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Relation between quenched and annealed random ferromagnetic systems
Volume 72A, number 1
PHYSICS LETTERS
11 June 1979
RELATION BETWEEN QUENCHED AND ANNEALED RANDOM FERROMAGNETIC SYSTEMS Moshe SCHWARTZ Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel Received 11 January 1979
A sufficient condition for the average of the product of two functions of many variables to be larger than the product of their averages is obtained. The theorem is used to show that the spin—spin correlation function in a ferromagnetic quenched Ising system is bound from above by the same correlation in the corresponding annealed system. Possible extension of this result to a much wider class of models is discussed.
We start by stating a theorem, that gives a sufficient condition for the average of the product of two functions to be larger than the product of the averages, Consider two functions f(x) and g(x) which are both monotonically non-decreasing or non-increasing, Let p (x) be a probability distribution. Clearly,
Definition 2. Two functions f(x1, ...,x~,...,x~)and g(x1, x,, x~)will be said to be co-monotonous in the variable x1 if both are monotonous in x1 and if both are non-decreasing or non-increasing in x1. (If the functions are differentiable in x, this means that (af/ax~)(x1, x~, Xn) (ag/axe) (x1, x1, x~) for all (x1, Xi, x~). The following theorem can be easily proven by induction on the number of variables. Theorem. Let p(x1, ..,x~)be a probability distribution, which can be factorized into probability distributions in each variable separately, ...,
...,
..,,
...,
...,
f f [f(x)
—
f(y)] [g(x)
—
g(y)] ~1,\
—~
~-
Xp~x,p(y)dxdy-O.
One obtains that, (fg>>~(f>(g),
,,
r f~x,p~x, ~~ ,-
(3\ ‘~ ~ -
-
.
The theorem we want to use is a generalization of eq. (2) to functions of many variables, Consider the following definitions: Definition 1. A function f(x1, ...,x1, ..,x~)will be said to be monotonous in x1 if for all x1, x~_1’ x~4.1, x~fixed, it is a monotonous function of xi and if it is non-decreasing (non-increasing) for a given set ofx1, ..,,x,_1 ~x~+i~.,.,x,~,it is also non-decreasing (non-increasing) for any other set of those variables. (1ffis differentiable with respect to x~this means that the sign of (af/ax,)(x1, ...,xz, ~ does not change from one point to another.) ...,
...,
-
-
n
p(x1,
-
...,
.
(2)
where the average is defined as ~——
.
.,.,
(4)
.~,Xn)= k=1
andletf(x1,...,x~)andg(x1,...,x~)becomonotonoU5 in all their variables, then (5)
(J~)~,
where the averages are taken with respect to p~ A non-correlated random bond, annealed, Ising system, is an Ising system in which each coupling strength is a dynamical variable governed by a “bare” distribution function P (J, ‘i~~ ~ Such a system can be described by the hamiltonian ...,
37
Volume 72A, number 1
Hr
—
Ej1.cj.a. (i,j)
/
PHYSICS LETTERS
Elnp(J.. ~
~
/
(1/)
EHo!3~’~lnp(J~1, ...)
‘~‘
) ~‘
‘‘
(6)
,
11 June 1979
calculate any spin average in that configuration and only then average over the bond configurations, because the coupling strengths do not play the roles of dynamical variables:
r tr0f(u1,
(1/)
where the summation is over pairs of sites. A nearest-neighbour system will be characterized by p(J~1,p1,...,p~)=~(J11),
(7)
for a non-nearest-neighbour bond. Only ifp(J11, 111, Pn) is (.J,~ J), the probability ~P(1J)(J,dJ)of finding the given bond (i,j) in the range between J and J + dJ is given by ,,,‘
~
—
9~1(J,ci.’) = p(~~1~ p1, p~)dJ (8) Otherwise the dressed ~ is given by 11B~(i,j)~“~Be ~H tr~1 ~j 1V, dJ)— n ci.’ e~ d.J~1—p(J11)ciJ, ...~
-
—
—
—
~ir a
B
B
(9) where the product over B is a product over all bonds. The annealed systems discussed usually in the literature [1,2] are systems where the knowledge of the “bare” distribution function is incomplete, while there is complementary knowledge about the dressed distribution function. The incomplete knowledge of the “bare” distribution function is manifest in the undetermined parametersp1, ...,p,~.The complementary knowledge of the dressed distribution function is knowledge of n independent characteristics of the dressed distribution function. For example, we may have a “bare” distribution function of the form p(J)p(J,p),
(10)
and knowledge of the average of J,J, in the system namely: J=f~(J)J~.
