Some general rules for solving the inverse problem of inhomogeneous ising models

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Some general rules for solving the inverse problem of inhomogeneous ising models

L. Šamaj

To cite this version:

L. Šamaj. Some general rules for solving the inverse problem of inhomogeneous ising models. Journal

de Physique, 1989, 50 (3), pp.273-282. �10.1051/jphys:01989005003027300�. �jpa-00210917�

273

Some general ^{rules for} solving the inverse problem ^of inhomogeneous Ising ^models

L. 0160amaj

Institute of Physics, EPRC, ^Slovak Academy ^of Sciences, ^{Dúbravská cesta} 9, ^{842 28} Bratislava, Czechoslovakia

(Reçu ^{le 18} juillet ^1988, accepté ^sous forme définitive le 14 octobre 1988)

Résumé.

Nous étudions des réseaux d’Ising ^de spin arbitraire, ayant ^des points d’articulation,

en présence d’une interaction de proche ^voisin inhomogène ^{et d’un} potentiel ^{externe. En} particulier, ^nous analysons ^le problème ^inverse qui consiste à trouver le potentiel requis pour

produire ^une forme donnée de la distribution d’équilibre ^{à un site. A cause de la} topologie spéciale ^{du réseau, nous} pouvons ^montrer que le potentiel appliqué ^{à un} site donné est une

fonctionnelle de la densité de probabilité ^de site à l’intérieur d’un sous-réseau fini limité par des

points d’articulation. De plus, ^{si le} potentiel ^est appliqué ^{à un} point d’articulation, ^le problème

inverse initial peut être réduit à un certain nombre de problèmes inverses locaux qui sont souvent exactement solubles. A titre d’exemple, ^nous présentons des résultats explicites pour le réseau de Bethe avec certains types de variables de spin ^et interactions de proches voisins. Nous montrons que les fonctions de corrélations directes ^sont à courte portée ^{dans tous les cas étudiés.}

Abstract.

We study higher spin Ising ^lattices having articulation points ^with spatially varying

both nearest-neighbour interactions and external potential. ^In particular, ^we analyse the inverse

problem ^{of the} potential required ^to produce ^a given profile ^of equilibrium single-site

distributions. Due to the special topology ^{of the} lattices, ^the potential applied ^{to a} given ^{site is}

shown to be a functional of the site probability density ^{inside a} finite sublattice limited by

articulation points. ^{Moreover, if the} potential ^is applied ^{to an} articulation site, ^the original ^inverse problem ^{can be reduced to a} number of local inverse problems ^{which are} often solvable exactly.

As an example, ^we present explicit ^results for the Bethe lattice with certain types ^{of the} spin

variable and nearest-neighbour interactions. Direct correlation functions in all considered cases are shown to be of short range.

J. Phys. ^{France 50} (1989) ^273-282 ler FÉVRIER 1989,

Classification Physics ^Abstracts

05.50 2013 75.10H

1. Introduction.

With regard ^to the character of the real physical world, increasing attention is now paid ^{to the}

influence of the spatial inhomogeneity ^{on the} thermodynamic properties of various systems.

The first step ⁱⁿ investigations of this kind is the study ^of low-dimensional models describing adequately ^{a lot of} phenomena associated with higher dimensions [1]. ^Their advantage ⁱⁿ equilibrium statistical mechanics consists in the fact that they ^are usually ^solvable exactly ^and

thus may be ^{a test} for general hypotheses ^and possible approximations ^{or make some new}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005003027300

suggestions. Unfortunately, ^{even for the} simplest nontrivial inhomogeneous ^{model in one} dimension, ^{i.e. the} Ising ^{model or lattice} gas with nonconstant nearest-neighbour interactions

or a position-dependent ^external field, the free energy ^can be expressed only ^{in terms of}

continued fractions [2] and nonlinear recurrence relations [3]. The closed form solutions are

given ^{for some} special examples, e.g., ^{at zero} temperature [4] ^or, building ^{the model’s} quasiperiodic ^or random fields from the known solution, ^{at a} specific temperature [5].

