Damage spreading

Physica A 168 (1990) 516-528 North-Holland

DAMAGE SPREADING

H.J. H E R R M A N N

SPhT, CEN Saclay, F-91191 Gif-sur-Yvette Cedex, France We introduce the concept of damage spreading in the context of Monte Carlo dynamics. This provides us with an efficient method to calculate correlation functions and generates the clusters of the fluctuations in magnetization. These clusters have at Tc the fractal dimension d -/3/~. In general, damage spreading defines new dynamical phase transitions but only in some cases it can be related to usual thermal critical points.

1. Introduction

Monte Carlo can be viewed as a dynamical process in phase space, i.e. starting from a given initial configuration one follows a trajectory in phase space under the application of the M o n t e Carlo procedure. This trajectory will of course depend on the specific type of M o n t e Carlo (heat bath, Glauber, Metropolis, Kawasaki, Creutz, etc.) but also on the sequence of r a n d o m numbers. The trajectories will always go towards equilibrium, which means that the configurations will be visited by the trajectory with a probability proportional to the Boltzmann factors. Once in equilibrium, the trajectories will stay there as ensured by the detailed-balance condition. The interesting question that we will ask now and which is the central issue of dynamical system theory is how much will a trajectory depend on the initial condition. More precisely: Suppose we m a k e a small perturbation in the initial condition, will the new trajectory be just slightly different or will it be totally different. The second case in which two initially close trajectories will b e c o m e very fast very different is generically called chaotic. In order that the concept of closeness of trajectories be meaningful one must define, what is a distance in phase space. Suppose we consider a system of Ising variables, then a useful metric can be given by the " H a m m i n g distance" or " d a m a g e " [1] 1

A(t) = ~-~

~/Io-M)- pi(t)l,

(1)

where {o's(t)) and { pi(t)} are the two (time-dependent) configurations in phase 0378-4371/90/$03.50 ~ 1990-Elsevier Science Publishers B.V. (North-Holland)

H.J. Herrmann / Damage spreading

517

space and N is the number of sites, labelled by i. This definition is certainly arbitrary and the results that we present in the following depend on the definition. Physically it just measures the fraction of sites for which the two configurations are different. The definition of eq. (1) can easily be generalized to continuous variables or to variable of q-states. It is now clear why the procedure that we will follow has been called "damage spreading". One creates at t - - 0 an initial damage A(0) in one configuration (which gives a second configuration) and watches if the damge spreads (chaotic situation) or if it heals (frozen situation). This kind of spreading has been investigated before [2] in cellular automata to describe for instance the influence of perturbations in genetic regulatory systems. The crucial idea in order to study damage spreading in Monte Carlo is therefore to take two configurations and to apply to them the s a m e sequence o f r a n d o m n u m b e r s in the Monte Carlo algorithm [1, 3-5]. In this way one is taking the same dynamics for both configurations. To get statistically meaningful results one must then average over many initial configurations of equal initial distance and over many different sequences of random numbers. It is also important to note that, as in deterministic dynamical systems, once the two configurations become identical they will always stay identical.

2. Relationship between damage and thermodynamic properties Damage can be related to correlation functions as has been shown in ref. [6]. In the following we will briefly report on these relationships. We will consider Ising variables o-i =--+1, which can also be expressed as usual by 7ri = 1 (o.i _ 1) = O, 1. We want to discuss the damage between configurations (O'A} and (o-B} and in order to produce a small damage we will fix the spin at the origin of configuration {O'B} to be always or0B = - 1 .

(2)

This condition is different from the ones considered before because now the damage is fixed constituting thus a source of damage. In principle two types of damage can be imagined at site i: damage + - where o-# = +1 and o-B = - 1 , or damage - + where o-A = - 1 and tr~ = +1. The probabilities of finding a certain type of damage at site i can then be expressed as d~+--

~---

(((I --

"/'/'iA )'/Ti B >)

and

d i-÷ = <(~r#(l -- zrB))>

(3)

518

H.J. Herrmann / Damage spreading

where (( . - . )) denotes a time average. Let us define the difference between the damages: Fs = d ,

+

-d,

+

= { ( ~ - ? ) ) - ((vry)),

(4)

where we have used eq. (3). We want to express the damage through thermodynamic quantities defined on an unconstrainted system {~ri} with averages taken over m a n y configurations, i.e. samples. So we translate condition (2) by a conditional probability and use ergodicity to go from time averages to thermal averages:

