Domain of Computation of a Random Field in ... - Semantic Scholar
Domain of Computation of a Random Field in Statistical Physics (Extended
Abstract)
Abbas Edalat Department of Computing Imperial College 180 Queen's Gate, London SW7 2BZ UK. Abstract
We present a domain-theoretic analysis of the invariant measure of the one-dimensional Ising model in a random external magnetic eld. The invariant measure is obtained as a xed point of the Markov transition operator of an iterated function system with probabilities acting on the probabilistic power domain of the upper space of a closed real interval. This enables us to use the generalised Riemann integral in combination with Elton's ergodic theorem to obtain an algorithm to compute the free energy density of the system. We also develop the generalised double Riemann integral, which we use, together with a two-dimensional version of Elton's theorem, to deduce algorithms to compute the magnetisation per spin and the Edwards-Anderson parameter of the system.
1 Introduction The Ising model was introduced by Ising as a model for ferromagnetism some seventy years ago; it also describes such systems as lattice gases, binary alloys and \melting" of DNA. The model deals with the con guration of a system of interacting objects, say two-valued spins, situated at the sites of a d-dimensional lattice (grid) in thermal equilibrium with a reservoir. The basic assumption in the Ising model is that objects interact only with their nearest neighbours as well as possibly with an external eld. An increase in the temperature can produce a radical change in the con guration of the system, i.e. a phase transition. See, for example, [17] for an introduction to this subject. The Ising model can be solved exactly in one and two dimensions; in two dimensions it already exhibits the phenomenon of phase transition, where the global con guration of the interacting objects changes. It is one of the few exactly soluble models which exhibits phase transition. In higher dimensions, the model has continued to be the subject of intensive theoretical and experimental research with the use of standard renormalisation techniques [15] and has been one of the main areas of study in the modern science of complex systems in the past decade. In recent years, the problem of a one-dimensional 21 spin Ising model in a random external eld has also been studied but with a di erent kind of renormalisation method [19, 10, 5, 2]. We will brie y review this work here. Consider a one-dimensional chain of N Ising spins hsn iNn=1 with sn = 1 for each 1 n N as in Figure one. To appear in \Theory and formal methods 1994: Proceedings of the second Imperial college workshop, Hankin et al Editors, Imperial College Press, 1995"
1
s1
s2
s3
sN
Figure 1. The one-dimensional 12 spin Ising model. Each state of the system is determined by a given set of values of the spins sn = 1. Assume now that there is a magnetic eld hn at each site n 1. Then, we can write the Hamiltonian of the system in any state hsn iNn=1 as N NX ?1 X HN = ? Jsn sn+1 ? hnsn n=1
n=1
where J > 0 is the coupling constant. This equation says that the only interactions which contribute to the energy of the system are between neighbouring spins on the one hand and between each spin and the magnetic eld on the other hand. In statistical physics, all physical quantities of a system, such as free energy and entropy, can be obtained from the canonical partition function X Z = e? i i
of the system, where i is the energy of the state i and the summation is over all the possible quantum states i of the system [13]. Here, = (kB T)?1 , where T is the temperature of the system and kB is the universal Boltzmann constant. The probability of nding the system in quantum state i is given by e? i =Z. For the one-dimensional Ising model of N spins, the canonical partition function is given by N NX ?1 X X (1) exp ( Jsn sn+1 + hn sn ): ZN = s1 ;:::;sN =1
n=1
n=1
Note that for each n = 1; : : :; N ? 1 and real variable x, we have: X exp (Jsn sn+1 + xsn) = 2 cosh (Jsn+1 + x) sn =1
= exp(A(x)sn+1 + B(x)); where the real functions A; B : R ! R are given by A(x) = (2 )?1 log(cosh (x + J)= cosh (x ? J)); (2) ? 1 B(x) = (2 ) log(4 cosh (x + J) cosh (x ? J)); (3) as one can easily check for the two possible values sn+1 = 1. Using the above formula, we carry out the summation in Equation (1) over the rst spin s1 to obtain: N NX ?1 X X (4) exp fB(h1 ) + Jsn sn+1 + (h2 + A(h1 ))s2 + hnsn g: ZN = s2 ;:::;sN =1
n=3
n=2
We now put 1 = h1 and
n = hn + A(n?1) for 2 n N. Therefore, Equation (4) can be rewritten as N NX ?1 X X ZN = exp fB(1 ) + Jsn sn+1 + 2 s2 + hnsn g: s2 ;:::;sN =1
n=3
n=2
(5)
If we continue to do the summation over the left spins, then after N ? 1 iterations we get [19]: NX ?1 X (6) exp (N sN + B(n )): ZN = sN =1
n=1
In other words, the partition function is reduced to that of a single spin sN . 2
If the magnetic eld is constant, i.e. hn = h for 1 n N, then it is easily shown that the smooth function g: R ! R x 7! h + A(x); 0 0 satis es jg (x)j = jA (x)j < 1 for all x 2 R and has a unique xed point x . This shows that x is an attractor and limn!1 gn (x) = x for all x 2 R. Therefore, in this case, limn!1 n = x , and one can compute the various physical quantities such as free energy and magnetisation per spin [10], as we will see in Section 5. In [10, 5, 2], the problem of a one-dimensional Ising model in a random magnetic eld has been studied as follows. Let h > 0 be a constant. Suppose hn is a random eld taking values h with equal probabilities, i.e. hhn in1 is a sequence of independent and identically distributed random variables with P(hn = h) = P(hn = ?h) = 1=2 for all n 1. Then, Equation (5) is a stochastic equation for the Markov process 1; 2 ; 3; : : : with transitional probabilities (7) P(n = h + A(n?1)) = P(n = ?h + A(n?1)) = 21 : The stochastic equation induces in the nth step a probability density pn(x) for the e ective random eld n satisfying the Perron-Frobenius (or Kolmogorov-Chapman) equation [20]: Z pn (y) = pn?1(x)f 21 (y ? h ? A(x)) + 21 (y + h ? A(x))g dx; (8) where (z) is the delta function centred at z = 0. Consider the two functions f+ ; f? : R ! R (9) de ned by f+ (x) = h + A(x) and f? (x) = ?h + A(x). Each f satis es jf0 (x)j < 1 and has a unique xed point x , where = , with x+ = ?x? = h=2 + (2 )?1 arcsinh(e2 J sinh h) > 0: (10) Furthermore f [x? ; x+] [x? ; x+] for = . Figure 2. The graphs of f+ and f? in the interval [x?; x+ ]
h+A(x)
x x-
o
x+
-h+A(x)
3
When the temperature T > 0, an analysis of iterates of the Perron-Frobenius Equation (8), including analysis by symbolic dynamics, has been presented in [1, 3] to show that the equation has a unique xed point, i.e. an invariant probability distribution with support in [x?; x+ ], which has a multifractal structure, and that starting with any initial probability distribution, the iterates tend to this invariant distribution. Various physical quantities such as free energy and magnetisation per spin can be expressed as integrals over this invariant distribution [10]. In this paper, we will present a domain-theoretic analysis of Equation (5), for T > 0, by interpreting it as the dynamics of an iterated function system (IFS) with probabilities, consisting of the maps f+ ; f? : R ! R with equal probabilities p+ = p? = 1=2. Following [6, 7], we will examine the dynamics of the IFS on the normalised probabilistic power domain of the upper space of [x? ; x+], i.e. instead of working in the space of probability distributions over [x?; x+ ], we work in the bigger space of probability measures, or normalised continuous valuations, on the upper space of [x? ; x+]. The domain-theoretic formulation enables us to show some further properties of the invariant probability distribution of the Perron-Frobenius equation. In particular, using the generalised Riemann integral, we obtain nite algorithms to compute various physical quantities of the Ising model in the random eld to any given threshold of accuracy. In Section 2, we will review the basic domain-theoretic framework for measures on compact metric spaces. In Section 3, we present some general results about the invariant measure of an IFS with probabilities. In Section 4, we extend the theory of generalised Riemann integration to multiple integration. Finally, in Section 5, we apply these general results to the one-dimensional random eld Ising model.
