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The application of the finite cluster approximation to the kinetic Ising model
Journal of Magnetism and Magnetic Materials 92 (1990) 245-250 North-Holland
245
The application of the finite cluster approximation Ising model M. Kerouad and M. Saber
to the kinetic
*
DGpartement de Physique, Facuh! des Sciences, B.P. 4050, Meknes, Morocco Received 15 December 1989; in revised form 2 April 1990
The Ising model provides a useful description for physical systems in which a localized variable can take on either of two discrete values. We have investigated the influence of bond and site dilution on the two-dimensional spin-l/2 Ising model as a dynamical model on a honeycomb lattice within the finite cluster approximation. We show that the dilute system presents a striking similarity with the pure system.
1. Introduction The magnetic properties of amorphous and disordered materials have been the object of much interest both from the experimental and from the theoretical point of view. We wish to point out in this paper the relevance of a simple formalism, the finite cluster approximation (FCA), recently introduced by Boccara [l], for the investigation of the dynamical aspects of cooperative systems As the model of such a system, we chose a simple one, namely an Ising spin system in which the interaction of spins with a heat bath induces spanteneous flips. The interaction of spins affects the probability of the transition process of a given spin, which is dependent on the configuration of the surrounding spins as it should be in order to assure the canonical distribution in the equilibrium state, but its functional form is assumed to be the simplest. We discuss in this paper the application of FCA to the kinetic diluted spin- $ Ising model on the honeycomb lattice. The choice of the lattice allows the algebra to be kept simple, but other lattices can be treated in a straightforward way. To make this paper self-contained we review in
section 2 the model and the fundamental equations. The pure spin-g Ising model is introduced and treated in section 3. The diluted model is discussed in section 4. Section 5 contains a brief discussion of the results.
2. We consider a system of interacting lsing spins which are represented by the spin variable (u, ). They also interact with a large heat bath, at a constant temperature which will not be treated explicitly. The heat bath functions only in giving rise to spontaneous flips of spins by exchanging the energy. The probability of flip of the ith spin will be denoted by w( a, ). This depends not only on a, but also on the variables of the surrounding spins, which are suppressed in this notation for the sake of simplicity. We assume no master equation exists for this model
= -Cw,(a,)P(a +Cw;(-U,)P(O
i Laboratoire de Magnetisme, Departement de Physique, Faculte des Sciences, B.P. 1014, Rabat, Morocco. 0304~8853/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
,‘“.‘O ,.“” ,,...,
ON. t)
-0 ,.....
OK:, P), (I)
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
246
where P(a,, . . . , uN, t) is the probability to find the spins in the configuration (q, . . . , oN ). The first term on the right hand side is the loss of the
probability by flipping out; and the second term the gain by flipping from the opposite direction. Now, the transition probabilities w(a,) must satisfy a certain relation in order that the master equation assures a thermal equilibrium so that we have the equality:
where PO(O,, . . . , Ui,. . . , a,) is the probability in equilibrium. The energy of the spins is assumed to be given by: H=--
c J,f"luj
(3)
u, = It19
(4.j)
where Ji, is the interaction between the pair (i, j). Eq. (2) gives:
W-J,) W-q)
exp( - PO, ) = exp( PE, 0, ) ’
where p = l/T
(4)
and the local field E, is defined
by:
4 = CQJ, and is a function of spins surrounding spin. Eq. (4) may be written as:
Y(g) V--q)
(5)
(6)
thus we may write the transition prbability as:
Here y on the the ith The by:
Y&(0,)
= -((a,)
+ (tanh(PEi)))*
(9)
Thus the calculation problem has been reduced to the problem of solving differential equations of the sort, eq. (9), subject to certain initial conditions.