((zr/A)) = (~'i)

and

((~-~)) - ( %U( -1--- ~ ~'o)) ,

(5)

where ( . . . ) denotes a thermal average. Inserting this into eq. (4) one finds

r~ = ( ~i ,To ) - ( ~ ) ( ~0 ) 1- (%)

c0, - 2(1-

m) '

(6)

m = (Cro)

(7)

where Co, = (o-~O-o) - (O-o) 2

and

are just the correlation function and the magnetization. Relation (6) gives us an equality of a certain combination of the damages with thermodynamical functions. In its derivation we used ergodicity but did not m a k e any assumptions on the dynamics, the r a n d o m numbers or the type of interaction and it is therefore of a very general validity. If one would have chosen another form of fixed d a m a g e the result would have changed slightly. If one would for instance fix or0B = - 1 and o"ff = + 1 then one would find F / = Col~(1 m2). If one considers variables with m o r e degrees of f r e e d o m than Ising variables, things can b e c o m e m o r e complicated but are still in principle feasible. As an example let us look at the A s h k i n - T e l l e r model where on each site one has two binary variables cr~ = +-1 and r i = + 1, which follow per site a Hamiltonian -

~ j = - K(crio) + riTj) - 2Lcr~o)r~rj.

(8)

In this case there are twelve possible damages per site, which we label such that left means configuration A, right configuration B, top means o- and b o t t o m means r; example _+w- is o"A= +1, rA = - - 1 , O'~ = --1 and r/B= +1. It can be

H . J . Herrmann / Damage spreading

519

shown [7] that if one fixes troB = - 1 and roa = - 1 then g

=

-

(9)

2(1 - (O-o%)) with Fii =

d~f- +

+

+ d

-

d [+ + d [ - +

+ d

,

(10)

and if one fixes o-0B = - 1 then = 2(1-

- C O-o)) '

(11)

with

=

d

- + d i+++d-++di

-

++di

(12)

This shows that both types of correlation functions that one has in the A s h k i n - T e l l e r model can be expressed as a combination of damages. The damage which we have defined in eq. (1) was not the quantity F~ but was the sum of all the damages: Z~ =

Ai

~

with A, = d T - + a,

-+

.

(13)

i

In order to express these quantities in terms of t h e r m o d y n a m i c quantities it is necessary to m a k e some assumptions. Let us therefore restrict ourselves now to ferromagnetic interactions, heat-bath dynamics and the use of the same r a n d o m numbers for both configurations. We consider again only Ising variables and fix the d a m a g e as in eq. (2), i.e. cr0B = - 1 . Since at t = 0 the only damage one has is the one of type + - at the origin we have p/A>_ B ~Pi

Vi,

(14)

where p/A is the value Pi = e x p ( 2 ~ h i ) / [ 1 + exp(2/3hi)] with h i = •nn O'j defined for configuration A. Suppose one would try to create a d a m a g e of type - + at site i. Then one would need, in order to produce o-/A = - 1 , a r a n d o m n u m b e r z which fulfills z ~>p/A according to heat bath. Since one is using the same r a n d o m n u m b e r for configuration B this means, using eq. (14), that z >~p~ = - 1 . It is therefore impossible to create a damage of type - + and therefore

H.J. Herrmann / Damage spreading

520

eq. (14) will be valid also at the next time step. By induction one can conclude now that eq. (14) always be valid and that a damage of type - + cannot be created under the conditions that we had imposed. Consequently we have proved that d -+ - - 0 and it follows that for the damage

'~-

C0i 2(1 - m)

and

A-

X 2(1 - m) '

(15)

where X = Z i C0i is the susceptibility. In fig. 1 we see how the exact relations of eq. (15) are realized numerically for the 2D Ising model. We see the correlation function obtained via eq. (15) (circles) and in the usual way (triangles) using for both methods roughly the same computational effort. One sees that in the usual method once the values I

1

1

I

I

Gtr)

1°-~~~'*,~., /T =2.6

T--3.0 ~

L

l O

I 1l

1 1E

I 20

r -=Fie. 1. C o r r e l a t i o n f u n c t i o n G ( ~il=r _C0i ( & for T = 2.6 a n d /~ for T = 3.0) a n d 2z~(r)(1 - m ) ( Q for T = 2.6 and O for T = 3.0) w h e r e A = r.l,l_ r A~ as a function of r in a s e m i - l o g plot. T h e d a t a c o m e f r o m 10 s y s t e m s of size 40 × 40 a n d w e r e t a k e n f r o m ref. [6] for a 2 D Ising m o d e l .