A greater progress has been made in the so-called inverse problem, ^{in which one finds the}

external potential ^{needed to} produce ^a given density profile. ^In solving ^this problem, ^direct

correlation functions, ^whose short range is the basis of many reasonable approximations ⁱⁿ

the theory ^{of fluids, are obtained} systematically by simple differentiation. The known exact

solutions for an arbitrary ^field [6, 7] ^{as well as its} generalization ^to varying both field and interactions [3, 8] ^are restricted to one dimension. Recently [9], ^{we have} proposed ^{a new} approach ^to one-dimensional inhomogeneous ^models which works with certain statistical variables having ^{a clear} physical meaning. ^We have shown that for a dichotomic spin ^variable

and bilinear nearest-neighbour interactions, the solution to the inverse problem ^{of a} potential

at a given site follows naturally from the statistical independence ^{of the} fragments resulting

upon the elimination of ^a spin ^at the considered site.

In the present work, ^{this fact} inspires ^{us to deal with more} complicated structures having

articulation points : the deletion of an arbitrary articulation point divides the lattice into two or more disconnected parts (for definitions in graph theory ^see, e.g. [10]). ^Some typical

lattices of this kind are pictured ⁱⁿ figures ^la-d. Here, the solid lines represent symmetric

nonconstant couplings 0 ij (si, sj ) ^{of spins at sites i and j. Since we} consider the case of general spin taking ^{one of the D + 1 values} s = 0, 1,..., D, they occupy ^a linear space of dimension

(D ⁺ 1 ) (D ⁺ 2 )/2. The external potential applied ^to arbitrary ^site i, wi (si ), ^{is a member of a}

linear space of dimension D ⁺ 1, ^{one dimension of} which is taken as a local reference and, consequently, ^{does not} affect the equilibrium statistics. For the usual choice wi (si)=

si ui, the local reference potential wi (0) =- ^{0 is} specified.

As has been indicated, ^{we do not} try ^{to find the} dependence ^{of the} single-site distribution

ni (s ), which is the probability ^of spin s at ^{site i} (si - s ), ^{on the} applied potential

Fig. ^1.

a) Simple ^decorated chain ; b) ^chain of squares ; c) Bethe lattice with coordination number

q ; d) triangle ^cactus.

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{ Wi (S )}. ^{On the contrary, we} look for the potential Wi (s ) evoking ^a given profile ^of single-site

distributions {ni (s )}. ^More specifically, taking ^{into account the} special topology ^{of the}

considered lattices, ^{we aim at :}

i) identifying the lattice regions ^{which are} insubstantial for solving the inverse problem ^{of a} specific potential, ^{i. e. ,} reducing ^the problem ^{to a finite} cluster of lattice sites (Sect. 2) ;

ii) simplifying the inverse problem ^{of the} potential applied ^{to an} articulation point (Sect. 3).

The successful realization of these aims will enable us to solve the inverse problem ^{for two} representative choices of exchange couplings ^and Ising spins ^on the Bethe lattice.

2. Réduction of the inverse problem ^{to a} finite cluster of sites.

In this part, ^{we are} concerned with the formal solution to the inverse problem ^{of the} potential applied ^{to a} given ^site (denoted simply ^as 0) ^{of the} lattice like those in figures ^{la-d. Let us take} advantage of the presence of articulation points ^and separate the lattice sites with respect ^to

the reference site 0 into two basic sets G1 ^and G2. ^{All the sites of} the first set

G1, ^{denoted as} 1, 2, ..., p, remain connected with site 0 through ^a continuous chain of interaction lines after eliminating ^an arbitrary ^lattice point. ^The remaining ^sites, belonging ^to

the second set G2, ^{can be} divided into disjunctive ^subsets li 0, il, - - -, i k]° (k

0, 1, ..., p ;

i 0’ il,..., i

^{0, 1,} ..., p with iO’--il""--ik) ^{defined as follows : a} given ^subset [i °, i? ..., ik]O contains all sites of G2 ^{which can} be connected by ^a continuous chain of interaction lines, walking through ^a sequence of sites belonging ^to G2, ^with only points io, il, ---, ik ^{of the set} {0,1, ^2, ..., p}. ^With regard ^{to the} connectivity ^{of the} lattice, ^an arbitrary ^{site of} G2 has the connection of this type ^{with at least one of the} points 0, 1, ...,p, and ^so the above evidently disjunctive decomposition ^covers the whole set

G2.

