Ergodicity,non-ergodic processes and aging processes

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1 Ergodicity,non-ergodic processes and aging processes By Amir Golan Outline: A. Ensemble and time averages - Definition of ergodicity B. A short overvie of aging phenomena C. Examples: 1. Simple-glass and Spin-glass: Non-ergodic process ith aging (Ising model) 2. Aging in an infinite-range Hamiltonian system of coupled rotators 1

2 Ergodization: If e take a system ith some d-dimentions (d>1) ith energy H(x i,p i ) =E, i =1,2,3 d The system ill fill all of it's "energy envelope" ithin some typical time. Rectangular Box With in a short time the system ill have d frequencies for the system: 1 2 2

3 Sinai Billiard : ithin 6 hits ith the alls, e have complete chaos. If e define V = arctan V y x θ, here V 2 x +V 2 y = 2E/m Ergodicity is reached hen q has covered all angles. 3

4 Elipsoid : In this special shape there are ergodic and nonergodic areas: Focusing areas: Whispering Gallery Puankare sections q - angle of impact S - location along the circumference These sections denote specific areas here there is ergodization and no-ergodicity. 4

5 Sin(q) 1 0 S -1 In the green section, there is complete ergodization. In that case, all measurement of the system ill yield the same result. This can be described as a stationary state, another charachteristic of ergodicity. ρ = t [ H, ρ] = 0 5

6 Ensemble and time averages average of y at a given time over all systems of the ensemble. y 1 N () t y( t) N k = 1 y (k) (t) Where y (k) (t) is the value assumed by y(t) in the kth system of the ensemble and here N is the very large total number of systems in the ensemble. 2. average of y for a given system of the ensemble over some very large time interval 2q (here q ). 1 θ { (k) y (t)} 2θ θ y (k) (t + t')dt' 6

7 vertical average over some systems horizontal average over one system long time 7

8 The averages commute: (k) { y (t)} 1 N N 1 2θ θ k = 1 θ y (k) (t + t')dt' = 1 2θ θ 1 N N θ k = 1 y (k) (t + t') dt' = 1 2θ θ θ y(t + t') dt' = (k) { y (t)} = { y(t + t') } Consider no a situation hich is "stationary" ith respect to y. This means that there is no preferred origin in time for the statistical description of y i.e., the same ensemble ensues hen all member functions y (k) (t) of the ensemble are shifted by arbitrary amounts in time. (In an equilibrium situation this ould, of course, be true for all statistical quantities.) For stationary ensembles e have a connection beteen ensemble and time averages if e assume ergodization. 8

Since q is very large, the behavior of y (k) (t) in each such section ill then be independent of its behavior

9 The ergodic assumption: Each system of the ensemble ill in the course of a sufficiently long time pass through all the values accessible to it. Since q is very large, the behavior of y (k) (t) in each such section ill then be independent of its behavior in any other section. Some large number of M such sections should then constitute as good a representative ensemble of the statistical behavior of y as the original ensemble average. 9

10 From the above, e can understand that the time interval q must be independent of the time t. The ergodic assumption implies that the time average must be the same for essentially all systems of the ensembles. Thus, { (t)} = {} y y (k) independent of k. Similarly, it must be true that in such a stationary ensemble the ensemble average of y must be independent of time. Thus, y(t) = y independent of t. 10

11 by taking the ensemble average (independent of k), e can get the relation: { y (k) (t)} = { y} If e take the time average of the second e ill get: { y } (k) (t) = y Hence from the above e can conclude that for a stationary ergodic ensemble : {} y = y 11

12 Aging History-dependent relaxation Aging in an infinite-range Hamiltonian system of coupled rotators The hemiltonian of the system is: H 1 = 2 1 L + 2 2N i [ 1 cos( θ θ )] = K U i i j + i, j Aging can be characterized by measuring the totime autocorrelation function If the state of the system in phase space can be completely characterized giving a state vector x, then the to-time autocorrelation function is defined as follos: C( t + t, t ) = x( t + t ) x( t ) σ t + t σ t x( t + t ) x( t ) σ t ' - standard deviations <...> - average over several realizations x ( θ, L) - The hamiltonian decomposes in coordinates and their conjugate momenta 12

13 "Freezing" conditions: Keeping all angles at zero hile giving the system a random momenta form a uniform distribution, such that the system has a total energy K+U. The system "remembers" the aiting time. This is the aging phenomena. Can e scale all these plots to fit one another? 13

14 Form: Marcelo A. Montemurro,1, Francisco A. Tamarit, and Celia Anteneodo PHYSICAL REVIEW E 67, (2003) λ β = + t t t t f t t t C ), ( 0.74 = λ 90 = 0. β 14

Simple-glass: Glass as e kno it is an amorphous SiO 2 hich ould, according to its loest free energy be in a crystalline shape (like sand) although it is "frozen" in a metastable state, hich is in a

15 Simple-glass: Glass as e kno it is an amorphous SiO 2 hich ould, according to its loest free energy be in a crystalline shape (like sand) although it is "frozen" in a metastable state, hich is in a higher free energy level. This state ill, in a finite (but VERY long) time decay into crystalline state by nucleating sufficiently large domains of the crystalline phase, hich ill then gro and cover all the material. This system is none ergodic since it has not yet reached it's equilibrium. Helmholtz's free energy : F = E - TS 15

16 Spin-glass: Non-ergodic process ith aging (Ising model) DF Ensemble of randomly oriented spins, hich are frozen due to short and infinite-range correlations. In other ords, in order for the system to get to "ergodization" it has to move though all it's possible states The glass structure is making this process very long. 16

17 Discovery of aging effects in spin glasses 20 years ago. Dependence of the memory decay functions. Memory effects in the thermoremanent magnetization (TRM) (or zero-field cooled magnetization) The freezing process: The sample is cooled through its spin glass transition temperature in a small magnetic field (zero field) and held in that particular field and temperature configuration for a aiting time t. The decay: At time t, a change in the magnetic field produces a very long time decay in the magnetization. The decay is dependent on the aiting time. memory of the time it spent in the magnetic field. 17

18 Experimental results from: G. F. Rodriguez and G.G. Kenning And R. Orbach Physical revie letters, Volume 91, number 3, (2003) 18

19 A mathematical analysis: The free energy: F V d L Time scale for a "flip" τ e F / KT The time to cross the barrier beteen the to phases diverges exponentially in the thermodynamical limit N ( or d L ) The name e call such systems that for them τ max as N, is: nonergodic. DF 19

The glass transition as a spin glass problem

The glass transition as a spin glass problem Mike Moore School of Physics and Astronomy, University of Manchester UBC 2007 Co-Authors: Joonhyun Yeo, Konkuk University Marco Tarzia, Saclay Mike Moore (Manchester)