3. The finite cluster approximatiaq
of the
two
dimensional model
Let us describe the application of FCA to a pure spin-i Ising model with nearest-neighbour interactions. The I-Iamiltonian is: H=
-J
c Q,U,, (i.j)
(10)
where J > 0, ui = + 1 and the summation runs over all pairs of nearest-neighbours. If (uo& denotes the mean value of a0 for a given configuration c for all other spins, i.e. when all other spins a, (i f 0) have fixed values, the equation of motion of (u&)) at lattice site 0 in the kinetic Ising model has the form [2]:
the i th
1 - a, tanh( /Xii) = 1 + U, tanh( PEi ) ’
yb,) = & [ 1 - u, tanh( PEE,)I.
where the sum is taken over all the 2N possible configurations of spin system. It is straightforward to see from eqs. (1) and (7) that the expectation value (u,) satisfies the differential equation:
(7)
is a constant which may in general depend temperature and on the spins other than one. expectation value of the it
where y is the arbitrary time scale, K = EJJ and the summation is extended over the z nearestneighbours u,( i = 1, 2.. . . , z) of uo. The equation of motion is obtained by averaging tanh( KXi= ,ui) over all configurations. This is a formidable task and the mean field approximation corresponds to the very crude estimate: (tanh[Ki,u,])
= tanh(zKm).
(12)
which corresponds to the probability distribution: (1% i
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
247
As the random variables si take the values - 1 and + 1, a better choice [l] is: kA(@i)) =
n(
~d(O,-l)+~~(*i+l)),
i
which still neglects correlations between different spins but takes exactly into account relations like Ui*= 1. To average tanh( KC:= lai) when the ui are distributed according to this approximate probabiltiy law, it is easier to use the following theorem: the set of all bounded real functions of ut, u2, . . . , uz is a 2*-dimensional Euclidean space. The set (1, u1’“” u:, u,u2,..., u,u2 . . . CL), which contains all the products of different spins, is an orthonorma1 basis for the inner product defined by: (fi/f2)
=
Y$ Tr
fi(~~,..*,u=)Jf~((~~~...,u~).
t Fig. 1. The variation of m( t ) as function of t.
The solution of eq. (17) for T z T, is given by:
U,,...,Uz
(15) If, for instance, have:
2 = 3 (honeycomb
lattice) we
m(t)
= Q+
(l--;;
-;I
e”‘-f”,Y
]I”
(‘*)
tanh[ K( ui + u2 + Us)] = a[tanh(3K) + i(tanh(3K)
+ tanh(K)](a,
and for T = TC by:
+ a2 + u3)
- 3 tanh(K))u,u2u.1
(16)
and when we average the right hand side of eq. (11) approximating (u,u2u~) by m3, the equation of motion has the form: d v-;iiTm(‘i
= -(I
-r)m(t)
+ Qm3(t),
(17)
where c = f [ tanh( 3K ) + tanh( K )] Q = $ [ tanh( 3K ) - 3 tanh( K
)] .
m(t)=
1
--i m’(0)
2Qt Y
- 1,‘2 1
’
where m(O) is the initial value of m( t ). Fig. 1 shows relaxation for T > T,, T < TCand for T = T,. For small deviations from equilibrium at T> T,, thz relaxation can be described by a simple relaxation time. However, as the temperature approaches the critical point, the linear term of relaxation tends to vanish, so that the higher order term becomes important.
of eq, (17) for The st_at_ionary stl!u!ion dm( t )/d t = 0 gives the simplest finite cluster ap-
proximation solution for equilibrium. Within this approximation the critical temperature is the solution of: 1 - c( T,) = 0. i.e., T, = 2.10 J, which is to be compared with the mean field result, T, = 3 J and the exact one T, = 1.519 J [3].