2 Domain-theoretic framework
Let X be a compact metric space, and let (UX; ) be its upper space, i.e. the non-empty compact subsets of X ordered by reverse inclusion, which is a bounded complete !-continuous dcpo (directed complete partial order). The singleton map embeds X onto the set of maximal elements of UX. Similarly, the set of probability measures on X can be embedded into the set of normalised measures on UX. To make this precise, we need some basic de nitions and results. A valuation on a topological space Y is a map : (Y ) ! [0; 1) which satis es: (i) (a) + (b) = (a [ b) + (a \ b), (ii) (;) = 0, and (iii) a b ) (a) (b). A normalised valuation further satis es (Y ) = 1. A continuous valuation [14, 12, 11] is a valuation such that whenever A (Y ) is a directed set (wrt ) of open sets of Y , then [ ( O) = supO2A (O): O2A
by
For any b 2 Y , the point valuation based at b is the valuation b : (Y ) ! [0; 1) de ned if b 2 O b(O) = 01 otherwise:
Any nite linear combination
n X i=1
ri bi
of point valuations bi with constant coecients ri 2 [0; 1), (1 i n) is a continuous valuation on Y ; we call it a simple valuation. 4
The normalised probabilistic power domain, PY , of a topological space Y consists of the set of normalised continuous valuations on Y with (Y ) = 1 and is ordered as follows: v i for all open sets O of Y , (O) (O): The F partial order (PY; v) is a dcpo in which the lub of a directed set hiii2I is given by i i = where for any open subset O 2 (Y ) we have (O) = supi2I i (O): If Y is an !-continuous dcpo with bottom ?, then PY is also an !-continuous dcpo with bottom ? and has a basis consisting of simple valuations [7]. Furthermore, any 2 PY extends uniquely to a Borel measure on Y [16, Theorem 3.9]. For convenience, we denote this unique extension by as well. For two simple valuations X X 1 = rb b 2 = sc c c2C
b2B
in P1 Y , where B; C are nite subsets of Y , we have by the splitting lemma [12, 7]: 1 v 2 i , for all b 2 B and all c 2 C, there exists a nonnegative number tb;c such that X X tb;c = rb tb;c = sc (11) c2C
b2B
and tb;c 6= 0 implies b v c. We can consider any b 2 B as a source with mass rb, any c 2 C as a sink with mass sc , and the number tbc as the ow of mass from b to c. Then, the above property can be regarded as conservation of total mass. Since (UX; ) is an !-continuous dcpo with bottom X, P1UX is an !-continuous dcpo with bottom X . The singleton map s : X ! UX x 7! fxg embeds X onto the set s(X) of maximal elements of UX. Furthermore, s(X) UX is a Borel subset; in fact any Borel subset B X induces a Borel subset s(B) UX [6, Corollary 5.10]. A valuation 2 P1UX is said to be supported in s(X) if its unique extension to a Borel measure on UX satis es (s(X)) = 1. It can be shown that 2 P1UX i (s(a)) = (a)
(12)
for all open subsets a X. If is supported in s(X), then the support of is the set of points y 2 s(X) such that (O) > 0 for any Scott-neighbourhood O UX of y. Any element of P1UX which is supported in s(X) is a maximal element of P1UX [6, Proposition 5.18]; we denote the set of all valuations which are supported in s(X) by S1X. We can identify S1X with the set M1 X of normalised Borel measures on X as follows. Let e : M1 X ! S1 X 7! s?1 and j : S1 X ! M1 X 7! s:
Theorem 2.1 [6, Theorem 5.21] The maps e and j are well-de ned and induce an isomorphism between S X and M X . 1
1
5
For 2 M1 X and an open subset a X, Equation (12) implies (a) = e()(s(a)) = e()(a): (13) Furthermore, it follows from Theorem 2.1 that there exists an increasing chain of simple valuations hi ii0 with G s?1 = i: (14) i0
In other words, any probability measure on X is the least upperbound of a chain of simple valuations on UX. This fundamental property is used in [7] to develop a theory of generalised Riemann integration, which we will be the basis of work in Section 4.
3 Invariant measure of an IFS
1 Suppose F A P UX is a directed set of simple valuations. We would like to know when the lub FA of A determines a Borel measure on X; in other words, under what conditions we have A 2 S1X. The following proposition gives the necessary and sucient condition for this. We will denote the diameter of any compact subset c X by jcj.
Proposition 3.1 The following conditions are equivalent. F (i) A 2 S X . P (ii) For all > 0 and all > 0, there exists a2A ra a 2 A with X ra < : 1
jaj
Suppose now we have an increasing chain hiii0 of simple valuations i in P1UX. Using Equation (11), it is easy to see that we can obtain a nitary branching tree whose nodes on the ith level are precisely the elements of Bi and a child of any node is a subset of that node. We call this a tree associated with the chain hi ii0. Note that a chain can have more than one associated tree. Clearly, the lub in UX of the nodes of any branch of an associated tree is an element of UX, which is to say that the intersection of a shrinking sequence of non-empty compact subsets of X is a non-empty compact subset. Using Proposition 3.1 and Konig's Lemma, we can prove the following useful result. Proposition 3.2 Suppose hiii0 2 P1UX is an increasing chain of simple valuations. If, for any associated tree of the chain, the lub of the nodes of any branch is a singleton subset of X , then the lub of the chain hiii0 is in S1 X .
Let (X; f1 ; : : :; fN ; p1; : : :; pN ) be an IFS with probabilities on the compact metric space X, i.e.PNfor each i = 1; : : :; N, the map fi : X ! X is continuous and we have 0 < pi < 1 with i=1 pi = 1. De ne the transition operator H : P1UX ! P1UX 7! H() PN by H()(O) = i=1 pi (fi?1 (O)). We know from [6,F Proposition 6.1] that H is Scott continuous and, therefore, has a least xed point = m0 H m X . The mth iterate H m X is the simple valuation given by H m X
=
N X i1 ;i2 ;:::;im =1
pi1 pi2 : : :pim fi1 fi2 :::fim X :
(15)
Moreover, the chain hH m X im0 has a unique associated tree which we can label with the transitional probabilities as in [8]. See Figure 3. We call this the IFS tree with transitional probabilities. 6
X p
p
1
fX
1
..
2
fX N
p
p 1
1
N
fX
1
ffX
p
2
p
N
..
ffX
1
p
1
N
..
..
..
.. ..
ffX
N 1
N
..
ffX
N N
Figure 3. The IFS tree with transitional probabilities. Assume now that the least xed point of H is in S1X. Then, by the remark following Equation (12), is a maximal element of P1UX and, therefore, the unique xed point of H. If 2 P1UX, then one can show using the monotonicity of H that is the maximal topological limit of H m () as m ! 1. Writing the maximal topological limit as limm!1 H m (), we have: Proposition 3.3 If 2 S1X , then is the unique xed point of H , and, for any 2 P1UX , we have limm!1 H m () = . A particular instance of the above result occurs if the IFS is hyperbolic, i.e. if fi : X ! X is contracting with contractivity factor si < 1. Then it follows from Proposition 3.2 that 2 S1X, since a typical node fi1 fi2 : : :fim X of the above tree on level m has diameter at most sm jX j which tends to 0 as m ! 1, where s = maxi si and jX j is the diameter of X. Corollary 3.4 For a hyperbolic IFS, H has a unique xed point and limm!1 H m() = for any 2 P1UX .
4 Multiple R-integration In this section, we extend the generalised Riemann integration, R-integration, introduced in [7] to multiple integration. We will only consider the double integral since the generalisation to any multiple integral is straightforward. The double R-integral is used in the next section to compute the magnetisation per spin and the Edwards-Anderson parameter in the Ising model.
