Equivalence between Glauber and heat bath dynanfics in damage spreading simulations

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Equivalence between Glauber and heat bath dynanfics in damage spreading simulations

Rita de Almeida

To cite this version:

Rita de Almeida. Equivalence between Glauber and heat bath dynanfics in damage spreading simu-

lations. Journal de Physique I, EDP Sciences, 1993, 3 (4), pp.951-956. �10.1051/jp1:1993175�. �jpa-

00246775�

Classification Physics Abstracts

05.40 05.50

Equivalence between Glauber and heat bath dynamics in damage spreading simulations

Rita M-C- de Almeida

Instituto de Fisica, UFRGS, C-P- 15051, 91501-970 Porto Alegre, RS~ Brazil

(Received

4 November 1992, accepted in final form 6 November

1992)

Abstract. The equivalence between Glauber and heat bath dynamics in spin glass damage

spreading simulations is demonstrated by considering explicitly the direction of the thermal noise

(stochastic,

localized

field)

acting _over the spins.

Slow relaxation _{processes are}

responsible

for many

interesting phenomena presented by complex

systems

(for

_a review _see Refs. [1,

2]). Although interesting,

these

phenomena

_are not

quite

well understood and _many different theoretical

approaches

and numerical

techniques

have been devised _to

study,

^for

example,

the

dynanfics

of

spin glasses

_or cellular automata. In

particular, damage spreading

simulations have been

widely applied

to

study

the

spin glass

transition and relaxation in _many models

(for

detailed

examples

_see Refs. [3]-[14] ^{and Refs.}

therein).

Damage spreading

simulations consist of

investigating

the evolution of two

initially

different

configurations

of _{a same}

sample

^of a system ^{in order} to obtain static and

dynamic properties.

In _some simulations the monitored

quantity

is the

Hamnfing

distance between the two

config-

urations that evolve

freely subject

to the _same thermal noise

[3]-[14].

^{The results show different} temperature

regimes,

^{that differ} in the

long

term value of the

Hamming

^distance between the two

configurations

and in its

dependence

on initial conditions. The available literature _on the

subject

accounts for results obtained with different

dynamics,

in

particular

heat bath and Glauber _ones, that show different behaviors in

spite

of

being physically equivalent (see,

for

example, refs.[8, 12]).

A second kind of

simulations, applied specifically

for

ferromagnets [15, 16],

considers two

configurations

A and B in thermal

equilibrium,

where the central

spin

So of _one of them is

kept

fixed in _one

given direction,

_say

S)(t)

=

S)(t

=

0),

for concreteness. The

configurations

are then let to evolve and two

quantities

_may ^{be monitored: the}

Hamming

distance and the time

dependent damage

difference

ro(I,t),

defined _as the difference between the

probability

df~

of

finding Sf(t)

=

S)(t

=

0)

and

Sf(t)

=

-S)(t

=

0)

^{and the}

probability d/+

of

finding Sf(t)

=

-S)(t

=

0)

and

Sf(t)

=

S)(t

=

0).

In _a recent paper Glotzer _et al. [16]

demonstrated that the time

dependent damage

^difference is related to the time

dependent spin

correlation function and is

independent

of the

microscopic dynanfics ruling

the evolution of the

952 JOURNAL DE PHYSIQUE I N°4

configurations: Glauber,

heat bath _{or any} other

equilibrium dynamics yield

the _same results for

ro(I, t).

However, different results _are obtained when the Glauber _or the heat bath

protocol

is used to measure other

quantities,

as the

Hamming

distance

D(t)

_or the accumulated

damage

difference

A(I, if),

defined _as the time average of

ro(I, t)

between t ₌ 0 and t ₌ if.

Here _we demonstrate that the differences obtained with heat bath and Glauber

protocols

for

dynamics-dependent quantities

in

damage spreading

simulations arise from _an

incomplete

account of the thermal noise in both

protocols.

A _more detailed

analysis

leads to the _same results for the two

dynamics.

To understand Glauber and heat bath

dynamics,

consider _a

given configuration

S =

(Si,52, ,SN)

of _a system ^{with N}

Ising spins,

described

by

the Hamiltonian

HIS).

