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: T high T low <T g aging-experiment: quench measure some dynamical quantity related to the  -relaxation

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-t w measurement of different quantities: quench: t=0 t=t w measurement: time

meaning of quench: quench: sudden change of temperature from T high to T low experiment: simulations: theory: qualitative features are often independent of the cooling rate initial temperature: experiment: > T g simulation: 'high' theory: 'high'

experimental results – one-time quantities: polystyrene, T g =94.8 o C volume recovery (Simon, Sobieski, Plazek 2001) data well reproduced by TNM model quench from 104 o C

TNM model (used quite frequently in the analysis): Y=enthalpy, volume, etc. (t w =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) T g =185K but  ''

T=0.4 simulation results – two-time quantities: binary Lennard-Jones liquid Kob, Barrat 2000 T c =0.435 quench T>T c  T<T c C st usually not observable in lab glasses

model calculations: domain-growth e.g. 2d Ising-model t1t1 t2t2 t3t3 t4t4 self similar structures but no equilibrium

ferromagnet: t w T=0.5 C(t w + ,t w )  / s plateau value: C 0 short times: stationary dynamics within domains long times: domainwall motion

main messages: typical aging experiment: quench from some T high to T low after t w : measurement frequent observations: relaxation time increases with t w equilibrium is reached for (extremely) long t w glasses: re-equilibration is determined by  -relaxation models: two-step decay of C(t,t w ) some debate about t-t w superposition and other details

violations of the FDT: dynamic quantity: correlation: response: apply a small field H  linear response H0H0 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: t w (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 T eff : p-spin models (1-RSB): coarsening models: (~MCT – glassy dynamics) examples: definition of an effective temperature: (there are other ways to define T eff )

FDR – experimental examples: Supercooled liquid dielectric Polarization noise Grigera, Israeloff, 1999 glycerol, T=179.8 K (T g =196 K) spinglas SQUID-measurement of magnetic fluctuations Herisson, Ocio 2002 CdCr 1.7 In 0.3 S 4, T=13.3 K (T g =16.2 K)

FDR – example from simulations: binary Lennard-Jones system Kob, Barrat 2000

T eff is a temperature - theory consider a dynamical variable M(t) coupled linearly to 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 T eff - cont: dynamic quantities a  0: a=0:

calculation of T eff – still cont: second term: t  t', thermometer in equilibrium at T x : -  t C x =T x R x first term: fast thermometer – R x decays fast

calculation of T eff – result: 'protocol': connect thermometer to a heat bath at T x disconnect from heat bath and connect to the glas if the heat flow vanishes: T eff =T x

fluctuation-dissipation relations – theoretical models: slow dynamics: solution of Newtonian dynamics impossible on 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 s i, i=1,...,N Hamiltonian H Langevin equation: deterministic force stochastic force stochastic force: 'gaussian kicks' of the heat bath

Langevin equations – FDR Langevin equation for variable x(t): correlation function: causality: x cannot depend on 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)=x 0 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 of initial state equilibrium correlation

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: 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<T c ): correlation function: d=3: T c =3.9568J/k stationary regime: short times aging regime: short times response: stationary regime: aging regime:

correlation function: t w T=0.5 C(t w + ,t w )  / s short times: stationary dynamics within domains long times: domainwall motion quasi-equilibrium at T c coarsening: domains grow and shrink

fluctuation dissipation ratio (T<T c ): stationary regime: FDT at T c 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) energy population of ‘states‘ (configurations): dynamics: transitions

FDR for master equations - cont: detailed balance: lossgain transition rates: master equation: example: Metropolis:

master equations - example: 1 dim. random walk: W0W0 W0W0 xkxk x k+1 x k-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 energy 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 t w during (t-t w ) T>T 0 : stationarity: as FDT holds!

results: correlation function T<T 0  (t w +t,t w ) t/t w scaling law:

results: response T<T 0 asymmetry: scaling regime:

results: T eff 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 T eff as a measure of the non-equilibrium state BUT:

problems - KCM: negative fluctuation-dissipation ratios in kinetically constrained models: Mayer et al 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: 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 ('97) (discussion of FDT violation) LF. Cugliandolo et al.; Phys. Rev. E ('97) (theory of T eff ) Garriga and F. Ritort; Eur. Phys. J. B ('01) (detailed calculation of T eff ) C. Monthus and J-P. Bouchaud; J. Phys. A ('96) (classical paper on the trap model) W. Kob and J-L. Barrat; Eur. Phys. J. B ('00) (FDT violations in binary LJ-system)...