4.1 The double R-integral
Recall that given Borel measures 2 MX and 2 MY on topological spaces X and Y , the product measure 2 M(X Y ) satis es ( )(B1 B2 ) = (B1 )(B2) for all Borel subsets B1 and B2 of X and Y respectively [18]. Similarly, given dcpo's D and E, and valuations 2 PD and 2 PE, we have the product valuation 2 P(D E) which for any \rectangle" O1 O2 , where O1 and O2 are open subsets of D and E respectively, satis es ( )(O1 O2 ) = (O1)(O2) [11]. Moreover, the product operation is continuous. In other words, we have the following property. Proposition 4.1 Suppose = Fi2I i 2 PD and = Fi2I i 2 PE, where hi ii2I and hiii2I are directed sets in PD and PE respectively. Then, G = i i: i2I
7
Furthermore, the product of two simple valuations is another simple valuation.
Proposition 4.2 Let
=
X a2A
ra a
and
=
X b2B
sb b
be simple valuations on dcpo's D and E . Then, the product valuation is a simple valuation on D E and is given by
=
X
a2A;b2B
ra sb (a;b):
Let X and Y be compact metric spaces, 2 M1 X, 2 M1 Y . We have the product measure 2 M1 (X Y ). On the other hand, by the remark following Theorem 2.1, there exist increasing chains hi ii0 and hi ii0 of simple valuations i 2 P1UX and i 2 P1UY for i 0 such that G G sX?1 = i and s?Y 1 = i i0
i0
where sX : X ! UX and sY : Y ! UY are the singleton maps. For each i 0, we have the product valuation i i 2 P1(UX UY ) which is a simple valuation by Proposition 4.2. Furthermore, by Proposition 4.1, we have G s?X1 s?Y 1 = i i: (16) i0
Note that UX UY is a subspace of U(X Y ). Let i : UX UY ! U(X Y ) be the inclusion map. Then, i is Scott continuous and we have the continuous embedding ^i : P1(UX UY ) ! P1 U(X Y ) 7! i?1 : It is easy to check that ^i( s?X1 s?Y 1 ) = ( ) s?X1Y where sX Y : X Y ! U(X Y ) is the singleton map. Furthermore, the product of the two simple valuations X X = ra a 2 P1 UX = sb b 2 P1 UY (17) a2A
satis es
b2B
^i( ) = ( ) i?1 X = ( ra sb (a;b) ) i?1 a2X A;b2B = ra sb i(a;b) a2X A;b2B = ra sb (a;b) a2A;b2B
= By Equation (16) and the continuity of ^i, we nally deduce:
Proposition 4.3
( ) sX?1Y =
G i0
8
i i:
Let f : X Y ! R be a bounded function. For the simple valuations and given by Equation (17), we have the lower sum of f with respect to de ned by X S ` (f; ) = ra sb inff(a; b); a2A;b2B
where f(a; b) is the image of (a; b) X Y under f. Similarly, the upper sum of f with respect to is de ned by X S u (f; ) = ra sb sup f(a; b): a2A;b2B
Furthermore, for a given choice of xa 2 a and yb 2 b for each a 2 A and b 2 B, the generalised Riemann sum of f with respect to is de ned by X S(f; ) = ra sb f(xa ; yb ): a2A;b2B
See [7] for details. The main result of this section is the following. Theorem 4.4 If f : X Y ! R is continuous almost everywhere with respect to the product measure , then Z Z S ` (f; i i) % f d ; S u (f; i i ) & f d ; X Y X Y Z S(f; i i) ! f d X Y
as i ! 1.
4.2 Integration over the invariant measure of an IFS
Suppose F now we have an IFS with probabilities (X; f1 ; : : :; fN ; p1; : : :; pN ) such that = m0 H m X , where H m X is given by Equation (15), is a maximal element of P1 UX. If g : X ! R is continuous almost everywhere with respect to = s, then for any point x 2 X we have the generalised Riemann sum for H m X given by N X pi1 : : :pim g(fi1 : : :fim x) S(g; H m X ) = i1 ;:::;im =1
and it follows that
S(g; H m X ) !
Z
g d
as m ! 1. If the maps fi are contacting and g satis es a Lipschitz condition, then we can obtain a nite algorithm to calculate the integral to any given accuracy as follows [8]. Suppose there exist k > 0 and c > 0 such that g satis es
jg(x) ? g(y)j c(d(x; y))k for all x; y 2 X. (When k = 1, we call c a Lipschitz constant for g.) Let > 0 be given. Then Z jS(g; H m X ) ? gd j for m = dlog((=c)1=k =jX j)= logse, where s is the contractivity of the IFS and dae is the least integer greater than or equal to a. 9
Next, assume that h : X X ! R is continuous almost everywhere with respect to . Then, for any x 2 X, we have the following generalised Riemann sum for H m X H m X : N N X X pi1 : : :pim pj1 : : :pjm h(fi1 : : :fim x; fj1 : : :fjm x) S(h; H m X H m X ) = and it follows that
i1 ;:::;im =1 j1 ;:::;jm =1
S(h; H m X H m X ) !
Z Z
h(x; y) d (x)d(y)
(18)
as m ! 1. RIf h satis es a Lipschitz condition then again we can obtain a nite algorithm to estimate h d to within any given threshold. We also have ergodic theorems for IFSs with probabilities. Let (X; f1 ; : : :; fN ; p1; : : :; pN ) be a hyperbolic IFS with probabilities with invariant measure . Let g : X ! R be a continuous function. Suppose i1 ; i2; : : : is is a sequence of independent identically distributed random variables on f1; 2; : : :; N g with probabilities P(in = k) = pk (1 k N) for all n 1. Let i1 2 X, and put in+1 = fin (in ) for all n 1. Then Elton's ergodic theorem states: Theorem 4.5 [9] For almost all sequences i1 ; i2; : : :, and all 1 2 X , Z Z k 1X g(x) d (x) = klim g( ): !1 k n=1 in It is also possible to deduce a two-dimensional version of Elton's Theorem. Let h : X X ! R be a continuous function. Let i1 ; i2; : : : and j1 ; j2; : : : be two sequences of independent identically distributed random variables on f1; 2; : : :; N g with probabilities P(in = k) = P(jn = k) = pk (1 k N) for all n 1. Let i1 ; j1 2 X X, and put in+1 = fin (in ) and jn+1 = fjn (jn ) for all n 1. Then, we have: Theorem 4.6 For almost all sequences i1 ; i2; : : : and j1; j2; : : :, and all 1; 1 2 X , Z Z k 1X h( ; ): h(x; y) d (x)d (y) = klim !1 k n=1 in jn
5 Domain-theoretic analysis of the Ising model In this section, we apply the results of the previous two sections to the one-dimensional random eld Ising model (1d RFIM) that we described in the introduction. We consider a more general case and allow the external magnetic eld be asymmetric, i.e. hn = h , for real numbers h+ and h? with h+ ?h? > 0. We assume, however, for convenience that the two value of the eld are still equally probable, i.e. P(hn = h+ ) = P(hn = h? ) = 1=2 for all n 1.
5.1 Invariant measure of the 1d RFIM
Consider the IFS ([x? ; x+ ]; f?; f+ ; 1=2; 1=2) where the two maps f+ ; f? : R ! R are de ned by f (x) = h + A(x), with A(x) = (2 )?1 log(cosh (x + J)= cosh (x ? J)), as given by Equations (9) and (2). Their unique xed points x+ ; x? are given by x = h =2 + (2 )?1 arcsinh(e2 J sinh h ), with x+ ?x? > 0, cf. equation (10). A simple calculation shows that the rst and the second derivatives of the two maps are given by A0 (x) = 21 (tanh (x + J) ? tanh (x ? J)) 1 ? ): A00(x) = 2 ( 2 1 2 cosh (x + J) cosh (x ? J) It follows that 1 > A0 (x) > 0 for all x 2 [x? ; x+] (in fact for all x 2 R). Furthermore, since A00 is negative in (0; x+] and A(x) = ?A(?x), we easily see that A0 , and hence f+0 and f?0 , 10
have maximum value
s = A0 (0) = tanh J (19) in the interval [x?; x+ ] with 0 < s < 1. By the intermediate value theorem, it follows that A; f+ ; f? are contracting with contractivity factor s. Therefore, by Corollary 3.4, the IFS has a unique invariant measure and starting with any initial valuation 2 P1UX we have limm!1 H m () = . Moreover, there is a nite algorithm to generate on any digitised screen [8]. An analysis of the multifractal nature of the invariant measure = s?1 is given in [1, 3]. When f+ [x?; x+ ] \ f? [x?; x+ ] = ; the support of is totally disconnected and homoeomorphic to the Cantor set. On the other hand, when f+ [x?; x+ ] \ f? [x? ; x+] 6= ;, the support of is the whole interval [x?; x+ ] and is a fat fractal as in Figure 4, which is taken from [2].