In an arbitrary ^subset [io, ile ^..., iklo, ^{there are} certainly the sites bounded directly by ^one

interaction line to points io, /i,

or i k (of ^{the set} G, ^{or identical with the Oth} point). ^If

k 1, these sites must also belong ^to G1 because there is no articulation point ^{whose deletion}

breaks all indirect bonds of arbitrary ^one of them with the Oth point. ^Since [io, il, ^..., ik 10

cannot contain any site of G1 ^{it holds} [io, i 1, ... , i k 10 = ^{0 for k 1.} Similarly, ^{the subset} [0 ]0 ^{includes at least one} point ^connected directly by ^one interaction line to site 0 what implies

[0]0 = 0. ^{For these} reasons we conclude that only ^subsets [i 10 (i = 1, 2,..., p ) ^can

constitute G2, ^{i. e.}

The separation of lattice sites into two basic groups together ^{with the} disjunctive decomposi-

tion of the second one G2 ^is represented schematically ⁱⁿ figure ² (for ^a practical realization

Fig. ^2.

Classification of lattice sites.

for certain lattices see Figs. ¹ a-d). According ^{to the} previous analysis, ^the disjunctive ^sets

[1 ]°, [2]°, ^..., [p ]° ^{have no direct or} indirect bound with the Oth site and with each other after

eliminating ^the respective points 1, 2, ...,p.

The Hamiltonian of the spins ^at sites classified in this way may be written ^as

where Je (Si’ [Si ]°) is the usual Hamiltonian of the isolated spin set {si’ [si JO} ^{including all pair}

interactions as well as one-site potentials. ^The thermodynamic properties ^{of our} system ^are determined by ^the partition ^function

where /3 is the inverse temperature ^{and the summation} goes ^over all possible spin configurations. Using ^{the standard notation}

it can be expressed ^{in a} convenient form

Here, ^the introduced quantities

are respectively ^the single-site distribution of spin ^at point j ^{and the} partition function, ^both

evaluated for the isolated spin system {Sj’ [Sj]O}. ^The probabilities nj(Sj) ^are constrained by

the evident condition

In the final expression ^{for the} partition ^function (4), the absence of direct bonds _among

disjunctive point ^sets { [j ]°}p-1 is reflected through ^the simple product ^of mutually

277

independent single-site distributions {fïj(Sj)}= 1. ^{Then,.the way for solving formally the}

inverse problem ^{is natural and} straightforward. ^{It is} only necessary ^to express within the

proposed ^{formalism the} probability ^of spin s (s

^{0, 1, ...,} D ) ^{at site 0,}

and at sites i (= ^1, 2, ... , p ) belonging ^{to the} first group of the above classification,

Substituting ^from (4) ^into (7a, b) ^{the ratio}

and respecting ^the constrains {s} no (s ) = X ,,) ^{ni (s ) = 1, we obtain (1 +p) D} independent

nonlinear equations. They constitute a complete ^{set to} be solved for D independent potentials applied ^{to site 0} and pD independent single-site distributions iij (s). ^{In this manner we have}

proved that the external potential ^at site 0 does not depend ^{on the} distribution profiles ^{and the}

model’s parameters (namely coupling ^constants and one-site potentials) of the lattice subsets

[110, [2 ]0, ..., [p ]0.

This fact substantially simplifies the inverse problem of the Oth-site potential ^{because we}

may restrict ourselves ^{to a} finite cluster of sites 0, 1, ..., p. It ^can be eliminated from the lattice system by neglecting the bonds between spins ^{at sites 1 and} [1 ]°, ^{2 and} [2 ]°, ..., p and [p ]°. Keeping ^the single-site distributions inside the cluster unchanged, ^this separation ^{has no influence on} the value of the investigated potential ^{at site 0} (but may modify potentials applied ^to other cluster sites 1, 2, ^..., p).

3. Simplification of the inverse problem ^{of the} potential applied ^{to an} articulation point.

In general, the solution to the inverse problem with the aid of nonlinear equations (7a, b), (8)

is very complicated ^{also for a finite} cluster of lattice sites. A substantial simplification ^comes along if the reference point 0 is the articulation point.

