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-1

g

log(

rela

xatio

n tim

e)

1 / T

typical protocol:

: ensemble average

one-time quantities:energy, volume, enthalpy etc.

equilibrium: t-independent

two-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 temperature

x: 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-growth

e.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

0.0

0.2

0.4

0.6

0.8

1.0

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 liquiddielectric

Polarization noiseGrigera, 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: correlation

response

(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,...,NHamiltonian H

Langevin equation:

deterministicforce

stochasticforce

stochastic force: 'gaussian kicks' of the heat bath

Langevin equations – FDR

Langevin 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 ferromagnet

spins on a lattice:

Ising: spherical:

globalconstraint

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

0.0

0.2

0.4

0.6

0.8

1.0

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

(fre

e)

en

erg

y

population of ‘states‘ (configurations):

dynamics: transitions

FDR for master equations - cont:

detailed balance:

loss gain

transition 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

en

erg

y

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 function

correlation:

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=10

5

tw=10

-7

T / T0=0.3

P(t

w+t

,tw)

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 T

FD-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+,t

w)

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+,t

w) 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)...