Figure 4. Distribution of the local magnetisation (kT = 1, J = 1). The multifractal can be fat (a) for h+ = ?h? = :6 or thin (b) for h+ = ?h? = 1:5.
5.2 Expected value of physical quantities
We will now use the results of Section 4, to obtain nite algorithms to compute the expected values of various physical quantities to any given threshold.
5.2.1 Free energy density
The free energy F plays the same fundamental role in thermal physics that ordinary energy plays in mechanics. It is given in terms of the partition function Z by F( ) = ? 1 log Z. See for example [13]. Therefore, for the Ising model of N spins, Equation (6) gives FN ( ) = ? 1 log ZN NX ?1 = ? 1 log(exp f 1 log(2 cosh N ) + B(n )g) n=1 NX ?1 1 = ? log(2 cosh N ) ? B(n ); n=1
where the function B is given by Equation (3). Therefore, the free energy density f( ) is given in the limit N ! 1 by 1 log ZN FN ( ) f( ) = Nlim !1 N !1 N = ? Nlim ?1 1 log(2 cosh ) ? lim 1 NX B(n ) = ?Nlim N N !1 N n=1 !1 N ?1 1 NX = ?Nlim !1 N n=1 B(n ) 11
If the eld hn is constant then limn!1 n = x and by the continuity of B we have f( ) = ?B(x ) as in [10]. For the random eld Ising model, Elton's ergodic Theorem 4.5 implies Z NX ?1 1 B(n ) = ? B d ; f( ) = ? Nlim !1 N n=1 since the IFS is hyperbolic and B is a continuous function. A simple calculation, similar to the case of A, shows that B : [x?; x+ ] ! [x?; x+ ] is contracting with contractivity factor (20) c = B 0 (x+ ) = 21 (tanh (x+ + J) + tanh (x+ ? J)): Therefore, for the choice of 0 2 [x? ; x+], we consider the generalised Riemann sums X B(fi1 : : :fin (0)): S(B; H n [x? ;x+ ] ) = 21n i1 ;:::;in = Given > 0, we have
Z
jS(B; H n x? ;x+ ) ? B d j < ; with n = dlog((=c)=(x ? x? ))= logse, where s is the contractivity of the IFS given by Equation (19), c is given by Equation (20) and dae is, as before, the least integer greater [
]
+
than or equal to a.
5.2.2 Magnetisation per spin
The magnetisation per spin is given by N 1X hsk iHN ; m( ) = Nlim !1 N k=1
where hsk iHN is the average value of sk with respect to the Hamiltonian HN of the N spins hsn iNn . In order to nd hsk iHN , one reduces the partition function ZN to that of a typical spin sk with 1 k N. This is done by applying the technique of summation, described =1
in the introduction, over spins from left and right as in [4]. Let n be as before, and let N = hN and n?1 = hn?1 + A(n ) for 1 n N ? 1. Then, it is easy to deduce that kX ?1 kX ?1 X exp ((k + A(k ))sk + B(n ) + ZN = B(n )) sk =1
= 2 cosh (k + A(k )) exp ( Therefore,
X
hsk iHN = Z1 f N sk
n=1 kX ?1
n=N kX ?1
n=1
n=N
B(n ) +
sk exp( (k + A(k )))g exp (
= 1
kX ?1
B(n )):
B(n ) +
n=1 kX +1
kX +1 n=N
B(n ))
kX ?1 = Z1 f2 sinh(k + A(k ))g exp ( B(n ) + B(n )) N n=1 n=N = tanh (k + A(k )): If the eld hn = h is constant, then k ; k ! x , where x is the unique solution of A(x) = h + x, and hence m( ) = tanh (2x + h) as in [10].
12
In order to compute the magnetisation per spin for the case of the random eld, we use the two-dimensional version of Elton's Ergodic Theorem 4.6 as follows. N 1X hs i m( ) = Nlim !1 N k=1 k HN N 1X = Nlim tanh (k + A(k )) Z !1 Z N k=1 = tanh (x + A(y)) d (x)d (y); where the double integral can be computed by Equation (18). We can deduce a nite algorithm to compute m( ) to within an accuracy of any given > 0. The function tanh (?) : R ! R has Lipschitz constant . Hence for x1; x2; y1; y2 2 [x? ; x+] we have j tanh (x1 + A(y1 )) ? tanh (x2 + A(y2 ))j (jx1 ? x2 j + jA(y1 ) ? A(y2 )j) (jx1 ? x2 j + sjy1 ? y2 j) where s = tanh T is the contractivity of A : [x?; x+ ] ! [x?; x+ ]. For convenience, we put h(x; y) = tanh (x+A(y)) and n = H n[x+ ;x? ] where H is the transition operator. We choose 0 2 [x? ; x+] for the generalised Riemann sum. It follows that if sn (x+ ? x? ) = (1 + s), i.e. if s)(x+ ? x? )) e; n = d log(= (1 +logs then Z Z jS(h; n n) ? m( )j = jS(h; n n) ? tanh (x + A(y)) d (x)d(y)j ; where
S(h; n n) = 212n
X
X
i1 ;:::;in = j1 ;:::;jn =
tanh (fi1 : : :fin (0) + A(fj1 : : :fjn (0))):
5.2.3 Edwards-Anderson parameter
Similarly, we can obtain a nite algorithm for computing the Edwards-Anderson parameter qEA given by Z Z N X 1 2 qEA = Nlim (tanh (x + A(y)))2 d (x)d(y): hs i = !1 N k=1 k HN The function tanh2 (?) : R ! R p has Lipschitz constant 4 3=9. It follows that if p + s)(x+ ? x? )) e; n = d log(9=4 3 (1logs then Z Z jS(h2 ; n n) ? qEAj = jS(h2 ; n n) ? tanh2 (x + A(y)) d (x)d (y)j ; where
S(h2 ; n n) = 212n
X
X
i1 ;:::;in = j1 ;:::;jn =
tanh2 (fi1 : : :fin (0) + A(fj1 : : :fjn (0))):
13
Acknowledgements I am grateful to Ulrich Behn for introducing me to this subject and for various informative discussions. This work has been founded by EPSRC grant \Foundational Structures in Computer Science" at Imperial College.
