))] = exp ( L ( q
↵ ,
( x )))
A

Replica  Field  Theory
P
C
1
,..., C m
P e
C
1
,..., C m
H ( C
1
) ··· H ( C m
) ⌦
( B ) q
↵ , e H ( C
1
) ··· H ( C m
) ⌦
=1 q
↵ ,
O ( q
1 ,m
)
( B )
=1
0
@
X
C
1
,..., C m e
1
H ( C
1
) ··· H ( C m
)
[ q
↵ ,
( x ) q ( C
↵
, C , x ))] = exp ( L ( q
↵ ,
( x )))
A
R q
↵ , R
( x ) |
B
=1 q
↵ ,
D q
↵ , exp ( L ( q
↵ ,
( x ))) O ( q
( x ) |
B
=1
D q
↵ , exp ( L ( q
↵ ,
( x )))
1 ,m
)