(11)
In fact, eq. (11) combined with eq. (9) serves to determine p. A generalization to a “bare” distribution depending on n parameters and n constraints on the n first moments of the “dressed” distribution is straightforward. In a quenched random system on the other hand [3] one has to consider a given configuration of bonds, ,
38
(~‘(o~, ..., G~))=~
UI
X
...,
a~)e~H0
tr e~~o (12)
dJBP(JB),
where the distribution p is temperature independent. In a quenched system, there is only one distribution and there is nothing that can “dress” it. Since we will be interested in a system, that will be a reference system to the quenched system, we will consider an annealed system, which is not designed to describe the physical distribution is known situation to some in extent which while the “dressed” the “bare” distribution is not known to some extent. We will consider a system in which there are only ferromagnetic nearest-neighbour interactions governed by a known temperature-independent probability distribution. The system can be described by the hamiltonian
E
1
—
H—
(i
1)111/
—
(2i3)
~
lnp(J~,)
~H0{J11} _j3_1 ~lnP(Ji1)
(13)
“I
where the summation is over nearest-neighbour pairs and where the probability distribution p is normalized, fp(J)dJ=l
(14)
,
—~
and is different from zero for ferromagnetic couplings only: p(J)O forJ
f f UI p(J..)dJ
—
‘‘
tr e~~0~i/}
(i ~
2, (16) in where Z is the averaged partition function over the cou~ —
Volume 72A, number 1
PHYSICS LETTERS
pling strength distribution. Spin averages are computed using the following equation:
f..-f 11~1~P(J11) dJ~trA {ok} e
~
f...f11(i/)P(Jif)dijtre~H0~~~1} ‘
(17) It may be easily verified that
I”
(A {~k}~ ~‘‘
“
JP(J~1)dJ~,
X
(18)
,
where Z is the partition function of an Ising ~ tem defined by the fixed couplings {J~~} and =
2’
f f fl ~(J..) d.J~,j
1
gim {J~,}z {J.~}
,
(19)
(ii)
where the superscript a stands for annealed, Since we are dealing with ferromagnetic interactions we see at once that Z {J,1 } is an increasing function of each ~ji1separately and so is g according to the Griffiths [4] inequality. By use of the theorem proven earlier we obtain now
~ f-.. f “ P(-~~)d.Ji/gj~{Jj/ } Z {Jj/} (20) ~a2 x 2’ f... f {J..} =
(ii)
p(Jj~)cJJijgim
=
g~l
where the superscript q stands for quenched, since the right-hand side of the inequality is the same correlation function in the corresponding quenched system. Since the spin—spin correlation function of the quenched system is bound from above by the correlation functionof ofthe theannealed annealedsystem, system, Ta, the istransition temperature an upper bound on the transition temperature, Tq, of the quenched system: Tq ~ Ta (21) Me the above results limited to the model we con.