References [1] U. Behn, J. L. van Hemmen, R. Kuhn, A. Lange, and V. A. Zagrebnov. Multifractality in forgetful memories. Physica D, 68:401{415, 1993. [2] U. Behn and A. Lang. 1d random eld Ising model and nonlinear dynamics. In From phase transitions to chaos, Topics in Modern Statistical Physics, pages 217{228. World Scienti c, 1992. [3] U. Behn, J. L. van Hemmen, R. Kuhn, A. Lang, and V. A. Zagrebnov. Multifractal properties of discrete stochastic mappings. In M. Fannes, C. Maes, and A. Verbevre, editors, On three levels: micro-, meso-, and macro-approaches in physics, volume 324 of NATO ASI, pages 399{409. Plenum Press, 1994. [4] U. Behn and V. Zagrebnov. One dimensional random eld Ising model and stochastic mappings. Technical Report E17-87-138, Physics department, Leipzig University, 1987. [5] U. Behn and V. Zagrebnov. One dimensional Markovian- eld Ising model: Physical properties and characteristics of the discrete stochastic mapping. J. Phys. A: Math. Gen., 21:2151{2165, 1988. [6] A. Edalat. Dynamical systems, measures and fractals via domain theory (Extended abstract). In G. L. Burn, S. J. Gay, and M. D. Ryan, editors, Theory and Formal Methods 1993. Springer-Verlag, 1993. Full paper to appear in Information and Computation. [7] A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163{193, 1995. [8] A. Edalat. Power domains and iterated function systems. Information and Computation, 124:182{197, 1996. [9] J. Elton. An ergodic theorem for iterated maps. Journal of Ergodic Theory and Dynamical Systems, 7:481{487, 1987. [10] G. Gyorgyi and P. Rujan. Strange attractors in disordered systems. J. Phys. C: Solid State Phys., 17:4207{4212, 1984. [11] C. Jones. Probabilistic Non-determinism. PhD thesis, University of Edinburgh, 1989. [12] C. Jones and G. Plotkin. A Probabilistic Powerdomain of Evaluations. In Logic in Computer Science, pages 186{195. IEEE Computer Society Press, 1989. [13] C. Kittel and H. Kroemer. Thermal Physics. W. H. Freeman, second edition, 1980. [14] J. Lawson. Valuations on Continuous Lattices. In Rudolf-Eberhard Ho mann, editor, Continuous Lattices and Related Topics, volume 27 of Mathematik Arbeitspapiere. Universitat Bremen, 1982. [15] T. Niemeijer and J. van Leeuwen. Renormalization theory for Ising-like spin systems. In C. Domb and M. Green, editors, Phase Transitions and Critical Phenomena, volume 6, chapter 7. Academic Press, 1976. 14
[16] T. Norberg. Existence theorems for measures on continuous posets, with applications to random set theory. Math. Scand., 64:15{51, 1989. [17] L. E. Reichl. A Modern Course in Statistical Physics. Edward arnold, 1980. [18] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1966. [19] P. Rujan. Calculation of the free energy of Ising systems by a recursion method. Physica, 91(A):549{562, 1978. [20] A. N. Shiryayev. Probability. Graduate texts in mathematics. Springer, 1984.
15
Abstract)
Abbas Edalat Department of Computing Imperial College 180 Queen's Gate, London SW7 2BZ UK. Abstract
We present a domain-theoretic analysis of the invariant measure of the one-dimensional Ising model in a random external magnetic eld. The invariant measure is obtained as a xed point of the Markov transition operator of an iterated function system with probabilities acting on the probabilistic power domain of the upper space of a closed real interval. This enables us to use the generalised Riemann integral in combination with Elton's ergodic theorem to obtain an algorithm to compute the free energy density of the system. We also develop the generalised double Riemann integral, which we use, together with a two-dimensional version of Elton's theorem, to deduce algorithms to compute the magnetisation per spin and the Edwards-Anderson parameter of the system.
1 Introduction The Ising model was introduced by Ising as a model for ferromagnetism some seventy years ago; it also describes such systems as lattice gases, binary alloys and \melting" of DNA. The model deals with the con guration of a system of interacting objects, say two-valued spins, situated at the sites of a d-dimensional lattice (grid) in thermal equilibrium with a reservoir. The basic assumption in the Ising model is that objects interact only with their nearest neighbours as well as possibly with an external eld. An increase in the temperature can produce a radical change in the con guration of the system, i.e. a phase transition. See, for example, [17] for an introduction to this subject. The Ising model can be solved exactly in one and two dimensions; in two dimensions it already exhibits the phenomenon of phase transition, where the global con guration of the interacting objects changes. It is one of the few exactly soluble models which exhibits phase transition. In higher dimensions, the model has continued to be the subject of intensive theoretical and experimental research with the use of standard renormalisation techniques [15] and has been one of the main areas of study in the modern science of complex systems in the past decade. In recent years, the problem of a one-dimensional 21 spin Ising model in a random external eld has also been studied but with a di erent kind of renormalisation method [19, 10, 5, 2]. We will brie y review this work here. Consider a one-dimensional chain of N Ising spins hsn iNn=1 with sn = 1 for each 1 n N as in Figure one. To appear in \Theory and formal methods 1994: Proceedings of the second Imperial college workshop, Hankin et al Editors, Imperial College Press, 1995"
1
s1
s2
s3
sN
Figure 1. The one-dimensional 12 spin Ising model. Each state of the system is determined by a given set of values of the spins sn = 1. Assume now that there is a magnetic eld hn at each site n 1. Then, we can write the Hamiltonian of the system in any state hsn iNn=1 as N NX ?1 X HN = ? Jsn sn+1 ? hnsn n=1
n=1
where J > 0 is the coupling constant. This equation says that the only interactions which contribute to the energy of the system are between neighbouring spins on the one hand and between each spin and the magnetic eld on the other hand. In statistical physics, all physical quantities of a system, such as free energy and entropy, can be obtained from the canonical partition function X Z = e? i i
of the system, where i is the energy of the state i and the summation is over all the possible quantum states i of the system [13]. Here, = (kB T)?1 , where T is the temperature of the system and kB is the universal Boltzmann constant. The probability of nding the system in quantum state i is given by e? i =Z. For the one-dimensional Ising model of N spins, the canonical partition function is given by N NX ?1 X X (1) exp ( Jsn sn+1 + hn sn ): ZN = s1 ;:::;sN =1
n=1
n=1
Note that for each n = 1; : : :; N ? 1 and real variable x, we have: X exp (Jsn sn+1 + xsn) = 2 cosh (Jsn+1 + x) sn =1
= exp(A(x)sn+1 + B(x)); where the real functions A; B : R ! R are given by A(x) = (2 )?1 log(cosh (x + J)= cosh (x ? J)); (2) ? 1 B(x) = (2 ) log(4 cosh (x + J) cosh (x ? J)); (3) as one can easily check for the two possible values sn+1 = 1. Using the above formula, we carry out the summation in Equation (1) over the rst spin s1 to obtain: N NX ?1 X X (4) exp fB(h1 ) + Jsn sn+1 + (h2 + A(h1 ))s2 + hnsn g: ZN = s2 ;:::;sN =1
n=3
n=2
We now put 1 = h1 and
n = hn + A(n?1) for 2 n N. Therefore, Equation (4) can be rewritten as N NX ?1 X X ZN = exp fB(1 ) + Jsn sn+1 + 2 s2 + hnsn g: s2 ;:::;sN =1
n=3
n=2
(5)
If we continue to do the summation over the left spins, then after N ? 1 iterations we get [19]: NX ?1 X (6) exp (N sN + B(n )): ZN = sN =1
n=1
In other words, the partition function is reduced to that of a single spin sN . 2
If the magnetic eld is constant, i.e. hn = h for 1 n N, then it is easily shown that the smooth function g: R ! R x 7! h + A(x); 0 0 satis es jg (x)j = jA (x)j < 1 for all x 2 R and has a unique xed point x . This shows that x is an attractor and limn!1 gn (x) = x for all x 2 R. Therefore, in this case, limn!1 n = x , and one can compute the various physical quantities such as free energy and magnetisation per spin [10], as we will see in Section 5. In [10, 5, 2], the problem of a one-dimensional Ising model in a random magnetic eld has been studied as follows. Let h > 0 be a constant. Suppose hn is a random eld taking values h with equal probabilities, i.e. hhn in1 is a sequence of independent and identically distributed random variables with P(hn = h) = P(hn = ?h) = 1=2 for all n 1. Then, Equation (5) is a stochastic equation for the Markov process 1; 2 ; 3; : : : with transitional probabilities (7) P(n = h + A(n?1)) = P(n = ?h + A(n?1)) = 21 : The stochastic equation induces in the nth step a probability density pn(x) for the e ective random eld n satisfying the Perron-Frobenius (or Kolmogorov-Chapman) equation [20]: Z pn (y) = pn?1(x)f 21 (y ? h ? A(x)) + 21 (y + h ? A(x))g dx; (8) where (z) is the delta function centred at z = 0. Consider the two functions f+ ; f? : R ! R (9) de ned by f+ (x) = h + A(x) and f? (x) = ?h + A(x). Each f satis es jf0 (x)j < 1 and has a unique xed point x , where = , with x+ = ?x? = h=2 + (2 )?1 arcsinh(e2 J sinh h) > 0: (10) Furthermore f [x? ; x+] [x? ; x+] for = . Figure 2. The graphs of f+ and f? in the interval [x?; x+ ]
h+A(x)
x x-
o
x+
-h+A(x)
3
When the temperature T > 0, an analysis of iterates of the Perron-Frobenius Equation (8), including analysis by symbolic dynamics, has been presented in [1, 3] to show that the equation has a unique xed point, i.e. an invariant probability distribution with support in [x?; x+ ], which has a multifractal structure, and that starting with any initial probability distribution, the iterates tend to this invariant distribution. Various physical quantities such as free energy and magnetisation per spin can be expressed as integrals over this invariant distribution [10]. In this paper, we will present a domain-theoretic analysis of Equation (5), for T > 0, by interpreting it as the dynamics of an iterated function system (IFS) with probabilities, consisting of the maps f+ ; f? : R ! R with equal probabilities p+ = p? = 1=2. Following [6, 7], we will examine the dynamics of the IFS on the normalised probabilistic power domain of the upper space of [x? ; x+], i.e. instead of working in the space of probability distributions over [x?; x+ ], we work in the bigger space of probability measures, or normalised continuous valuations, on the upper space of [x? ; x+]. The domain-theoretic formulation enables us to show some further properties of the invariant probability distribution of the Perron-Frobenius equation. In particular, using the generalised Riemann integral, we obtain nite algorithms to compute various physical quantities of the Ising model in the random eld to any given threshold of accuracy. In Section 2, we will review the basic domain-theoretic framework for measures on compact metric spaces. In Section 3, we present some general results about the invariant measure of an IFS with probabilities. In Section 4, we extend the theory of generalised Riemann integration to multiple integration. Finally, in Section 5, we apply these general results to the one-dimensional random eld Ising model.