Let the multiplicity ^of articulation point ^{0 be} equal ^to q. Upon ^its deletion, ^the remaining

p cluster sites ^are divided into q disjunctive ^{and isolated sets} of connected sites denoted as

[1]*, [2 ] *, - ^..., [q ] ^{*. Site 0} together ^{with an} arbitrary ^one of these groups constitutes a star which has no articulation point. The Hamiltonian of spins ^localized inside the considered cluster is then given by

Here, ^the introduced Hamiltonian Je (O)(So, [si ] *) ^includes pair interactions of all spins

{SO, ^{[s; ]} *} ^{as well as external} potentials applied ^{to all sites} [i ] ^* (but ^{not to site} 0). ^{Note that}

these potentials ^{need not} coincide with the original ones {Wi (s )} .

The absence of direct bonds between an arbitrary couple ^{of sets} [i ] ^{* and} [y] ^* permits ^{us to}

express the partition ^{function as}

where

Using ^the explicit ^{forms of two} different Oth-site distributions

the formal solution ^to the inverse problem ^reads

The above defined quantity Eoi(so) is, similarly ^to JC(o)(so, [-yj*). ^a function of pair

interactions between all spins {so, [sij* and extemal potentials applied ^{to sites} [i ) *.

According ^{to section} 2, ^these potentials depend only ^{on the} single-site distributions and two-

spin couplings inside the star {0, [i ] *} because of the articulation character of the Oth point.

Therefore, separating ^the star {O, [i ] *} from the cluster and keeping ^the profile ^of single-site

distributions inside it, _eOi (so) ^remains unchanged. ^{On the} contrary, ^the potential ^{at site} 0,

wo(so)

[- ln Wo(so)]/f3, ^{modifies to} WJi)(SO)

[- ln WJi)(SO)]/f3 ^{yielded by}

what implies

In this way, the original ^inverse problem ^{of the} potential ^at site 0 is transformed to q independent ^inverse problems ^{of the} potential ^{at site} 0, ^{site 0} being consecutively ^the

member of q ^{different stars.}

279

To be ^more particular, ^{we take as an} example ^{the case} of the Bethe lattice with coordination number q (see Fig. lc) ^{where the sets} [1]*, [2]*, ^..., [q]* introduced above

correspond ^to single ^sites 1, 2, ..., q, respectively. ^{Each of} the q ^stars {0,1}, {O, 2 },..., {0, q} ^{contains two points. Let us consider an isolated star,} say {0, i } . ^The kept single-site distributions at sites 0 and i can be written explicitly ^as

where

is the two-spin partition ^{sum. To} obtain the desired W,(’)(s), ^{we first use relation} (15a) ^to

express the quantity wli )(s) (being ^a function of the ith-site potential which is irrelevant in

our problem) ^{as follows}

Then, substituting (16) ^into (15b) ^and assuming ^the symmetric ^matrix eot (s, s’ ), ^{we have}

with s = 0, 1,..., D. Among ^{D + 1} nonlinear relations one is superfluous because of the

constrain L ^{ni (s )}

^{1 and one of D + 1 unknowns WJi)(S) is taken as a} local reference.

{s}

Therefore, (17) constitutes a closed system ^of equations ^for solving the inverse problem ^{of the}

Oth-site potential ^for the isolated star {O, i}. Having ^the explicit ^{forms of} WJi)(S)

(i

^{1, 2,} ..., q ) ^and substituting ^{them into} (14), ^{we arrive at} the final solution to the inverse

problem for the Bethe lattice.

Before proceeding further, ^{we discuss an} important problem concerning ^{the local reference} potential. ^{If a} local reference is specified, say Wo(O) == ^{1, it is} suitably ^to work with the

quantity F (s, s’ ; no, ni )

WJi )(s )/WJi )(s’) ^{for which it holds} F (s, ^{s’ ;} no, ni )

1/F (s’, s ; no, ni). Resulting equations ^for solving the inverse problem (14), (17) ^{are then}

rewritten as

If ^a local reference is not specified, ^{it is more} advantageous ^{to use}

WJi )(s). Consequently,

where the unknown ^constant a o is fixed by imposing ^a local reference for the potential. ^The

« two-site » equations (18b), (19b) appear, of course, also in the remarkable work by ^Percus [7] dealing ^{with the} ordinary one-dimensional chain (which corresponds ^{to the} Bethe lattice with coordination number q

2). The technical job ^of solving ^them depends ^{on the nature of}

the spin and the interaction.