11 June 1979
sidered? The only properties we used are the facts that Z{J,1} and g~rn{~~1} are increasing functions of each of the couplings separate’y. The same result holds for any Ising system described by ferromagnetic interaction among the spins, provided the probability distribution of couplings is separable. Cana higher we extend the above results to systems possessing symmetry? Will they hold, for example, for a system of n-component classical spins interacting via two-body ferromagnetic interaction? For the case n = 2 the answer is definitely yes. Z is an increasing function of all couplings and the Griffiths inequality needed to show that is also an increasing function of all the couplings was proven by Ginibre [5] for that case. For n > 2 it is easy to show, that Z is an increasing function of all couplings, when none of them is negative. The Griffiths inequality is not known to hold for n > 2 although the heuristic arguments given by Griffiths for the case n = I seem to be valid for any n. The conclusion is that our results (20) and (21)can be extended rigorously to very general ferromagnetic Ising and X— Y systems and on a heuristic basis to ferromagnetic systems of classical spins possessing higher symmetries. When is the upper bound on T obtained in eq. (21) useful numerically? The answer the bound on Tq is useful ii we can calculate easily Ta. A case where the calculation of Ta is easy in terms of the constant coupling model, is the Ising one. It may be easily yenfled by using the high-temperature expansion to all orders that if an average of a given function of the spins A ~°k} is given as
(22)
in the case of constant coupling, in the case of the random annealed problem described by eq. (13),
(23)
fp(J)sinhjYdJ
fp(J)coshjlJd.J so that the inverse transition temperature by laIr &‘ [tghj3~J1 ,
1~ais
(24) defined (25)
where 13c is the inverse transition temperature in the constant coupling model. Even though the above treatment seems to work 39
Volume 72A, number 1
PHYSICS LETTERS
only for Ising systems it looks as if it will be always easier to calculate physical quantities for the annealed problem than for the corresponding quenched one. References [1] M.E. Fisher, Phys. Rev. 176 (1968) 257.
40
II June 1979
[21 D. Bergman, Y. Imry and L. Gunther, J. Stat. Phys. 7 (1973) 337. [31 R. Brout, Phys. Rev. 115 (1959) 824. [4] R.B. Griffiths, J. Math. Phys. 8 (1968) 478. [5] I. Ginibre, Commun. Math. Phys. 16 (1970) 310.
PHYSICS LETTERS
11 June 1979
RELATION BETWEEN QUENCHED AND ANNEALED RANDOM FERROMAGNETIC SYSTEMS Moshe SCHWARTZ Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel Received 11 January 1979
A sufficient condition for the average of the product of two functions of many variables to be larger than the product of their averages is obtained. The theorem is used to show that the spin—spin correlation function in a ferromagnetic quenched Ising system is bound from above by the same correlation in the corresponding annealed system. Possible extension of this result to a much wider class of models is discussed.
We start by stating a theorem, that gives a sufficient condition for the average of the product of two functions to be larger than the product of the averages, Consider two functions f(x) and g(x) which are both monotonically non-decreasing or non-increasing, Let p (x) be a probability distribution. Clearly,
Definition 2. Two functions f(x1, ...,x~,...,x~)and g(x1, x,, x~)will be said to be co-monotonous in the variable x1 if both are monotonous in x1 and if both are non-decreasing or non-increasing in x1. (If the functions are differentiable in x, this means that (af/ax~)(x1, x~, Xn) (ag/axe) (x1, x1, x~) for all (x1, Xi, x~). The following theorem can be easily proven by induction on the number of variables. Theorem. Let p(x1, ..,x~)be a probability distribution, which can be factorized into probability distributions in each variable separately, ...,
...,
..,,
...,
...,
f f [f(x)
—
f(y)] [g(x)
—
g(y)] ~1,\
—~
~-
Xp~x,p(y)dxdy-O.
One obtains that, (fg>>~(f>(g),
,,
r f~x,p~x, ~~ ,-
(3\ ‘~ ~ -
-
.