Consider also _a

randomly

chosen

spin Si,

at _a

given

^simulation step, ^and

assign

the

probabili-

ties

P(S;)

and

P(-Si)

for the state of the

spin

after the

updating, given by

~j~~

^exP

i-flH)

j~~

exP

i-flH)

+ exP

i-fIHF)

~~~

~j_~~

^exP

i-fIHF)

j~~

exP

(-flH)

+ _exp

j-pH~)

where

fl

is the inverse of temperature and HF is the _energy of the system ^when Si has

flipped.

It is

straightforward

to obtain Glauber and heat bath

protocols

from

equations ii

and

(2):

I) ^Glauber

dynamics

considers the

flipping probability P~(Sj),

which is

given directly by equation (2),

that

is, P~(S;)

=

P(-Sj);

it)

Heat bath

dynamics

considers the

probability P~~(+I)

that

Si

= +I after the

updating,

which is

P~~(+I)

=

P(Si)

if Si ₌ +I and

P~~(+I)

=

Pi- Si)

if S; ₌ -I.

It is direct to

verify

from

equations ii)

and

(2)

that all four

possible

events

represented by

the

pairs

of

spin

states before and after the

updating, (+I, +I), (+I, -I), i-I, +I), (-I, -I),

have the _same

probabilities

for both

protocols: they

_are

physically equivalent

and

yield

the

same results when

simulating

the evolution of _a

single configuration.

Damage spreading simulations,

_on the other

hand, study

the evolution of _two

configurations,

and it has been

implemented

in such _{a way} that the _{same sequence} of random numbers is

compared

to the

(+I)-probability

for heat bath

or to the

flipping probability

for Glauber

dy- namics, yielding dynamics-dependent

final

results,

what is rather

puzzling

since both

dynamics

describe

equivalently

the _same

physical

situations.

The central idea in

damage spreading investigations

is _to somehow

study

the role

played by

the

complexity

of the

phase

_space in the relaxation _processes of _a system,

isolating

these effects from the

stochasticity implied by

the thermal noise. It is then clear that both

samples

must evolve with the _same thermal noise. For _a

given spin,

thermal noise is _a

local, instantaneously applied magnetic field;

for

Ising spins

it _means that the thermal noise may ^not

only

present

a random

intensity,

but also _{may occur} in two different orientations

(up

_or

down). When,

for

instance,

_an

up-spin

suffers _a

local, instantaneously applied

field in the _up

direction,

_no

matter how intense the field

is, nothing happens.

When

analyzing

the evolution of _a

unique configuration,

there is _a

probability

of 50~ that the thermal noise field is in the _same direction of the affected

spin

and the

only

difference that makes the

explicit

consideration of the vectorial

character of the stochastic field is _a renormalization of the relaxation time

by

_a factor 2 in Glauber and heat bath simulations

(in

both

protocols,

the field is

always

taken in the

opposite

direction of the

spin

because _a

change

ⁱⁿ the

spin

orientation is

always possible).

But when two

configurations

^{should evolve with the} same thermal

noise,

it _means that the thermal field should be in the _same direction for both

configurations:

simultaneous

spin flips

^from _up to

down in _one

configuration

and from down to up in the

other,

for

example,

_are forbidden.

However, both heat bath and Glauber standard

protocols,

in _some

instance,

allow such

flips:

this

neglect

of the directional

degree

of freedom of the thermal noise

originates

the differences found between Glauber and heat bath

damage spreading

simulations.

The

equivalence

between the _two

dynamics

when the orientation of the thermal noise is taken into account is demonstrated

by

direct enumeration of all

possible

events and related

probabilities.

Consider the

spins

of two

configurations, St

and

St, subject

to the thermal field

hf.

For Glauber

dynamics

_one considers the

flipping probabilities Pi

and

Pi

_for the i~~

spin

in

configurations

A and B

respectively,

that _are

given by

~~B

_"

_(I

_{+ ~XP}

(2~Sf'~ ~j#I ~ij Sl'~)

^~

=

l

₊

exP

1+2P£j~; Jo St~)1~~

13)

p+

A,B

with

Jij being

the

couplings

between

spins

and the

sign

+

depends

_on the value of the

spin

Sl'~

_The

probabdities Pfl~

and

Pf~

_in heat bath

dynamics

that the

i~~-spin

_assumes the

positive

value in each

configuration

is

~~i

" ~A,B

(~)

where

Pj~

is defined in

equation (3).

At each simulation step, these

probabilities

_are then

compared'to

a random number _z, which

is,

in _{some sense,} associated to the

intensity

of the stochastic field at that instant. As _an

example,

_we shall calculate _some of the

update joint probabilities

in both

protocols.

Consider that the chosen

spin

to

update

is such that

initially St

= +I and

St

= -I. For the final

configuration

in heat bath

dynamics

to be

St

= +I

and

St

=

+I,

the random number _z should be less than the smaller

+I-probability, given by equation (4),

that is _{z <}

ruin(Pi, Pi ).

^As _z is

uniformly

distributed the

joint probability

in the heat bath

dynamics

in this _case is

equal

to

min(Pi, _{Pi ).} Analogously,

for the final

configuration

in Glauber

dynamics

to be

St

= +I and

St

=

+I,

the random number _z should

simultaneously

be greater than the

St-flip-probability (Pf

= I

-Pj

and less than the

St-flip- probability (Pi ).

Such _an

update

is

only possible

^if

Pi

> I

P£

and the related

probability

is

given

by

max(0, Pi

I +

P().

These different

probabilities

for the _same event lead _to different

joint

evolutions when _{one or} other

protocol

is

applied.

Tables I and II enumerate all

possible

events and related

probabilities,

obtained _as in the above

example,

in

damage spreading

simulations for standard heat bath and Glauber

protocols respectively.

To take into account the direction of the thermal

perturbation acting

_over the

spins,

_{we prc-}

pose a "directional"

protocol

that _may be summarized

by

the

following spin update

routine:

choose first _a random direction

(+I

_or

-I)

for the stochastic

magnetic

field that is associated to the thermal noise and then _a random number _z to be

compared

to the

flip probability (Glauber)

_or the

+I-probability (heat bath).

When the field is in the _same direction of the

spin

of _a

given configuration, nothing happens

to it. When this is not the _case for _{one or} both

configurations,

the random number _z is

compared

to the

flip

or

+I-probability

to

update

the

spins.

To

illustrate,

consider the _case when

initially St

= +I and

St

=

-I,

and _we

shall calculate the

joint probability

that the

updated

states _are

St

= +I and

St

= +I. Now

we must take into account the direction of the thermal noise. When the thermal noise is in

the down

direction,

the

probability

that the

spin

in

configuration

B

flips

to the _up orientation is _zero, for _any

updating

protocol, ^and hence the

joint probability

in _{our case} is also _zero. When

954 JOURNAL DE PHYSIQUE I N°4

Table _I

Update joint probabilities

for the

spins

in two

configurations

A and _B

following

standard heat bath

dynamics.

The first two columns show the initial states and the

following

columns list the

probabilities

of the four final

possibilities.

The values of

P£~

are defined in

equation (3).

^'

S~ S~

p(+I, +I) p(+I, -I) p(-I, +I) p(-I, -I)

+I +I

minjPp, P~j

maxj0, P~ Pi j maxj0, Pi P~j minjl Pi, I Pi j +I -I

minjPp, Pij

maxj0, Pi

Ppj

maxj0, Pi

P~j

minjl P~, I

Ppj

~l +I