The extension of this method to the less trivial case of random models is straightforward. If. for instance, the reduced interaction K,, are random
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
248
variables, the equation given by:
of motion, for z = 3. is
nomial forms. This leads to the following d
yam(t)
-(u,,(t)), + (tanh(K,,q+ Ko2u2+ K03d)-
=
equa-
tion:
(20) The theormm from which we obtained ea. (17) L.
from eq. (11) can be used here. It
-
gives:
= - [I -+(p,
+ Qd
P,
K)lm(d
K)m3(t)t
(22)
where the coefficients, which are averaged over disorder, are functions of K and (n) = p, and they are given by: El3(P, K)=G%+A,+A,)
Y&&)),
ZZ- : [(4p
= -(u*(c))c+ (A+?, fAp2 +A,u3 + +72cJ,)
+
+p3
where: QB(P~
A, = $ [ tanh( K,, + Ko2 + K,,) + tanh( K,,
-%2+K03)
+ tanh(
+
+ tanh( A,
=
a [
K,,
-
Ko2 Ko2
p -
413)
K,,)],
+tanh(-K,,+ K,i-K,,)],
and the bond percolation
- tanh(
K,,-kx+~,,)
+ tanh(
Kc,,
ni
u
h,.,,-l
uullu-ul1uI~u
.-a;1..trr04
for
threshold
p;l;
by:
motion (22) takes the form: d
K,,+ K,,- K,,)
,\c WI
(22)
i.e., pg = 0.557 which is rather good compared with the exact result ,I$ = 0.6527 [4]. To put K + oo i.e, T+ 0, the equation of
K,,+ Ko3)
- tanh(
K,,
eq.
,
K)] =O,
lim [l -cB(p,
K-rcc
B=$[tanh(K,,+K,,+K,,)
-
of
>]
K)=O
l-~(p,
+
3K )- 3 tanh( K
is the boli Loncentration. The stationary solution
+ tanh( K,,+ K,,- K,,)
+ tanh( - K,,- Km + Kdl'
Y~Pa(f)= - [l -%I( P)lW) +QdP)GW~
-
c-.,FCQrn Y 3yJLblal I\(),
(24)
where
Ko3 )] . =
Y,
1x11,
..v 1,**a Wllblb
ii,
the occupation number of bond Oi. The coefficients of eq. (21) are functions of the discrete random variables n,, n 2 and n 3. To average first over all spin configurations and then over disorder, we use the same theorem: writing all functions of n,, n, and n3 in polyis
K)=(B)
t
-K,,-tKoz+Ko3)
+ tanh( -K,,
(23)
dm( )/dt = 0 gives the simplest finite cluster approximation solution for equilibrium. Within this approximation, the second order transition line is determined by:
4 = i[tanh(K,,, + Ko2+ K,,) + tanh( K,, -K0*+K,3)
c
tanh(3K )],
= fp3 [ tanh(
tanh( K,, + K,, -k K,,)
+tanh(
1
4( p2 - p”) tanh(2 K )
(21)
‘)
~ol
- 8p2 + 5p3) tanh( K)
1:Es( p/\ -- Kllll$
--+
B(~)=iimmQB(p, -9
p, 1’&-\=313,Ln,2+/4,\ / 4\LY TY K)=
-ff/
-Sp3
and PB( l) is the percolation probability in the percolation theory which is analogous to the magnetisation of a ferromagnct.
M. Kerouad, M. Saber / Kinetic Ising model and finite clwer approximation
249
t
t
Fig. 2. The variation of Pe( t ) as function of t.
Fig. 3. The variation of Ps(t) as function of t.
The eq. (24) describes the percolation effects in a Kinetic Ising model. The solution of eq. (24) for p # pg is given by:
problem except that they are multiplied by an extra factor p. The expressions of E and Q are given by:
&3(t) = (l-%(P)) i
1
x \Qd P) +
-
%(P,
K) =P%(P~
K),
Qsb
O=PQ,(P,
0
The second order transition determined by:
QB(P)
line is. in this case.
I - cS( p, K) = 0, p is now the site concentration.
and for p =pg
by:
.