2 Domain-theoretic framework
Let X be a compact metric space, and let (UX; ) be its upper space, i.e. the non-empty compact subsets of X ordered by reverse inclusion, which is a bounded complete !-continuous dcpo (directed complete partial order). The singleton map embeds X onto the set of maximal elements of UX. Similarly, the set of probability measures on X can be embedded into the set of normalised measures on UX. To make this precise, we need some basic de nitions and results. A valuation on a topological space Y is a map : (Y ) ! [0; 1) which satis es: (i) (a) + (b) = (a [ b) + (a \ b), (ii) (;) = 0, and (iii) a b ) (a) (b). A normalised valuation further satis es (Y ) = 1. A continuous valuation [14, 12, 11] is a valuation such that whenever A (Y ) is a directed set (wrt ) of open sets of Y , then [ ( O) = supO2A (O): O2A
by
For any b 2 Y , the point valuation based at b is the valuation b : (Y ) ! [0; 1) de ned if b 2 O b(O) = 01 otherwise:
Any nite linear combination
n X i=1
ri bi
of point valuations bi with constant coecients ri 2 [0; 1), (1 i n) is a continuous valuation on Y ; we call it a simple valuation. 4
The normalised probabilistic power domain, PY , of a topological space Y consists of the set of normalised continuous valuations on Y with (Y ) = 1 and is ordered as follows: v i for all open sets O of Y , (O) (O): The F partial order (PY; v) is a dcpo in which the lub of a directed set hiii2I is given by i i = where for any open subset O 2 (Y ) we have (O) = supi2I i (O): If Y is an !-continuous dcpo with bottom ?, then PY is also an !-continuous dcpo with bottom ? and has a basis consisting of simple valuations [7]. Furthermore, any 2 PY extends uniquely to a Borel measure on Y [16, Theorem 3.9]. For convenience, we denote this unique extension by as well. For two simple valuations X X 1 = rb b 2 = sc c c2C
b2B
in P1 Y , where B; C are nite subsets of Y , we have by the splitting lemma [12, 7]: 1 v 2 i , for all b 2 B and all c 2 C, there exists a nonnegative number tb;c such that X X tb;c = rb tb;c = sc (11) c2C
b2B
and tb;c 6= 0 implies b v c. We can consider any b 2 B as a source with mass rb, any c 2 C as a sink with mass sc , and the number tbc as the ow of mass from b to c. Then, the above property can be regarded as conservation of total mass. Since (UX; ) is an !-continuous dcpo with bottom X, P1UX is an !-continuous dcpo with bottom X . The singleton map s : X ! UX x 7! fxg embeds X onto the set s(X) of maximal elements of UX. Furthermore, s(X) UX is a Borel subset; in fact any Borel subset B X induces a Borel subset s(B) UX [6, Corollary 5.10]. A valuation 2 P1UX is said to be supported in s(X) if its unique extension to a Borel measure on UX satis es (s(X)) = 1. It can be shown that 2 P1UX i (s(a)) = (a)
(12)
for all open subsets a X. If is supported in s(X), then the support of is the set of points y 2 s(X) such that (O) > 0 for any Scott-neighbourhood O UX of y. Any element of P1UX which is supported in s(X) is a maximal element of P1UX [6, Proposition 5.18]; we denote the set of all valuations which are supported in s(X) by S1X. We can identify S1X with the set M1 X of normalised Borel measures on X as follows. Let e : M1 X ! S1 X 7! s?1 and j : S1 X ! M1 X 7! s:
Theorem 2.1 [6, Theorem 5.21] The maps e and j are well-de ned and induce an isomorphism between S X and M X . 1
1
5
For 2 M1 X and an open subset a X, Equation (12) implies (a) = e()(s(a)) = e()(a): (13) Furthermore, it follows from Theorem 2.1 that there exists an increasing chain of simple valuations hi ii0 with G s?1 = i: (14) i0
In other words, any probability measure on X is the least upperbound of a chain of simple valuations on UX. This fundamental property is used in [7] to develop a theory of generalised Riemann integration, which we will be the basis of work in Section 4.
3 Invariant measure of an IFS
1 Suppose F A P UX is a directed set of simple valuations. We would like to know when the lub FA of A determines a Borel measure on X; in other words, under what conditions we have A 2 S1X. The following proposition gives the necessary and sucient condition for this. We will denote the diameter of any compact subset c X by jcj.
Proposition 3.1 The following conditions are equivalent. F (i) A 2 S X . P (ii) For all > 0 and all > 0, there exists a2A ra a 2 A with X ra < : 1
jaj
Suppose now we have an increasing chain hiii0 of simple valuations i in P1UX. Using Equation (11), it is easy to see that we can obtain a nitary branching tree whose nodes on the ith level are precisely the elements of Bi and a child of any node is a subset of that node. We call this a tree associated with the chain hi ii0. Note that a chain can have more than one associated tree. Clearly, the lub in UX of the nodes of any branch of an associated tree is an element of UX, which is to say that the intersection of a shrinking sequence of non-empty compact subsets of X is a non-empty compact subset. Using Proposition 3.1 and Konig's Lemma, we can prove the following useful result. Proposition 3.2 Suppose hiii0 2 P1UX is an increasing chain of simple valuations. If, for any associated tree of the chain, the lub of the nodes of any branch is a singleton subset of X , then the lub of the chain hiii0 is in S1 X .
Let (X; f1 ; : : :; fN ; p1; : : :; pN ) be an IFS with probabilities on the compact metric space X, i.e.PNfor each i = 1; : : :; N, the map fi : X ! X is continuous and we have 0 < pi < 1 with i=1 pi = 1. De ne the transition operator H : P1UX ! P1UX 7! H() PN by H()(O) = i=1 pi (fi?1 (O)). We know from [6,F Proposition 6.1] that H is Scott continuous and, therefore, has a least xed point = m0 H m X . The mth iterate H m X is the simple valuation given by H m X
=
N X i1 ;i2 ;:::;im =1
pi1 pi2 : : :pim fi1 fi2 :::fim X :
(15)
Moreover, the chain hH m X im0 has a unique associated tree which we can label with the transitional probabilities as in [8]. See Figure 3. We call this the IFS tree with transitional probabilities. 6
X p
p
1
fX
1
..
2
fX N
p
p 1
1
N
fX
1
ffX
p
2
p
N
..
ffX
1
p
1
N
..
..
..