521

H.J. Herrmann / Damage spreading

are less than about 10 -3 the statistical noise takes over the curve flattens. O n the other hand, using eq. (15) one gets to much smaller values without feeling substantial noise. The reason why the use o f damage is superior numerically c o m e s from the fact that this m e t h o d looks just at the difference b e t w e e n two configurations subjected to the same thermal noise, so that this noise is effectively cancelled. A similar fact was already pointed out for continuous systems by Parisi [8].

3. D a m a g e clusters The damage that was fixed at the origin at time t = 0 acts as a source from which constantly damage spreads away. A t a given instant one can l o o k at all the sites that are d a m a g e d and one will find a cloud or cluster of sites around the origin. T h e s e clusters fluctuate in size and shape and are not necessarily connected. In fig. 2 we see two of these clusters for the two-dimensional Ising m o d e l at T c which are just 38 time steps apart. Since heat bath was used these

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Fig. 2. Damage clusters of the 2D Ising model at T~. The damage is fixed to be tr~ = -1 at the origin and the size of the system is L = 60. Heat bath and the same random number sequence were applied on both configurations (taken from ref. [6]).

522

H.J. Herrmann / Damage spreading

clusters represent according to eq. (15) the fluctuation of the magnetization due to the application of a local field at the origin. Using the equivalence of the Ising model to a lattice gas one can interprete these fluctuations as the fluctuations in local density which have actually been measured in a recent experiment [9]. Using eq. (15) it is actually possible to calculate the dependence between radius and number of sites for the damage clusters. One finds that at the critical point the clusters are fractal; that means that if L is the size of the smallest box into which the cluster fits and s is the number of sites in the cluster one has a relation S oc L df

with d f = d - ~3Iv.

(16)

The damage cloud is therefore reminiscent of the droplet [10] in an Ising model, where neighbouring spins in the same state belong to the same droplet with a probability PB = 1 - e -2j/kr. The numerical determination of the fractal dimension of the damage cloud in the 2D ferromagnetic Ising model at its critical temperature has given [6] a value de = 1.87 - 0.02, in good agreement with the exact value d r = d - f l / v = 15/8 of t h e fractal dimension of the droplets. If the situation for the two-dimensional Ising model is quite well established, the three-dimensional case still poses some open questions. A recent preprint [11], focusing on the finite size scaling analysis of the damage cloud for the 3D Ising model at the critical temperature using the heat-bath dynamics, indicates as fractal dimension a value close to 1.9 as opposed to the expected value 2.5 predicted by df = d - f l / v . In order to explain this discrepancy, it has been suggested that the fractal dimension of a cloud of damage allowed to grow up to the boundaries of a box of fixed size, is indeed equal to df = d - 2 / 3 / v . Moreover, such argument could also try to account for the predictions of the two-dimensional data, due to the small value of/3 in this case. However, the error bars estimated for the value of the fractal dimension seem actually to exclude the value 7/4 predicted by this argument. In order to clarify the question whether the fractal dimension of the damage cloud in the three-dimensional ferromagnetic Ising model is indeed numerically given by df = d - / 3 / v , we consider [12] a configuration of Ising spins on a cubic lattice of linear size L and with periodical boundary conditions following heat-bath dynamics. The spins belonging to each of the two sublattices, in which the cubic lattice splits, are updated in parallel. The initially random configuration is thermalized to equilibrium at the critical temperature T¢ = 4.51. The simulation is stopped when the damage finally touches the boundary of the lattice and a new configuration is generated. We analyse system sizes from L = 10 to L = 98, the thermalization time ranging from 2000 to 5000 time

H.J. Herrmann / Damage spreading

523

steps and the statistics varying from 15000 configurations for the smallest system size to 5 configurations for L -- 98. The whole simulation took about 40 hours of cpu time on the Cray XMP. To calculate the fractal dimension of the damage cloud, we first analyse the data by the box-counting method. The mass of the cloud, that is the number of damaged sites, within a box around the origin is calculated for concentric boxes. The double logarithmic plot of the mass as function of the box radius provides then the fractal dimension of the cloud. Fig. 3 shows the data for two different system sizes. In the intermediate region the curves have a straight-line behaviour, whose slope d f is substantially smaller than df = d - fl/u ~-2.5 but increases slowly and steadily with the system size. To account for this size dependence, we have analysed the corrections to scaling for the data. Fig. 4 shows the value of the measured fractal dimension d e for a given system size as function of l/In L, where a logarithmic correction to scaling of the form d e = dell + a(ln L) -l] has been assumed. The data do indeed show a good linear behaviour and tend asymptotically to the value 2.5, in good agreement with df = d - fl/v ~-2.52 [13]. We have also considered the