(26)
The corresponding diagram PB( t )/PB( t = 0), as a function of t for different values of p is represented in fig. 2. It shows a striking similarity to the diagram of the pure system given in fig. 1, .., h.ara hPrP 1 - yn p’aJ3 T%lm.rrc the% rs4P nf l hP lamnarnt¶.ra W11b1b *aLdakd A L11td 1”&b “1 Saab LL4IIyLdcLI”I b T. for a site diluted system K,, = #non, where no and n, are the occupation number of sites 0 and i. A simple calculation shows that the equation of motion is given by eq. (22) where the coefficients are functions of K and p. Their dependency on those variables is the same as that of the bond
The site percolation thershold p< is equal to 0.801 to be compared with the result obtained by series expansions p.$ = 0.698 [5]. The analogous equation to eq. (24) which describes the site dilution effects in the limit K + co, i.e. T --) 0 in the Kinetic Ising model is the same as eq. (24) except that the coefficients cB( p) and QB( p) are multiplied by an extra factor p. The corresponding phase diagram which describes percolation effects is given in fig. 3. tt has the same general features as the one we obtaine for bond dilution.
FCA has allowed us to recover, by means of comparatively simple calculation. the phase di-
250
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
agrams for the kinetic diluted Ising models. The pure system has the same qualitative features as the ones obtained by Suzuki et al. [2] and Ilkovic i61. FCA allows one to treat site percolation as easily as bond percolation. The FCA has, however, one essential limitation: one has to know the order parameter.
References [l] N. Boccara, Phys. Lett. A 94 (1983) 185. [2] M. Suzuki and R. Kubo, J. Phys. Sot. Japan 24 (1968) 51. [3] M.S. Green and A. Hurst, Order Disorder Phenomena, Monographs in Statistical Physics (Wiley, London, 1964). [4] M.F. Sykes and J.W. Essam, Phys. Rev. 133 A (1964) 310. [5] M.F. Sykes, D.S. Gaunt and M. Glen, J. Phys. A 9 (1976) [6] Y’Bkovic, Phys. Stat. Sol. (b) 149 (1988) K55.
245
The application of the finite cluster approximation Ising model M. Kerouad and M. Saber
to the kinetic
*
DGpartement de Physique, Facuh! des Sciences, B.P. 4050, Meknes, Morocco Received 15 December 1989; in revised form 2 April 1990
The Ising model provides a useful description for physical systems in which a localized variable can take on either of two discrete values. We have investigated the influence of bond and site dilution on the two-dimensional spin-l/2 Ising model as a dynamical model on a honeycomb lattice within the finite cluster approximation. We show that the dilute system presents a striking similarity with the pure system.
1. Introduction The magnetic properties of amorphous and disordered materials have been the object of much interest both from the experimental and from the theoretical point of view. We wish to point out in this paper the relevance of a simple formalism, the finite cluster approximation (FCA), recently introduced by Boccara [l], for the investigation of the dynamical aspects of cooperative systems As the model of such a system, we chose a simple one, namely an Ising spin system in which the interaction of spins with a heat bath induces spanteneous flips. The interaction of spins affects the probability of the transition process of a given spin, which is dependent on the configuration of the surrounding spins as it should be in order to assure the canonical distribution in the equilibrium state, but its functional form is assumed to be the simplest. We discuss in this paper the application of FCA to the kinetic diluted spin- $ Ising model on the honeycomb lattice. The choice of the lattice allows the algebra to be kept simple, but other lattices can be treated in a straightforward way. To make this paper self-contained we review in
section 2 the model and the fundamental equations. The pure spin-g Ising model is introduced and treated in section 3. The diluted model is discussed in section 4. Section 5 contains a brief discussion of the results.