.. ..
ffX
N 1
N
..
ffX
N N
Figure 3. The IFS tree with transitional probabilities. Assume now that the least xed point of H is in S1X. Then, by the remark following Equation (12), is a maximal element of P1UX and, therefore, the unique xed point of H. If 2 P1UX, then one can show using the monotonicity of H that is the maximal topological limit of H m () as m ! 1. Writing the maximal topological limit as limm!1 H m (), we have: Proposition 3.3 If 2 S1X , then is the unique xed point of H , and, for any 2 P1UX , we have limm!1 H m () = . A particular instance of the above result occurs if the IFS is hyperbolic, i.e. if fi : X ! X is contracting with contractivity factor si < 1. Then it follows from Proposition 3.2 that 2 S1X, since a typical node fi1 fi2 : : :fim X of the above tree on level m has diameter at most sm jX j which tends to 0 as m ! 1, where s = maxi si and jX j is the diameter of X. Corollary 3.4 For a hyperbolic IFS, H has a unique xed point and limm!1 H m() = for any 2 P1UX .
4 Multiple R-integration In this section, we extend the generalised Riemann integration, R-integration, introduced in [7] to multiple integration. We will only consider the double integral since the generalisation to any multiple integral is straightforward. The double R-integral is used in the next section to compute the magnetisation per spin and the Edwards-Anderson parameter in the Ising model.
4.1 The double R-integral
Recall that given Borel measures 2 MX and 2 MY on topological spaces X and Y , the product measure 2 M(X Y ) satis es ( )(B1 B2 ) = (B1 )(B2) for all Borel subsets B1 and B2 of X and Y respectively [18]. Similarly, given dcpo's D and E, and valuations 2 PD and 2 PE, we have the product valuation 2 P(D E) which for any \rectangle" O1 O2 , where O1 and O2 are open subsets of D and E respectively, satis es ( )(O1 O2 ) = (O1)(O2) [11]. Moreover, the product operation is continuous. In other words, we have the following property. Proposition 4.1 Suppose = Fi2I i 2 PD and = Fi2I i 2 PE, where hi ii2I and hiii2I are directed sets in PD and PE respectively. Then, G = i i: i2I
7
Furthermore, the product of two simple valuations is another simple valuation.
Proposition 4.2 Let
=
X a2A
ra a
and
=
X b2B
sb b
be simple valuations on dcpo's D and E . Then, the product valuation is a simple valuation on D E and is given by
=
X
a2A;b2B
ra sb (a;b):
Let X and Y be compact metric spaces, 2 M1 X, 2 M1 Y . We have the product measure 2 M1 (X Y ). On the other hand, by the remark following Theorem 2.1, there exist increasing chains hi ii0 and hi ii0 of simple valuations i 2 P1UX and i 2 P1UY for i 0 such that G G sX?1 = i and s?Y 1 = i i0
i0
where sX : X ! UX and sY : Y ! UY are the singleton maps. For each i 0, we have the product valuation i i 2 P1(UX UY ) which is a simple valuation by Proposition 4.2. Furthermore, by Proposition 4.1, we have G s?X1 s?Y 1 = i i: (16) i0
Note that UX UY is a subspace of U(X Y ). Let i : UX UY ! U(X Y ) be the inclusion map. Then, i is Scott continuous and we have the continuous embedding ^i : P1(UX UY ) ! P1 U(X Y ) 7! i?1 : It is easy to check that ^i( s?X1 s?Y 1 ) = ( ) s?X1Y where sX Y : X Y ! U(X Y ) is the singleton map. Furthermore, the product of the two simple valuations X X = ra a 2 P1 UX = sb b 2 P1 UY (17) a2A
satis es
b2B
^i( ) = ( ) i?1 X = ( ra sb (a;b) ) i?1 a2X A;b2B = ra sb i(a;b) a2X A;b2B = ra sb (a;b) a2A;b2B
= By Equation (16) and the continuity of ^i, we nally deduce:
Proposition 4.3
( ) sX?1Y =
G i0
8
i i:
Let f : X Y ! R be a bounded function. For the simple valuations and given by Equation (17), we have the lower sum of f with respect to de ned by X S ` (f; ) = ra sb inff(a; b); a2A;b2B
where f(a; b) is the image of (a; b) X Y under f. Similarly, the upper sum of f with respect to is de ned by X S u (f; ) = ra sb sup f(a; b): a2A;b2B
Furthermore, for a given choice of xa 2 a and yb 2 b for each a 2 A and b 2 B, the generalised Riemann sum of f with respect to is de ned by X S(f; ) = ra sb f(xa ; yb ): a2A;b2B
See [7] for details. The main result of this section is the following. Theorem 4.4 If f : X Y ! R is continuous almost everywhere with respect to the product measure , then Z Z S ` (f; i i) % f d ; S u (f; i i ) & f d ; X Y X Y Z S(f; i i) ! f d X Y
as i ! 1.
4.2 Integration over the invariant measure of an IFS
Suppose F now we have an IFS with probabilities (X; f1 ; : : :; fN ; p1; : : :; pN ) such that = m0 H m X , where H m X is given by Equation (15), is a maximal element of P1 UX. If g : X ! R is continuous almost everywhere with respect to = s, then for any point x 2 X we have the generalised Riemann sum for H m X given by N X pi1 : : :pim g(fi1 : : :fim x) S(g; H m X ) = i1 ;:::;im =1
and it follows that
S(g; H m X ) !
Z
g d
as m ! 1. If the maps fi are contacting and g satis es a Lipschitz condition, then we can obtain a nite algorithm to calculate the integral to any given accuracy as follows [8]. Suppose there exist k > 0 and c > 0 such that g satis es
jg(x) ? g(y)j c(d(x; y))k for all x; y 2 X. (When k = 1, we call c a Lipschitz constant for g.) Let > 0 be given. Then Z jS(g; H m X ) ? gd j for m = dlog((=c)1=k =jX j)= logse, where s is the contractivity of the IFS and dae is the least integer greater than or equal to a. 9
Next, assume that h : X X ! R is continuous almost everywhere with respect to . Then, for any x 2 X, we have the following generalised Riemann sum for H m X H m X : N N X X pi1 : : :pim pj1 : : :pjm h(fi1 : : :fim x; fj1 : : :fjm x) S(h; H m X H m X ) = and it follows that
i1 ;:::;im =1 j1 ;:::;jm =1
S(h; H m X H m X ) !
Z Z
h(x; y) d (x)d(y)
(18)
as m ! 1. RIf h satis es a Lipschitz condition then again we can obtain a nite algorithm to estimate h d to within any given threshold. We also have ergodic theorems for IFSs with probabilities. Let (X; f1 ; : : :; fN ; p1; : : :; pN ) be a hyperbolic IFS with probabilities with invariant measure . Let g : X ! R be a continuous function. Suppose i1 ; i2; : : : is is a sequence of independent identically distributed random variables on f1; 2; : : :; N g with probabilities P(in = k) = pk (1 k N) for all n 1. Let i1 2 X, and put in+1 = fin (in ) for all n 1. Then Elton's ergodic theorem states: Theorem 4.5 [9] For almost all sequences i1 ; i2; : : :, and all 1 2 X , Z Z k 1X g(x) d (x) = klim g( ): !1 k n=1 in It is also possible to deduce a two-dimensional version of Elton's Theorem. Let h : X X ! R be a continuous function. Let i1 ; i2; : : : and j1 ; j2; : : : be two sequences of independent identically distributed random variables on f1; 2; : : :; N g with probabilities P(in = k) = P(jn = k) = pk (1 k N) for all n 1. Let i1 ; j1 2 X X, and put in+1 = fin (in ) and jn+1 = fjn (jn ) for all n 1. Then, we have: Theorem 4.6 For almost all sequences i1 ; i2; : : : and j1; j2; : : :, and all 1; 1 2 X , Z Z k 1X h( ; ): h(x; y) d (x)d (y) = klim !1 k n=1 in jn
5 Domain-theoretic analysis of the Ising model In this section, we apply the results of the previous two sections to the one-dimensional random eld Ising model (1d RFIM) that we described in the introduction. We consider a more general case and allow the external magnetic eld be asymmetric, i.e. hn = h , for real numbers h+ and h? with h+ ?h? > 0. We assume, however, for convenience that the two value of the eld are still equally probable, i.e. P(hn = h+ ) = P(hn = h? ) = 1=2 for all n 1.