D 10 3

i

1

i

i

I

102

Q

1 0.2

0.5

1

2

I 5

I 10

I 20

50

R Fig. 3. l o g - l o g plot of the mass of the d a m a g e cloud D within a sphere of radius R versus R for two different system sizes: 1900 configurations of L = 30 (O) with 4000 time steps of thermalization and a slope d~ = 1.45; 187 configurations of L = 70 ( 0 ) with the same thermalization time and a d~' = 1.73 (taken from ref. [12]).

524

H.J. Herrmann / Damage spreading

d? 3t-

'

'

0.1

0.2

0.~,

0.3

11InL Fig. 4. The effective fractal dimension d~' measured via box-counting method as function of l/In L. The arrow indicates the extrapolated value df = d - fl/u =2.5 (taken from ref. [12]). possibility of a correction to scaling of the f o r m d f = dr(1 + a L - W ) . This f o r m is also well satisfied by the data for a quite small correction e x p o n e n t w -~ 0.3, which numerically is not inconsistent with a logarithmic correction. Since i n d e p e n d e n t calculations [11] have indicated that in a finite-size scaling plot df is r a t h e r closer to 1.9 than 2.5, we analyse next the scaling of the mass of the whole cloud at the time w h e n it first t o u c h e s the b o u n d a r y of the lattice as function of the system size. T h e effective fractal dimension at a given size as function o f l / I n L does indeed extrapolate asymptotically to a smaller exponent close to 1.9 (fig. 5a). As we k n o w f r o m fig. 4, the numerical results can be reconciled to the t h e o r y if o n e assumes logarithmic corrections to scaling. F o r this reason we have tried to fit the data of fig. 5a by a curve of the f o r m d f = d f + ( b / l n L ) +

1/ In/ink----L) 0

0.2 ,

tnD

[

~

0.4 r

I

elNi,II-.e~

InL_.Lz

0.6 i

0.8

I

I

l

°. ' ' ~ .

x ~

I 0.1

I O.2

I

I 0.3

_

x '----- x ~

x

I

0.~, l/InL

Fig. 5. The effective fractal dimension In D/ln L as function of l/In L (lower scale) giving an extrapolated value df --~1.9 (x). The same data plotted by using instead of L the variable L/ln L (upper scale) recovering as extrapolated value df = d - / 3 / ~ = 2.5 (@) (taken from ref. [12]).

H.J. Herrmann / Damagespreading

525

I n [ l + a/ln L ] / l n L, where the last term corresponds to the correction to scaling and df = d - fl/v. We find that it is not possible to find any pair of constants a and b to convincingly fit the data so that the deviations from the expected value observed in fig. 5a are not consistent in this case with logarithmic corrections to scaling. Another possible explanation for the extremely slow convergence might be the existence of a logarithmic prefactor, namely a relation M - ( L / I n L ) de for the mass of the cloud. In order to check this point, we have looked at the scaling of the mass of the damage cloud as function of this new variable L/lnL. The effective fractal dimension as function of 1/ln(L/ln L) does indeed extrapolate to the expected value 2.5 predicted by df = d - / 3 h , and obtained via the box-counting method (fig. 5b). If, as the data indicate, a logarithmic prefactor exists for the effective fractal dimension of the damage cloud at the touching time, one should consistently also find this prefactor with the box-counting method. For this reason we have also plotted the data of fig. 3 in a log-log plot as a function of R/ln R instead of R. We find that the data do not fall on a straight line, seemingly excluding the possibility of a logarithmic prefactor in this case. Moreover, a logarithmic prefactor cannot be explained by a relation as R = L/ln L between the radius of gyration and the system size. In fact, the radius of gyration R of a damage cloud touching a box of size L is proportional to L itself and not to L/ln L. As a consequence, the situation seems rather contradictory. There could be, however, a possibility to explain this scenario. It has recently been shown [14] that the dynamical scaling of the structure function of growing domains in a nucleation problem has a novel form of scaling called multiscaling. An infinity of growing exponents is then obtained by continuously varying a given characteristic parameter. We now assume the possibility of a similar type of multiscaling to hold also in the damage spreading problem. In order to do so, for each cloud of a given radius of gyration Rg we fiX the value of the parameter x and we calculate the mass re(x) contained within a shell confined between r=xRg and r ' - (x + 1)Rg. We then look at the scaling dependence of the quantity re(x) as function of Rg at a fixed value of x. If dynamical multiscaling holds, a scaling of the form