2. We consider a system of interacting lsing spins which are represented by the spin variable (u, ). They also interact with a large heat bath, at a constant temperature which will not be treated explicitly. The heat bath functions only in giving rise to spontaneous flips of spins by exchanging the energy. The probability of flip of the ith spin will be denoted by w( a, ). This depends not only on a, but also on the variables of the surrounding spins, which are suppressed in this notation for the sake of simplicity. We assume no master equation exists for this model
= -Cw,(a,)P(a +Cw;(-U,)P(O
i Laboratoire de Magnetisme, Departement de Physique, Faculte des Sciences, B.P. 1014, Rabat, Morocco. 0304~8853/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
,‘“.‘O ,.“” ,,...,
ON. t)
-0 ,.....
OK:, P), (I)
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
246
where P(a,, . . . , uN, t) is the probability to find the spins in the configuration (q, . . . , oN ). The first term on the right hand side is the loss of the
probability by flipping out; and the second term the gain by flipping from the opposite direction. Now, the transition probabilities w(a,) must satisfy a certain relation in order that the master equation assures a thermal equilibrium so that we have the equality:
where PO(O,, . . . , Ui,. . . , a,) is the probability in equilibrium. The energy of the spins is assumed to be given by: H=--
c J,f"luj
(3)
u, = It19
(4.j)
where Ji, is the interaction between the pair (i, j). Eq. (2) gives:
W-J,) W-q)
exp( - PO, ) = exp( PE, 0, ) ’
where p = l/T
(4)
and the local field E, is defined
by:
4 = CQJ, and is a function of spins surrounding spin. Eq. (4) may be written as:
Y(g) V--q)
(5)
(6)
thus we may write the transition prbability as:
Here y on the the ith The by:
Y&(0,)
= -((a,)
+ (tanh(PEi)))*
(9)
Thus the calculation problem has been reduced to the problem of solving differential equations of the sort, eq. (9), subject to certain initial conditions.
3. The finite cluster approximatiaq
of the
two
dimensional model
Let us describe the application of FCA to a pure spin-i Ising model with nearest-neighbour interactions. The I-Iamiltonian is: H=
-J
c Q,U,, (i.j)
(10)
where J > 0, ui = + 1 and the summation runs over all pairs of nearest-neighbours. If (uo& denotes the mean value of a0 for a given configuration c for all other spins, i.e. when all other spins a, (i f 0) have fixed values, the equation of motion of (u&)) at lattice site 0 in the kinetic Ising model has the form [2]:
the i th
1 - a, tanh( /Xii) = 1 + U, tanh( PEi ) ’
yb,) = & [ 1 - u, tanh( PEE,)I.
where the sum is taken over all the 2N possible configurations of spin system. It is straightforward to see from eqs. (1) and (7) that the expectation value (u,) satisfies the differential equation:
(7)
is a constant which may in general depend temperature and on the spins other than one. expectation value of the it
where y is the arbitrary time scale, K = EJJ and the summation is extended over the z nearestneighbours u,( i = 1, 2.. . . , z) of uo. The equation of motion is obtained by averaging tanh( KXi= ,ui) over all configurations. This is a formidable task and the mean field approximation corresponds to the very crude estimate: (tanh[Ki,u,])
= tanh(zKm).
(12)
which corresponds to the probability distribution: (1% i
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
247
As the random variables si take the values - 1 and + 1, a better choice [l] is: kA(@i)) =
n(
~d(O,-l)+~~(*i+l)),
i
which still neglects correlations between different spins but takes exactly into account relations like Ui*= 1. To average tanh( KC:= lai) when the ui are distributed according to this approximate probabiltiy law, it is easier to use the following theorem: the set of all bounded real functions of ut, u2, . . . , uz is a 2*-dimensional Euclidean space. The set (1, u1’“” u:, u,u2,..., u,u2 . . . CL), which contains all the products of different spins, is an orthonorma1 basis for the inner product defined by: (fi/f2)
=
Y$ Tr
fi(~~,..*,u=)Jf~((~~~...,u~).
t Fig. 1. The variation of m( t ) as function of t.