5.1 Invariant measure of the 1d RFIM
Consider the IFS ([x? ; x+ ]; f?; f+ ; 1=2; 1=2) where the two maps f+ ; f? : R ! R are de ned by f (x) = h + A(x), with A(x) = (2 )?1 log(cosh (x + J)= cosh (x ? J)), as given by Equations (9) and (2). Their unique xed points x+ ; x? are given by x = h =2 + (2 )?1 arcsinh(e2 J sinh h ), with x+ ?x? > 0, cf. equation (10). A simple calculation shows that the rst and the second derivatives of the two maps are given by A0 (x) = 21 (tanh (x + J) ? tanh (x ? J)) 1 ? ): A00(x) = 2 ( 2 1 2 cosh (x + J) cosh (x ? J) It follows that 1 > A0 (x) > 0 for all x 2 [x? ; x+] (in fact for all x 2 R). Furthermore, since A00 is negative in (0; x+] and A(x) = ?A(?x), we easily see that A0 , and hence f+0 and f?0 , 10
have maximum value
s = A0 (0) = tanh J (19) in the interval [x?; x+ ] with 0 < s < 1. By the intermediate value theorem, it follows that A; f+ ; f? are contracting with contractivity factor s. Therefore, by Corollary 3.4, the IFS has a unique invariant measure and starting with any initial valuation 2 P1UX we have limm!1 H m () = . Moreover, there is a nite algorithm to generate on any digitised screen [8]. An analysis of the multifractal nature of the invariant measure = s?1 is given in [1, 3]. When f+ [x?; x+ ] \ f? [x?; x+ ] = ; the support of is totally disconnected and homoeomorphic to the Cantor set. On the other hand, when f+ [x?; x+ ] \ f? [x? ; x+] 6= ;, the support of is the whole interval [x?; x+ ] and is a fat fractal as in Figure 4, which is taken from [2].
Figure 4. Distribution of the local magnetisation (kT = 1, J = 1). The multifractal can be fat (a) for h+ = ?h? = :6 or thin (b) for h+ = ?h? = 1:5.
5.2 Expected value of physical quantities
We will now use the results of Section 4, to obtain nite algorithms to compute the expected values of various physical quantities to any given threshold.
5.2.1 Free energy density
The free energy F plays the same fundamental role in thermal physics that ordinary energy plays in mechanics. It is given in terms of the partition function Z by F( ) = ? 1 log Z. See for example [13]. Therefore, for the Ising model of N spins, Equation (6) gives FN ( ) = ? 1 log ZN NX ?1 = ? 1 log(exp f 1 log(2 cosh N ) + B(n )g) n=1 NX ?1 1 = ? log(2 cosh N ) ? B(n ); n=1
where the function B is given by Equation (3). Therefore, the free energy density f( ) is given in the limit N ! 1 by 1 log ZN FN ( ) f( ) = Nlim !1 N !1 N = ? Nlim ?1 1 log(2 cosh ) ? lim 1 NX B(n ) = ?Nlim N N !1 N n=1 !1 N ?1 1 NX = ?Nlim !1 N n=1 B(n ) 11
If the eld hn is constant then limn!1 n = x and by the continuity of B we have f( ) = ?B(x ) as in [10]. For the random eld Ising model, Elton's ergodic Theorem 4.5 implies Z NX ?1 1 B(n ) = ? B d ; f( ) = ? Nlim !1 N n=1 since the IFS is hyperbolic and B is a continuous function. A simple calculation, similar to the case of A, shows that B : [x?; x+ ] ! [x?; x+ ] is contracting with contractivity factor (20) c = B 0 (x+ ) = 21 (tanh (x+ + J) + tanh (x+ ? J)): Therefore, for the choice of 0 2 [x? ; x+], we consider the generalised Riemann sums X B(fi1 : : :fin (0)): S(B; H n [x? ;x+ ] ) = 21n i1 ;:::;in = Given > 0, we have
Z
jS(B; H n x? ;x+ ) ? B d j < ; with n = dlog((=c)=(x ? x? ))= logse, where s is the contractivity of the IFS given by Equation (19), c is given by Equation (20) and dae is, as before, the least integer greater [
]
+
than or equal to a.
5.2.2 Magnetisation per spin
The magnetisation per spin is given by N 1X hsk iHN ; m( ) = Nlim !1 N k=1
where hsk iHN is the average value of sk with respect to the Hamiltonian HN of the N spins hsn iNn . In order to nd hsk iHN , one reduces the partition function ZN to that of a typical spin sk with 1 k N. This is done by applying the technique of summation, described =1
in the introduction, over spins from left and right as in [4]. Let n be as before, and let N = hN and n?1 = hn?1 + A(n ) for 1 n N ? 1. Then, it is easy to deduce that kX ?1 kX ?1 X exp ((k + A(k ))sk + B(n ) + ZN = B(n )) sk =1
= 2 cosh (k + A(k )) exp ( Therefore,
X
hsk iHN = Z1 f N sk
n=1 kX ?1
n=N kX ?1
n=1
n=N
B(n ) +
sk exp( (k + A(k )))g exp (
= 1
kX ?1
B(n )):
B(n ) +
n=1 kX +1
kX +1 n=N
B(n ))
kX ?1 = Z1 f2 sinh(k + A(k ))g exp ( B(n ) + B(n )) N n=1 n=N = tanh (k + A(k )): If the eld hn = h is constant, then k ; k ! x , where x is the unique solution of A(x) = h + x, and hence m( ) = tanh (2x + h) as in [10].
12
In order to compute the magnetisation per spin for the case of the random eld, we use the two-dimensional version of Elton's Ergodic Theorem 4.6 as follows. N 1X hs i m( ) = Nlim !1 N k=1 k HN N 1X = Nlim tanh (k + A(k )) Z !1 Z N k=1 = tanh (x + A(y)) d (x)d (y); where the double integral can be computed by Equation (18). We can deduce a nite algorithm to compute m( ) to within an accuracy of any given > 0. The function tanh (?) : R ! R has Lipschitz constant . Hence for x1; x2; y1; y2 2 [x? ; x+] we have j tanh (x1 + A(y1 )) ? tanh (x2 + A(y2 ))j (jx1 ? x2 j + jA(y1 ) ? A(y2 )j) (jx1 ? x2 j + sjy1 ? y2 j) where s = tanh T is the contractivity of A : [x?; x+ ] ! [x?; x+ ]. For convenience, we put h(x; y) = tanh (x+A(y)) and n = H n[x+ ;x? ] where H is the transition operator. We choose 0 2 [x? ; x+] for the generalised Riemann sum. It follows that if sn (x+ ? x? ) = (1 + s), i.e. if s)(x+ ? x? )) e; n = d log(= (1 +logs then Z Z jS(h; n n) ? m( )j = jS(h; n n) ? tanh (x + A(y)) d (x)d(y)j ; where
S(h; n n) = 212n
X
X
i1 ;:::;in = j1 ;:::;jn =
tanh (fi1 : : :fin (0) + A(fj1 : : :fjn (0))):
5.2.3 Edwards-Anderson parameter
Similarly, we can obtain a nite algorithm for computing the Edwards-Anderson parameter qEA given by Z Z N X 1 2 qEA = Nlim (tanh (x + A(y)))2 d (x)d(y): hs i = !1 N k=1 k HN The function tanh2 (?) : R ! R p has Lipschitz constant 4 3=9. It follows that if p + s)(x+ ? x? )) e; n = d log(9=4 3 (1logs then Z Z jS(h2 ; n n) ? qEAj = jS(h2 ; n n) ? tanh2 (x + A(y)) d (x)d (y)j ; where
S(h2 ; n n) = 212n
X
X
i1 ;:::;in = j1 ;:::;jn =
tanh2 (fi1 : : :fin (0) + A(fj1 : : :fjn (0))):
13
Acknowledgements I am grateful to Ulrich Behn for introducing me to this subject and for various informative discussions. This work has been founded by EPSRC grant \Foundational Structures in Computer Science" at Imperial College.
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