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T. Odagaki and T. EkimotoDepartment of Physics, Kyushu University
Ngai Fest September 16, 2006
Free Energy Landscape
Order parameter
High T
Low T
Phase transition
Fre
e en
ergy
Phase Transition
Configuration
High T
Fre
e en
ergy
Glass transition
Low T
: Diverging mean waiting time gT
PhenomenologyPhenomenology
Fundamental TheoryFundamental Theory
Dynamics
Single particle: Gaussian to non-Gaussian transitionSlow and fast relaxations
Specific heat: Annealed to quenched transition
Thermodynamics
Cooling-rate dependence
Construction of free energy landscape
Dynamics
Thermodynamics
Slow and fast relaxations
Separation of time scalesSeparation of time scales
Total
Microscopic Relaxation
Free energy landscape
])(exp[})({ 2 i
iiii RrCRr For practical calculation
dRrrHN
RNVTZ iiii })({})]({exp[!
1}){,,,(
}){,,,(ln}){,,,( iBi RNVTZTkRNVT
Dynamics on the FEL
)(})({ tQRdt
dRiiR
ii
)(tQi : Random force
[Ansatz]
)()( tuRtr iii ttt 0for )(trR ii where
tt 0Separation of time scales
)()sin(
)( tQdx
xdTg
dt
dx
0)( tQ )(4)()( 210
21 ttT
TtQtQ
BkT /20
Dynamics
random force
and
A toy model for the dynamics on the FEL
Scaled equation
)()cos()( tQxTgdt
dx
)(2 Tg
)(Tg
0/TT
Three models for g(T)
TT /0
)]1tanh(1/[)]/tanh(1[ 0 TT
movie
1)( Tg
The dynamical structure factor of Model 1
0.01T0
0.1T0
1T0
10T0
100T0
1000T0
k=0.5ωS(k,ω)
ω
Oscillatory motion
Jump motion
The dynamical structure factor of Model 2ωS(k,ω)
ω
k=0.5
0.01T0
0.1T0
0.3T0
10T0
1000T0
Jump motion
Oscillatory motion
The dynamical structure factor of Model 3k=0.5
ωS(k,ω)
ω
100T0
10T0
1T0
0.3T00.1T0
0.01T0
Jump motion
Oscillatory motion
T/1
Characteristic time scales
Phenomenology
Fundamental Theory
Dynamics
Single particle: Gaussian to non-Gaussian transitionSlow and fast relaxations
Specific heat: Annealed to quenched transition
Thermodynamics
Cooling-rate dependence
Construction of free energy landscape
Dynamics
Thermodynamics
Unified Theory for Glass Transition
))()()(()(2
1
))(()(
log)(][)]([
212121 ll
ll
l
cdd
dd
rrrrrr
rrr
rrr
: Direct correlation function)(rc
Ramakrishnan-Yussouff free energy functional
}){,(])(exp[)( 2i
iiC RRrr
})]{,([})({ ii RR as a function of }{ iR
Free energy landscape
No of atoms in the core ： 32555.0 362
Ｓｔｒｉｎｇ ｍｏｔｉｏｎ and CRR
Simultaneously and cooperatively rearranging regions
SRR: Difference between two adjacent basins
CRR: Atoms involved in the transition state
108
523.0
N
return
Phenomenological understanding : Heat capacity
T. Tao &T.O(PRE 2002),T.O et al (JCP 2002),T. Tao et al (JCP2005)
aE
),( tTPa
Energy of basin a
Probability of being in basin a at t
),(),( tTPEtTE aa
a
0
0000
),(),(),(
TT
tTEtTEttTC
)0,(TC
),( TC
： Quenched
： Annealed
a
)10,10,10( 642coolCt
)10( 2heatCt
Annealed-to-quenched transition and cooling rate dependence
• 20 basins:Einstein oscillators
slow
fast
T. Tao, T. O and A. Yoshimori: JCP 122, 044505 (2005)
return
Trapping Diffusion ModelTrapping Diffusion Model
return
)2()( tt
)(
)()(
gcg
gcgc
TsT
TsTTTs
Waiting time distribution for jump motion
Unifying concept
0t1 0T 0)( 0 Tsc
2t xT10 gTt 0D
2)(
)(
0
0
TT
TT
TsT
TsT
g
X
gcg
XcX
)/(
12
0TTT
T
gg
x
Characteristic Temperature Equation
Characteristic Temperature Equation
V B Kokshenev & P D Borges, JCP 122, 114510 (2005)
g
C
T
T
0/TTg
g
C
T
T
0/TTg
return
Waiting time distribution for slow relaxation
g
dgCgp0
])(exp[)()( Prob. of activation free energy
2)( tt
)(
)()(1
)(*
gcg
gcgcc
TsT
TsTTTs
S
TkTs
Waiting time distribution
gneww 0
)(/* TsSn c :Size of CRR by Adam and Gibbs
SRR
CRR
return
Non-Gaussian parameter Susceptibility
return
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