p(r, Rg) = r-Id-df(r/RP]f( r--) Rg

(17)

should be found for the density p(r, Rg) and the fractal dimension df should continuously depend on the parameter x = r/Rg. The multiscaling would then reflect the rich internal structure of the cloud where shells with different x do

H.J. Herrmann / Damage spreading

526

exhibit independent scaling exponents. The expected value of df would then be recovered in the limit x--~ 0. One can evaluate the total mass of the cloud M by integrating eq. (17),

L

L/R

M°c f P(r, Rg) rd-l dr= f Rdf(X)xdf(x)- lf(x) dx , o o and assuming that

Rdf(0) M oc ~-------~ (In R) '

df(x)= df(0)- [3X2,

(18)

one finds by saddle-point integration

(19)

where y depends on the form of f(x). This is actually the general expression for a scaling variable of the type log-prefactor discussed above. If on the contrary df(x) is constant for small x by similar arguments one finds a logarithmic correction. The form of dr(x ) could well differ if the averages are taken over all the clouds or only over the touching ones. This difference in averaging could very well explain the discrepancy between the data of fig. 3 and fig. 5. The hypothesis of multiscaling made to obtain this result is presently object of further investigation. Since the physical picture behind the multiscaling behaviour is that the outer regions of the spreading damage cloud have lower fractal dimensions than the inner regions, we measure the number of sites in the damage cloud at the touching distance, i.e. on the boundary of the system (fig. 6). As function of the system size, the number of damaged sites on the boundary does remain constant and is about equal to one. This is no longer valid if, for instance, one uses as touching condition the requirement that the cloud touches the boundaries in all three directions at the same time. In this case, the damage at the boundary does show some power-law dependence on the system size with an exponent roughly equal to 0.75 (fig. 6). It is, however, worth stressing that the scaling behaviour of the whole damage cloud is independent on the touching condition, which only affects the amplitude factor in the mass-radius relation. The fact that different touching conditions show different scaling behaviours hints to the existence of the multiscaling given in eq. (17). In conclusion, we have found that the damage clouds introduced in ref. [6] show a very slow convergence toward the expected fractal dimension df = d - [3/v. This can be explained either by a logarithmic correction or by a logarithmic prefactor. The scenario is compatible with a multiscaling behaviour of the radial density of the clouds, a point which is presently investigated further.

H.J. Herrmann / Damage spreading

527

D boundor y

100

I

I

I

/

lO

X

I

i

I

I

xf

5O

20

I

I

J

f

3 2

o

1

0

o

o

o

o

I

I

I

I

10

20

0

1

I I I I

5O

100

L Fig. 6. L o g - l o g plot of the mass of the damage cloud on the boundaries of the lattice at the time of touching versus L. The symbols correspond to the two touching conditions: (©) at least one site on the boundary is damaged; ( x ) at least one site is damaged on each of the t h r e : boundary planes (100, 010, 001). For the touching condition in all three directions the scaling exponent is equai to about 0.75 (taken from ref. [12]).

4. Outlook

We saw that damage spreading fixing just one spin in a ferromagnetic model and using heat-bath dynamics can be related to thermodynamic correlation functions. This gives an efficient method to calculate these correlation functions and the fractal dimension df of the damage clusters at T c. The slow convergence towards df suggests the existence of multiscaling in the cluster density. If one uses other dynamics like Glauber [1, 3, 4, 15] or Metropolis [16] the relation to thermodynamic quantities does not hold anymore and novel, yet unexplained dynamical phases and phase transitions have been found. Also in spin glasses an interesting behaviour is observed in the case of heat-bath dynamics [5, 17, 18] which might help identifying the spin-glass phase.

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H.J. Herrmann / Damage spreading

Acknowledgements I thank Lucilla de Arcangelis, Antonio

C o n i g l i o a n d A n a n i a s M a r i z for

many interesting discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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