Violations of the fluctuation dissipation theorem
in non-equilibrium systems
Introduction:glassy systems: very slow relaxation
quench from high to low temperature:equilibrium is reached only after very long times
physical aging
out-of equilibrium dynamics:•energy depends on time•response and correlation depend on 2 times
(aging not restricted to glassy systems)
physical aging:glassforming liquids:
Thigh Tlow <Tg
aging-experiment:quench
measure some dynamical quantity
related to the -relaxation
T-1g
log(
rela
xatio
n tim
e)
1 / T
typical protocol:
: ensemble average
one-time quantities:energy, volume, enthalpy etc.
equilibrium: t-independenttwo-time quantities:
correlation functions equilibrium: Y depends on time-difference t-tw
measurement of different quantities:
quench:
t=0 t=tw
measurement:time
meaning of quench:quench: sudden change of temperature from Thigh to Tlow
experiment: simulations:
theory:
qualitative features are often independent
of the cooling rate
initial temperature:experiment: > Tg
simulation: 'high'theory: 'high'
experimental results – one-time quantities:
polystyrene , Tg=94.8oCvolume recovery
(Simon, Sobieski, Plazek 2001)
data well reproduced by TNM model
quench from 104oC
TNM model (used quite frequently in the analysis):
Y=enthalpy, volume, etc. (tw=0) Tool-Narayanaswamy-Moynihan (TNM):
: fictive temperaturex: nonlinearity parameter
(=const.)
experimental results – dielectric response: dielectric relaxation of various glassforming liquids:
Leheny, Nagel 1998 Lunkenheimer, Wehn, Schneider, Loidl, 2005
re-equilibration is always determined by the
-relaxation (independent of frequency)
Tg=185K
but
''
''
T=0.4
simulation results – two-time quantities: binary Lennard-Jones liquid
Kob, Barrat 2000
Tc=0.435quench T>Tc T<Tc
Cst usually not observable in lab glasses
model calculations: domain-growthe.g. 2d Ising-model
t1
t2
t3
t4
self similar structures but no equilibrium
ferromagnet:
10-2 10-1 100 101 102 103 104 105
0.00.20.40.60.81.0
tw
T=0.5
C(t w
+ ,t w
)
/ s
plateau value:C0
short times:stationary dynamics
within domains
long times:domainwall
motion
main messages:
typical aging experiment:•quench from some Thigh to Tlow•after tw: measurement
frequent observations:•relaxation time increases with tw•equilibrium is reached for (extremely) long tw•glasses: re-equilibration is determined by -relaxation•models: two-step decay of C(t,tw)
some debate about t-tw superpositionand other details
violations of the FDT:dynamic quantity:
correlation:
response: apply a small field H linear response
H0 H=0
FDT – equilibrium:
FDT:
stationarity:
equilibrium statistical mechanics:
impulse response
:
equilibrium – integrated response:(step response function):
FD-plot:
C
slope: 1/T
non-equilibrium:T='high' T(working)quench at t=0
start of measurement: tw
(fluctuation-dissipation ratio)FDT violation:
fluctuation-dissipation relation FDR:(definition of X)
fluctuation-dissipation ratio:
some models in the scaling regime:
integrated response:
equilibrium:
FD-plots:
SK model (RSB)p-spin model (1RSB)spherical model, Ising model
examples:
typical behavior of the FDR:
short times:
FDT: X=1
long times:
X<1
effective temperature Teff:
p-spin models (1-RSB):
coarsening models:
(~MCT – glassy dynamics)
examples:
definition of an effective temperature:
(there are other ways to define Teff)
FDR – experimental examples:Supercooled liquid
dielectric Polarization noise
Grigera, Israeloff, 1999
glycerol, T=179.8 K (Tg=196 K)
spinglasSQUID-measurement
of magnetic fluctuationsHerisson, Ocio 2002
CdCr1.7In0.3S4, T=13.3 K (Tg=16.2 K)
FDR – example from simulations:binary Lennard-Jones systemKob, Barrat 2000
Teff is a temperature - theory
consider a dynamical variable M(t) coupled linearlyto a thermometer with variable x(t)linear coupling:
net power gain of the thermometer:
calculation of : linear response theory
thermometer: correlationresponse
(Cugliandolo et al. 1997)
calculation of Teff - cont:
dynamic quantities a0:
a=0:
calculation of Teff – still cont:
second term: tt' , thermometer in equilibrium at Tx: -tCx=TxRx
first term: fast thermometer – Rx decays fast
calculation of Teff – result:
'protocol':• connect thermometer to a heat bath at Tx • disconnect from heat bath and connect to the glas
if the heat flow vanishes: Teff=Tx
fluctuation-dissipation relations – theoretical models:
slow dynamics:
solution of Newtonian dynamics impossibleon relevant time scales
standard procedures:
consider stochastic models:
Langevin equations (Fokker-Planck equations)master equations
Langevin equations:consider some statistical mechanical model
(very often spin models)dynamical variables si, i=1,...,N
Hamiltonian H
Langevin equation:
deterministicforce
stochasticforce
stochastic force: 'gaussian kicks' of the heat bath
Langevin equations – FDRLangevin equation for variable x(t):
correlation function:
causality: x cannot dependon the noise to a later time
time-derivatives:
Langevin equations – FDR – cont.