~~~l~i'~i1

~~1°'~i ~il

~~~1°'~i ~il

~~~l~ ~i' ^~

~il

-I -I minjP~,P~j maxj0, P~ P~j maxj0,P~ P~j minjl P~, I P~j

Table II

Update joint probabilities

for the

spins

in two

configurations

A and B

following

standard Glauber

dynamics.

The first two columns show the initial states and the

following

columns list the

probabilities

of the four final

possibilities.

The values of

Pi

~ are defined in

equation (3).

^'

S,

_S~

p(+I, +I) p(+I, -I) pi-I,

+

I) p(-

I,

-I)

+I +I

minjPi,

P~j maxj0, P~ Pij maxj0, Pi P~j mini I

Pi,

I Pi j

+I -1 maxj0, Pi + P~ lj minjl Pi, P~j

minjPj,

I P~j maxj0,1- P~

Pij

I +1 maxj0, P~ + Pi lj

minjP~,

I

Pij

minjl

Pi,

Pi j maxj0,1-

Pi P~j

I -I minjP~, P~j maxj0, P~ P~j maxj0, P~ P~ j minjl P~, I P~ j

Table III

Update joint probabilities

for the

spins

in two

configurations

A and B

following

Glauber _or heat bath

dynamics,

where the orientation of the thermal noise field has been

explicitly

considered. The first two columns show the initial states, the third indicates the direction of the thermal noise field

h/

and the

following

columns list the

probabilities

of the four final

possibilities.

The values of

Pj~

are defined in

equation (3).

