t. odagaki and t. ekimoto department of physics, kyushu university
DESCRIPTION
Ngai Fest September 16, 2006. Fast and Slow Dynamics in the Free Energy Landscape. T. Odagaki and T. Ekimoto Department of Physics, Kyushu University. High T. High T. Low T. Free energy. Low T. Free energy. Phase transition. Order parameter. Configuration. - PowerPoint PPT PresentationTRANSCRIPT
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