Weak Ergodicity Breaking in Weak Ergodicity Breaking in Continuous Time Random Continuous Time Random

WalkWalk

Golan Bel (UCSB)

Eli Barkai (BIU)

June 28, 2006

ErgodicityErgodicity

• Ensemble of non interacting particles

eqf f p

Thermal equilibrium

eq Bp p

/ /E TBp e Z

/E TZ e

Partition function

/eq eqp N N

• Single particle, time measurement

0

' 't

t

f t dt

f f pt

/tp T t

Occup ati on timeT

Measure ment e timt

Conditions for ErgodicityConditions for Ergodicity

• All phase space is visited.• The fraction of occupation time is

proportional to the fraction of phase space volume.

• Microscopic time scale exists.• Sections of the measured signal are

independent.• Independent of initial condition in the long

time limit.

Ergodicity BreakingErgodicity Breaking

• Strong non-ergodicity: Phase space is apriori divided into mutually inaccessible regions. Dynamics is limited.

• Weak non-ergodicity: Phase space is connected, but the fraction of occupation time is not equal to the fraction of phase space occupied.Dynamics exists over the whole phase space.

J. P. Bouchaud, J. De Physique I (1992).

MotivationMotivation

• Single molecule experiments remove the problem of ensemble average.

• In many single molecule experiments the microscopic time scale diverges.

• What replace Boltzmann-Gibbs statistical mechanics in this case?

Weitz’s ExperimentWeitz’s ExperimentI. Y. Wong et al, Phys. Rev. Lett. (2004)

Trajectory

Power law waiting time PDF

Anomalous DiffusionAnomalous Diffusion 2x t t

Subdiffusion in living yeast cellsSubdiffusion in living yeast cellsI. M. Tolic-Norrelikke et al, Phys. Rev. Lett. (2004)

Continuous Time Random walkContinuous Time Random walk

1a

lq x 1 lq x

2x t

0 1

1 for 1

TrajectoriesTrajectories

Fraction of occupation time Fraction of occupation time histogramhistogram

1 2

3 4

1,2,3, Non - Ergodic 4, Ergodic

Two States ProcessTwo States Process

11fpt fpt nx

nx l Lx l Rx

aq x q x

1x

x

a

*1 3 1

2 4

...

...x n

nx n

T

T

First Passage TimeFirst Passage Time

0

1 1n

CT DisDisn

u uS u u S n S u

u u

0

nDis Dis

n

S z z S n

Relation between the Survival probability in discrete time RW and CTRW

0

,CT Disn

S t w n t S n

In Laplace space

1

,nu

w n u uu

0 1 1fpt

CT DisCTu uS u S t S u u

Lamperti’s PDFLamperti’s PDF

, x x xx x

nx

T T af R Rt t a

11

22 2

sin 1,

1 2 1 cos

RR

R R

For unbiased and uniformly biased CTRW an exact solution of the FPT PDF exists, allows to determine .nxa

Fraction of Occupation Time PDFFraction of Occupation Time PDF

PDF of the fraction of occupation PDF of the fraction of occupation time in unbiased CTRWtime in unbiased CTRW

0 1

1 2 0

1 1

1 2

1

1 1

2 1 1

1 1 1 1 1

1 2 1 1

1 2 1

l

l

y l y l y

L l L L

L l L

p n q p n

p n q p n p n

p n q y p n q y p n

p n q L p n p n

p n q L p n

The master equation

Consider the CTRW as function of visits number

Visitation FractionVisitation Fraction

The master equation describes both discrete time RW and CTRW thus the visitation fraction in both cases is equal and given by

/x eqn n p x

1 x x eqp n p n p x n

Detailed balance

= eq Bp x p x

1 /

1 1E x E x Tl

l

q xe

q x

VF in unbiased CTRW (periodic boundary conditions)

VF in unbiased CTRW (reflecting b.c.)VF in unbiased CTRW (reflecting b.c.)

Visitation Fraction and Ensemble Visitation Fraction and Ensemble Average in Harmonic PotentialAverage in Harmonic Potential

Derivation of Lamperti’s PDF Using Derivation of Lamperti’s PDF Using the Visitation Fractionthe Visitation Fraction

1 *, i 1

11

,

1x

x x

n t x x n ni x

nn n n

n s

f T T I t t t

u sf u u s s

u s

0, i 1

0

, 1x x

n t x x n ni x

n n n

n s

f T T I t t t

u s sf u s

s

1 0

, , ,1eq eqn s n s n sf u p x f u p x f u

x eqn p x n

1

,

1 11

eq eqn

p x p x

eq eqn s

u s sf u p x p x u s s

u s s

Summing over n

1

1 1 11

1eq eq

eq eqs p x p x

u s sf u p x p x

u s s u s s

Tauberian theorem

1 1

, 0

1

1

eq eq

s s ueq eq

p x u s p x sf u

p x u s p x s

Inverting the double Laplace transform

/ , / 1

eqx x x x

eq

p xf T t R T t R

p x

This solution recovers the exact solution for the uniformly biased CTRW

Visitation Fraction and Ensemble Visitation Fraction and Ensemble Average in Harmonic PotentialAverage in Harmonic Potential

Fraction of Occupation Time PDF Fraction of Occupation Time PDF on the bottom of harmonic potentialon the bottom of harmonic potential

Fraction of occupation time PDF on Fraction of occupation time PDF on the bottom of harmonic potentialthe bottom of harmonic potential

0.3 3T

0.5 3T

0.8 3T

• In ergodic system microscopic time scale exists, thus the visitation fraction is equal to the fraction of occupation time, which in turn is equal to the equilibrium probability in ensemble sense.

• In the case where the visitation fraction is equal to the equilibrium probability, but due to divergence of the microscopic time scale, the fraction of occupation time is not equal to the equilibrium probability, the system is said to exhibit weak ergodicity breaking.

• If both the visitation fraction and the fraction of occupation time are not equal to the equilibrium probability, the system exhibits strong ergodicity breaking.

ConclusionConclusion

• CTRW with power law PDF of sojourn times exhibits weak ergodicity breaking.

• Weak non-ergodicity in the context of CTRW was precisely defined.

• The weak non-ergodicity was quantified by the universal probability density function of the fraction of occupation time.

• Generalization of Boltzmann-Gibbs statistical mechanics to weakly non-ergodic system is possible.

ReferencesReferences

• G. Bel, E. Barkai, PRL 94, 240602 (2005).

• G. Bel, E. Barkai, PRE 73, 016125 (2006).

• G. Bel, E. Barkai, EPL 74, 15 (2006).