The solution of eq. (17) for T z T, is given by:
U,,...,Uz
(15) If, for instance, have:
2 = 3 (honeycomb
lattice) we
m(t)
= Q+
(l--;;
-;I
e”‘-f”,Y
]I”
(‘*)
tanh[ K( ui + u2 + Us)] = a[tanh(3K) + i(tanh(3K)
+ tanh(K)](a,
and for T = TC by:
+ a2 + u3)
- 3 tanh(K))u,u2u.1
(16)
and when we average the right hand side of eq. (11) approximating (u,u2u~) by m3, the equation of motion has the form: d v-;iiTm(‘i
= -(I
-r)m(t)
+ Qm3(t),
(17)
where c = f [ tanh( 3K ) + tanh( K )] Q = $ [ tanh( 3K ) - 3 tanh( K
)] .
m(t)=
1
--i m’(0)
2Qt Y
- 1,‘2 1
’
where m(O) is the initial value of m( t ). Fig. 1 shows relaxation for T > T,, T < TCand for T = T,. For small deviations from equilibrium at T> T,, thz relaxation can be described by a simple relaxation time. However, as the temperature approaches the critical point, the linear term of relaxation tends to vanish, so that the higher order term becomes important.
of eq, (17) for The st_at_ionary stl!u!ion dm( t )/d t = 0 gives the simplest finite cluster ap-
proximation solution for equilibrium. Within this approximation the critical temperature is the solution of: 1 - c( T,) = 0. i.e., T, = 2.10 J, which is to be compared with the mean field result, T, = 3 J and the exact one T, = 1.519 J [3].
The extension of this method to the less trivial case of random models is straightforward. If. for instance, the reduced interaction K,, are random
M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
248
variables, the equation given by:
of motion, for z = 3. is
nomial forms. This leads to the following d
yam(t)
-(u,,(t)), + (tanh(K,,q+ Ko2u2+ K03d)-
=
equa-
tion:
(20) The theormm from which we obtained ea. (17) L.
from eq. (11) can be used here. It
-
gives:
= - [I -+(p,
+ Qd
P,
K)lm(d
K)m3(t)t
(22)
where the coefficients, which are averaged over disorder, are functions of K and (n) = p, and they are given by: El3(P, K)=G%+A,+A,)
Y&&)),
ZZ- : [(4p
= -(u*(c))c+ (A+?, fAp2 +A,u3 + +72cJ,)
+
+p3
where: QB(P~
A, = $ [ tanh( K,, + Ko2 + K,,) + tanh( K,,
-%2+K03)
+ tanh(
+
+ tanh( A,
=
a [
K,,
-
Ko2 Ko2
p -
413)
K,,)],
+tanh(-K,,+ K,i-K,,)],
and the bond percolation
- tanh(
K,,-kx+~,,)
+ tanh(
Kc,,
ni
u
h,.,,-l
uullu-ul1uI~u
.-a;1..trr04
for
threshold
p;l;
by:
motion (22) takes the form: d
K,,+ K,,- K,,)
,\c WI
(22)
i.e., pg = 0.557 which is rather good compared with the exact result ,I$ = 0.6527 [4]. To put K + oo i.e, T+ 0, the equation of
K,,+ Ko3)
- tanh(
K,,
eq.