without proof:
definition of the asymmetry:
FDR:
Langevin equations – FDR – equilibrium
FDT:
time reversal symmetry:
stationarity:
FDR – Ornstein-Uhlenbeck process:Langevin equation for diffusion:diffusion in a potential:
diffusion in a harmonic potential: Ornstein-Uhlenbeck
solution: (inhomogeneous differential equation)
decay of initial condition
x(0)=x0
inhomogeneity
OU-process excercise: calculate
response:
asymmetry:
FD ratio:
correlation function:
OU-process correlation function:
OU-process cf - cont:
equilibrium:
equilibrium is reached for s due to the decay
decay ofinitial state
equilibriumcorrelation
OU-process response:
OU-process FDR:asymmetry:
OU-process:
OU-process FDR cont:
independent of t
equilibrium is reached after long times
another example: spherical ferromagnetspins on a lattice:
Ising: spherical:global
constraint
S-FM: exact solution for arbitrary d
stochastic dynamics:Langevin equations:
stochastic forces: Gaussian
Lagrange multiplier z(t)=2d+(t)
solution of the L-equations:Fourier-transform
all dynamical quantities
correlation and response (T<Tc):correlation function: d=3: Tc=3.9568J/k
stationary regime: short times
aging regime: short times
response:stationary regime: aging regime:
correlation function:
10-2 10-1 100 101 102 103 104 105
0.00.20.40.60.81.0
tw
T=0.5C
(t w+
,t w)
/ sshort times:stationary dynamics
within domains
long times:domainwall
motionquasi-equilibrium
at Tc
coarsening: domainsgrow and shrink
fluctuation dissipation ratio (T<Tc):
stationary regime: FDT at Tc
aging regime:
limiting value:
domain walls are in disordered state
typical for coarsening systems
FDR for master equations:stochastic evolution in an energy landscape
order parameter
(free
) ene
rgy
population of ‘states‘ (configurations):dynamics: transitions
FDR for master equations - cont:
detailed balance:
loss gaintransition rates:
master equation:
example: Metropolis:
master equations - example:
1 dim. random walk:W0W0
xk xk+1xk-1
for nearest neighbor transitions
solution: Fourier transform
gaussian
propagator:
e.g. quench:T= T(working)
preparation in initial states:
time evolution at T:
same master equation as for populations
calculation of dynamical quantities:
response:
coupling to an external field H
?
correlation function:
perturbed transition rates:equilibrium:
choice: (Ritort 2003)
detailed balance:
(not sufficient to fix the transition rates)
typically ==1/2
response:perturbation theory
FDR for Markov processes
'asymmetry'
asymmetry is not related to measurable quantities:
looks similar to the FDR for Langevin equations
example: trap model
order parameter
ener
gy
distribution of energies:
random choice of arrival trap
activated jump out
of initial trap
transition rates: (global connectivity)
trap populations:solution of the master equation:
stationary solution:
distribution of trap energies existence of stationary solution or not
exponential distribution:
agingequilibrium relaxation
dynamical quantities:
choose dynamic quantity M(t):
transition:? ?simple assumption:
M randomizes completely (takes on any value out of distribution)
standard for trap models (Monthus and Bouchaud 1996)
results: correlation functioncorrelation:
probability that the system has not left the trap occupied at tw during (t-tw)
T>T0:stationarity:
as
FDT holds!
results: correlation function T<T0
10-2 101 104 107 1010 1013
10-4
10-3
10-2
10-1
100
(t / tw)-x
tw=105
tw=10-7
T / T0=0.3
P(t w
+t,t
w)
t / tw
(t w
+t,t w
)
t/tw
scaling law:
results: response T<T0
asymmetry:
scaling regime:
results: Teff
long-time limit:effective temperature:
resume:violations of the fluctuation-dissipation theorem in non-equilibrium systems:•simulated glasforming liquids•spinglas models•random manifolds•coarsening systems•sheared liquids•Spin models without randomness•oscillator models
definition of Teff as a measure of the non-equilibrium state
BUT:
problems - KCM:
negative fluctuation-dissipation ratios in kinetically constrained models: Mayer et al. 2006
Mayer et al. 2006
problems – models with stationary solution:trap model with a gaussian distribution of energies
the system reaches equilibrium for all TFD-plot:
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
tw>>eq
tw=0
(tw+,tw)
T=0.2 ; ==0.5
T(
t w+
,t w)
0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
tw>>eq
tw=0T(
t w+
,t w)
T=0.2 ; ==0.5
(tw+,tw) wrong slope from FD-plot
analytical calculation:
X=1/2
some references:A. Crisanti and R. Ritort; J. Phys. A 36 R181 ('03) (a comprehensive review article)LF. Cugliandolo et al.; Phys. Rev. Lett. 79 2168 ('97) (discussion of FDT violation)LF. Cugliandolo et al.; Phys. Rev. E 55 3898 ('97) (theory of Teff)• Garriga and F. Ritort; Eur. Phys. J. B 21 115 ('01) (detailed calculation of Teff)C. Monthus and J-P. Bouchaud; J. Phys. A 29 3847 ('96) (classical paper on the trap model)W. Kob and J-L. Barrat; Eur. Phys. J. B 13 319 ('00) (FDT violations in binary LJ-system)...