Let us consider the case of a dichotomic spin variable s

0,1 ^{and chose} 0 oi (s, s’)

Joi. ^{s .} s’, wo(s) ^{= s .} uo. As Wo(0) - 1, the formulation (18a, b) ^is convenient, ^i.e.

where is given by

Using ^the explicit forms of the matrices

and respecting ^the positivity ^of F (1, 0 ; no, ni), ^{we have}

where and

Now, ^we present ^a general ^{class of} higher-spin Bethe lattices for which the formulation

(19a, b) ^is appropriate. The considered spin ^{variable assumes D + 1 values} (s

^{0, 1, ...,} D )

and nearest-neighbour interactions are given by

For this special choice of the matrix eoi (s, s’), ^the procedure ^for finding ^the explicit ^{form of} G (s ; no, ni) ^{from the « two-site »} equation (19b) is described in detail in [7]. ^{We write down}

the final result :

with K(no, ni ) ^{defined by}

The solution to the inverse problem ^{is then} yielded by (19a).

281

The application ^{of the method to} other lattices is evident. It is only necessary ^to find all above defined ^stars {0, [1]*} , {0, [2]*} , ^..., {0, [q] *} ^{and to} solve the inverse problem ^of

the Oth-site potential for every isolated star with kept single-site distributions. Substituting

these solutions ^into (14), ^{we arrive at} the final solution to the inverse problem ^{of the Oth-site} potential.

4. Concluding ^remarks.

It is instructive to recapitulate briefly ^the proposed procedure ^for solving the inverse problem

of the potential ^{at a} given ^site 0 in view of its practical application. ^{As a first} step, taking advantage of the presence of articulation points ^{on the} investigated lattices, ^{we have} performed ^a classification of lattice sites with respect ^{to the reference site 0.} On the basis of this classification we have proved ^{that the} problem ^can be reduced to a finite cluster of sites

0, 1, ..., p. They ^remain connected with the Oth site through ^a continuous chain of interaction lines after eliminating ^an arbitrary lattice site. As a second step, ^a substantial simplification

comes along ^{from the} supposed articulation character of site 0 itself. The remaining ^cluster

sites 1, 2, ..., pare then divided into q disjunctive groups of connected sites arising upon its deletion. Together with the Oth site, each of these groups constitutes ^{a star} having ^no

articulation point. Using conclusions of the first step ^we have shown that for solving ^the original ^inverse problem ^{it is} sufficient to know the solutions to the inverse problem ^{of the} potential ^{at site 0 for} these q ^{different stars} (see ^relation (14)). ^The investigated ^{case of the}

Bethe lattice, leading ^{to a number of} relatively simple ^two-site problems, demonstrates the usefulness of the proposed procedure.

As has been mentioned in reference [7], ^the (modified) direct correlations

are ill-defined within the proposed formalism because of the constrain , {S’} ^{ni (s’ )}

^{1. This}

difficulty may be avoided either by eliminating ^one component ^or by restricting derivatives to

the surface tangent ^to consider

Since , we

In accordance with section 2, wo (s ) depends ^{on the} single-site distributions inside the cluster of lattice sites 0, 1,..., p. Consequently, the direct correlation functions (24) ^{of all the}

considered Ising ^{lattices have} only short range support.

In conclusion, ^we would like to emphasize ^an important fact. It is well known that the

rigorous results in the theory ^of inhomogeneous ^models obtained till now are restricted to one

dimension. The present approach implies the solution to the inverse problem ^{and the exact}

direct correlation functions for a large ^{class of} inhomogeneous Ising lattices whose

homogeneous ^versions undergo ^a phase transition at nonzero temperature. We believe that these qualitatively ^{new results will} be useful in the future study ^of spatially inhomogeneous systems.

Acknowledgment.

The author is grateful ^to the Joint Institute for Nuclear Research (Dubna, USSR) ^{for kind}

hospitality during ^the early stage of this work.

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[5] DERRIDA, B., MENDÈS-FRANCE, ^{M. and} PEYRIÈRE, J., ^{J. Stat.} Phys. ⁴⁵ (1986) ^439.

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[8] ^BORZI, C., ORD, ^{G. and} PERCUS, J. K., J. Stat. Phys. ⁴⁶ (1987) ^51.

[9] 0160AMAJ, L., Physica ^A (in press).

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