,
K)] =O,
lim [l -cB(p,
K-rcc
B=$[tanh(K,,+K,,+K,,)
-
of
>]
K)=O
l-~(p,
+
3K )- 3 tanh( K
is the boli Loncentration. The stationary solution
+ tanh( K,,+ K,,- K,,)
+ tanh( - K,,- Km + Kdl'
Y~Pa(f)= - [l -%I( P)lW) +QdP)GW~
-
c-.,FCQrn Y 3yJLblal I\(),
(24)
where
Ko3 )] . =
Y,
1x11,
..v 1,**a Wllblb
ii,
the occupation number of bond Oi. The coefficients of eq. (21) are functions of the discrete random variables n,, n 2 and n 3. To average first over all spin configurations and then over disorder, we use the same theorem: writing all functions of n,, n, and n3 in polyis
K)=(B)
t
-K,,-tKoz+Ko3)
+ tanh( -K,,
(23)
dm( )/dt = 0 gives the simplest finite cluster approximation solution for equilibrium. Within this approximation, the second order transition line is determined by:
4 = i[tanh(K,,, + Ko2+ K,,) + tanh( K,, -K0*+K,3)
c
tanh(3K )],
= fp3 [ tanh(
tanh( K,, + K,, -k K,,)
+tanh(
1
4( p2 - p”) tanh(2 K )
(21)
‘)
~ol
- 8p2 + 5p3) tanh( K)
1:Es( p/\ -- Kllll$
--+
B(~)=iimmQB(p, -9
p, 1’&-\=313,Ln,2+/4,\ / 4\LY TY K)=
-ff/
-Sp3
and PB( l) is the percolation probability in the percolation theory which is analogous to the magnetisation of a ferromagnct.
M. Kerouad, M. Saber / Kinetic Ising model and finite clwer approximation
249
t
t
Fig. 2. The variation of Pe( t ) as function of t.
Fig. 3. The variation of Ps(t) as function of t.
The eq. (24) describes the percolation effects in a Kinetic Ising model. The solution of eq. (24) for p # pg is given by:
problem except that they are multiplied by an extra factor p. The expressions of E and Q are given by:
&3(t) = (l-%(P)) i
1
x \Qd P) +
-
%(P,
K) =P%(P~
K),
Qsb
O=PQ,(P,
0
The second order transition determined by:
QB(P)
line is. in this case.
I - cS( p, K) = 0, p is now the site concentration.
and for p =pg
by:
.
(26)
The corresponding diagram PB( t )/PB( t = 0), as a function of t for different values of p is represented in fig. 2. It shows a striking similarity to the diagram of the pure system given in fig. 1, .., h.ara hPrP 1 - yn p’aJ3 T%lm.rrc the% rs4P nf l hP lamnarnt¶.ra W11b1b *aLdakd A L11td 1”&b “1 Saab LL4IIyLdcLI”I b T. for a site diluted system K,, = #non, where no and n, are the occupation number of sites 0 and i. A simple calculation shows that the equation of motion is given by eq. (22) where the coefficients are functions of K and p. Their dependency on those variables is the same as that of the bond
The site percolation thershold p< is equal to 0.801 to be compared with the result obtained by series expansions p.$ = 0.698 [5]. The analogous equation to eq. (24) which describes the site dilution effects in the limit K + co, i.e. T --) 0 in the Kinetic Ising model is the same as eq. (24) except that the coefficients cB( p) and QB( p) are multiplied by an extra factor p. The corresponding phase diagram which describes percolation effects is given in fig. 3. tt has the same general features as the one we obtaine for bond dilution.
FCA has allowed us to recover, by means of comparatively simple calculation. the phase di-
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M. Kerouad, M. Saber / Kinetic Ising model and finite cluster approximation
agrams for the kinetic diluted Ising models. The pure system has the same qualitative features as the ones obtained by Suzuki et al. [2] and Ilkovic i61. FCA allows one to treat site percolation as easily as bond percolation. The FCA has, however, one essential limitation: one has to know the order parameter.
References [l] N. Boccara, Phys. Lett. A 94 (1983) 185. [2] M. Suzuki and R. Kubo, J. Phys. Sot. Japan 24 (1968) 51. [3] M.S. Green and A. Hurst, Order Disorder Phenomena, Monographs in Statistical Physics (Wiley, London, 1964). [4] M.F. Sykes and J.W. Essam, Phys. Rev. 133 A (1964) 310. [5] M.F. Sykes, D.S. Gaunt and M. Glen, J. Phys. A 9 (1976) [6] Y’Bkovic, Phys. Stat. Sol. (b) 149 (1988) K55.