The theorem we want to use is a generalization of eq. (2) to functions of many variables, Consider the following definitions: Definition 1. A function f(x1, ...,x1, ..,x~)will be said to be monotonous in x1 if for all x1, x~_1’ x~4.1, x~fixed, it is a monotonous function of xi and if it is non-decreasing (non-increasing) for a given set ofx1, ..,,x,_1 ~x~+i~.,.,x,~,it is also non-decreasing (non-increasing) for any other set of those variables. (1ffis differentiable with respect to x~this means that the sign of (af/ax,)(x1, ...,xz, ~ does not change from one point to another.) ...,
...,
-
-
n
p(x1,
-
...,
.
(2)
where the average is defined as ~——
.
.,.,
(4)
.~,Xn)= k=1
andletf(x1,...,x~)andg(x1,...,x~)becomonotonoU5 in all their variables, then (5)
(J~)~
where the averages are taken with respect to p~ A non-correlated random bond, annealed, Ising system, is an Ising system in which each coupling strength is a dynamical variable governed by a “bare” distribution function P (J, ‘i~~ ~ Such a system can be described by the hamiltonian ...,
37
Volume 72A, number 1
Hr
—
Ej1.cj.a. (i,j)
/
PHYSICS LETTERS
Elnp(J.. ~
~
/
(1/)
EHo!3~’~lnp(J~1, ...)
‘~‘
) ~‘
‘‘
(6)
,
11 June 1979
calculate any spin average in that configuration and only then average over the bond configurations, because the coupling strengths do not play the roles of dynamical variables:
r tr0f(u1,
(1/)
where the summation is over pairs of sites. A nearest-neighbour system will be characterized by p(J~1,p1,...,p~)=~(J11),
(7)
for a non-nearest-neighbour bond. Only ifp(J11, 111, Pn) is (.J,~ J), the probability ~P(1J)(J,dJ)of finding the given bond (i,j) in the range between J and J + dJ is given by ,,,‘
~
—
9~1(J,ci.’) = p(~~1~ p1, p~)dJ (8) Otherwise the dressed ~ is given by 11B~(i,j)~“~Be ~H tr~1 ~j 1V, dJ)— n ci.’ e~ d.J~1—p(J11)ciJ, ...~
-
—
—
—
~ir a
B
B
(9) where the product over B is a product over all bonds. The annealed systems discussed usually in the literature [1,2] are systems where the knowledge of the “bare” distribution function is incomplete, while there is complementary knowledge about the dressed distribution function. The incomplete knowledge of the “bare” distribution function is manifest in the undetermined parametersp1, ...,p,~.The complementary knowledge of the dressed distribution function is knowledge of n independent characteristics of the dressed distribution function. For example, we may have a “bare” distribution function of the form p(J)p(J,p),
(10)
and knowledge of the average of J,J, in the system namely: J=f~(J)J~.
(11)
In fact, eq. (11) combined with eq. (9) serves to determine p. A generalization to a “bare” distribution depending on n parameters and n constraints on the n first moments of the “dressed” distribution is straightforward. In a quenched random system on the other hand [3] one has to consider a given configuration of bonds, ,
38
(~‘(o~, ..., G~))=~
UI
X
...,
a~)e~H0
tr e~~o (12)
dJBP(JB),
where the distribution p is temperature independent. In a quenched system, there is only one distribution and there is nothing that can “dress” it. Since we will be interested in a system, that will be a reference system to the quenched system, we will consider an annealed system, which is not designed to describe the physical distribution is known situation to some in extent which while the “dressed” the “bare” distribution is not known to some extent. We will consider a system in which there are only ferromagnetic nearest-neighbour interactions governed by a known temperature-independent probability distribution. The system can be described by the hamiltonian
E
1
—
H—
(i
1)111/
—
(2i3)
~
lnp(J~,)
~H0{J11} _j3_1 ~lnP(Ji1)
(13)
“I
where the summation is over nearest-neighbour pairs and where the probability distribution p is normalized, fp(J)dJ=l
(14)
,
—~
and is different from zero for ferromagnetic couplings only: p(J)O forJ