+1 >0 0 0 0

+I +1 _< 0

min(Pi,Pjj

max(o, Pi Pjj max(o, Pi Pij min(i Pi, i Pjj

+I -1 >0 Pi

i-'Pj

0 0

+1 -1 <0 0 Pi ⁰ 1-Pi

-I +1 > 0 Pi ^{0 1} Pi ⁰

-1 +1 <0 0 0 Pi i-Pi

-I -1 _> 0 min(Pi,Pjj max(o, Pi Pjj max(o, Pi Pij min(i Pi, i

Pjj

-I -1 <0 0 0 0 1

the thermal noise is in the _up

direction, nothing happens

to the

spin

in the

configuration

A

probability

reduces to the

probability

that

St flips

in the Glauber

protocol (Pi),

which is

equal

to the heat bath

probability

that

St

= +I. Table III shows the results for the other

possibilities

when the orientation of the thermal noise

h/ acting

_over the

i~~-spin

is taken into account: for all

possible

events both

dynamics (flip

_or +I

probabilities compared

to the

random number

z) yield rigorously

the _same

probabilities.

In

damage spreading

simulations the evolution of the

Hamming

distance D between two

configurations

is _an

important quantity

to be measured.

Hence,

it is

clarifying

to calculate from tables I to 3 the values of the variation AD of the

Hamming distance,

and the related

averaged probabilities P,

in _a

single update

for the three

protocols. They

_are

AD~~=+I; P=@<(P[-Pp(>

AD~~=0; P=1-(-'~j/~~<(P[-Pp(> ₍₅₎

AD~~=-I; P=((I-<(P[-Pj(>)

for standard heat bath

dynamics,

AD~=+I; P=@<(P[-Pp(>

AD~=0; P=1-'~~~<(P[-Pp(>-§<max(0,Pj+Pp-1)> (6)

AD~=-I; P=(<max(0,P[+Pj-1)>

for standard Glauber

dynamics

and

finally, AD~~~

= +I P ₌

l§j/~

<

[Pi Pi

>

~~~~~

_~

'

~

§ ij/

_~

_{'~A ~B}

_>

_(~)

&

~DIR

_{_~ p} ^D

' 2N

for the "directional"

protocol,

where the

symbols

< > stand for _{an average over} all

possible pairs

and

DIN

and

IN D) IN

_are the

probabilities

that

St

=

-St

and

St

=

St respectively.

The above

equations

show that in the

high

temperature

limit,

for which the

probabilities Pfl~

go ^to

0.5,

the heat bath and "directional"

protocols yield

the _same evolution of the

Hammiig

distance

ID

-

0),

the

physically expected behavior,

and

they

also show the difference for the Glauber

dynamics

limit: for _an initial

Hamming

distance D ₌

0.5,

the

long time, high

temperature value in this _case is 0.5

(AD

₌

₀₎

_{[8, 12].}

This

equivalence

between Glauber and heat bath

(non-standard) dynamics,

obtained

through

the consideration of the orientation of the

noise,

which is

compatible

with the vectorial char-

acter of _a thermal _source that _may act over

spins, brings

to _an end the

puzzling

difference

observed between two

dynamics

that should be

physically equivalent.

It is _a

rigorous equiv-

alence: when the orientation of the noise is taken into account both Glauber and heat bath

protocols

not

only yield

the _same result to

dynamics-independent quantities,

but they are ex-

actly

the _same

dynamics.

The two standard

protocols

have been used to

investigate

several

properties

of

complex

systems and have somehow

managed

to grasp many

interesting

features

but,

_as shown

above,

not all

degrees

of freedom of the thermal noise

acting

_over the _con-

figurations

_were considered and hence _some external _sources of

damage

_were not discarded:

probably

_some deviations from the former results will be obtained when this _new

protocol

is used to

produce quantitative predictions.

It _seems that the

neglect

of the orientation of the stochastic field acts

against

the thermal

noise,

whose effect is to lower the

Hamming

distance

D,

in the

sense that it allows

(forbidden) flips

that tends _to separate ^the

configurations.

Conse-

quently,

transition temperatures _may have been overestimated. The directional

protocol

could

956 JOURNAL DE PHYSIQUE I N°4

then

yield

more

precise

results when

determining quantitatively

transition temperatures ^and relaxation

behaviors,

for

example.

Finally,

the orientation of the thermal noise may also _{appear as a} relevant

ingredient

in dam- age

spreading

simulations of other systems ^besides

spin glasses:

in

binary alloys,

for

example,

where order-disorder transition is

studied,

the

driving

force for atom diffusion is _a chemical

potential

that _may also present ^different orientations.

Acknowledgements.

1acknowledge

J-R-

Iglesias,

L-G-

Brunnet,

J-J-

Arenzon,

G.

Martinez,

and A-T- Bernardes for

helpful

discussions and careful

reading

of the

manuscript.

This work has been

partially

supported by

brazilian

agencies CNPq (Conselho

Nacional de Desenvolvimento Cientifico _e

Tecno16gico)

and FINEP

(Financiadora

de Estudos _e

Projetos).

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