f f UI p(J..)dJ
—
‘‘
tr e~~0~i/}
(i ~
2, (16) in where Z is the averaged partition function over the cou~ —
Volume 72A, number 1
PHYSICS LETTERS
pling strength distribution. Spin averages are computed using the following equation:
f..-f 11~1~P(J11) dJ~trA {ok} e
~
f...f11(i/)P(Jif)dijtre~H0~~~1} ‘
(17) It may be easily verified that
I”
(A {~k}~ ~‘‘
“
JP(J~1)dJ~,
X
(18)
,
where Z is the partition function of an Ising ~ tem defined by the fixed couplings {J~~} and =
2’
f f fl ~(J..) d.J~,j
1
gim {J~,}z {J.~}
,
(19)
(ii)
where the superscript a stands for annealed, Since we are dealing with ferromagnetic interactions we see at once that Z {J,1 } is an increasing function of each ~ji1separately and so is g according to the Griffiths [4] inequality. By use of the theorem proven earlier we obtain now
~ f-.. f “ P(-~~)d.Ji/gj~{Jj/ } Z {Jj/} (20) ~a2 x 2’ f... f {J..} =
(ii)
p(Jj~)cJJijgim
=
g~l
where the superscript q stands for quenched, since the right-hand side of the inequality is the same correlation function in the corresponding quenched system. Since the spin—spin correlation function of the quenched system is bound from above by the correlation functionof ofthe theannealed annealedsystem, system, Ta, the istransition temperature an upper bound on the transition temperature, Tq, of the quenched system: Tq ~ Ta (21) Me the above results limited to the model we con.
11 June 1979
sidered? The only properties we used are the facts that Z{J,1} and g~rn{~~1} are increasing functions of each of the couplings separate’y. The same result holds for any Ising system described by ferromagnetic interaction among the spins, provided the probability distribution of couplings is separable. Cana higher we extend the above results to systems possessing symmetry? Will they hold, for example, for a system of n-component classical spins interacting via two-body ferromagnetic interaction? For the case n = 2 the answer is definitely yes. Z is an increasing function of all couplings and the Griffiths inequality needed to show that is also an increasing function of all the couplings was proven by Ginibre [5] for that case. For n > 2 it is easy to show, that Z is an increasing function of all couplings, when none of them is negative. The Griffiths inequality is not known to hold for n > 2 although the heuristic arguments given by Griffiths for the case n = I seem to be valid for any n. The conclusion is that our results (20) and (21)can be extended rigorously to very general ferromagnetic Ising and X— Y systems and on a heuristic basis to ferromagnetic systems of classical spins possessing higher symmetries. When is the upper bound on T obtained in eq. (21) useful numerically? The answer the bound on Tq is useful ii we can calculate easily Ta. A case where the calculation of Ta is easy in terms of the constant coupling model, is the Ising one. It may be easily yenfled by using the high-temperature expansion to all orders that if an average of a given function of the spins A ~°k} is given as
(22)
in the case of constant coupling, in the case of the random annealed problem described by eq. (13),
(23)
fp(J)sinhjYdJ
fp(J)coshjlJd.J so that the inverse transition temperature by laIr &‘ [tghj3~J1 ,
1~ais
(24) defined (25)
where 13c is the inverse transition temperature in the constant coupling model. Even though the above treatment seems to work 39
Volume 72A, number 1
PHYSICS LETTERS
only for Ising systems it looks as if it will be always easier to calculate physical quantities for the annealed problem than for the corresponding quenched one. References [1] M.E. Fisher, Phys. Rev. 176 (1968) 257.
40
II June 1979
[21 D. Bergman, Y. Imry and L. Gunther, J. Stat. Phys. 7 (1973) 337. [31 R. Brout, Phys. Rev. 115 (1959) 824. [4] R.B. Griffiths, J. Math. Phys. 8 (1968) 478. [5] I. Ginibre, Commun. Math. Phys. 16 (1970) 310.