Duality and exact results for conductivity of 2D isotropic heterophase systems in magnetic field

0.17 MB·English

by  S. A. Bulgadaev

5 DUALITY AND EXACT RESULTS FOR CONDUCTIVITY
0 OF 2D ISOTROPIC HETEROPHASE SYSTEMS
0 IN MAGNETIC FIELD
2

n S.A.Bulgadaev 1, F.V.Kusmartsev 2
a
J
LandauInstituteforTheoretical Physics,Chernogolovka, Russia,142432
0
DepartmentofPhysics,LoughboroughUniversity,Loughborough, LE113TU,UK
3

]
n Using a fact that the effective conductivity σˆe of 2D random heterophase sys-
n tems in the orthogonal magnetic field is transformed under some subgroup of
- thelinearfractionalgroup,connected withagroupoflineartransformations of
s twoconservedcurrents,theexactvaluesforσˆe ofisotropicheterophasesystems
i
d arefound. Asknown,forbinary(N=2)systemsadeterminationofexactvalues
. ofbothconductivities (diagonalσed andtransverseHallσet)ispossibleonlyat
at equal phase concentrations and arbitrary values of partial conductivities. For
m heterophase(N ≥3)systemsthismethodgivesexactvaluesofeffectiveconduc-
tivities, when their partial conductivities belong to some hypersurfaces in the
- space of these partial conductivities and the phase concentrations are pairwise
d equal. Inallthesecasesσe doesnotdependonphaseconcentrations. Thecom-
n plete, 3-parametric, explicit transformation, connecting σe in binary systems
o withamagneticfieldandwithoutit,isconstructed.
c
[

2 PACS: 75.70.Ak, 72.80.Ng, 72.80.Tm, 73.61.-r
v
5
1. Introduction
6
3
2 The properties of the electrical transport of classical macroscopically inho-
1 mogeneous (random or regular) heterophase systems, consisting of N (N 2)
≥
4 phases with different conductivities σ (i=1,2,...,N), have had always an im-
i
0 portance for practice [1]. Last time new materials appeared which are very
/
t perspective from the high technology point of view. They often have the het-
a
erophaseinhomogeneousstructureonsmallerscales(frommicroscopictillmeso-
m
scopic) and some unusual transport properties. For example, the magnetore-
-
d sistance of oxide materials with a perovskite type structure becomes very large
n (thesocalledcolossalmagnetoresistanceinsuchmaterialsasmanganites)[2]or
o growsapproximatelylinearlywithmagneticfielduptoveryhighfields(insilver
c
chalcogenides) [3]. There is an opinion that these properties take place due to
:
v inhomogeneitiesofthesematerials[3]. Forthis reasonacalculationoftheeffec-
i
X tive conductivity σe of inhomogeneous heterophase systems without and with
magnetic field at arbitrary partial conductivities and phase concentrations is
r
a very important problem. Unfortunately, the existing effective medium approxi-
mations(EMA)describeswellthe effectiveconductivity ofrandomheterophase
systems without magnetic field only for binary (N =2) systems and at not too
large inhomogeneities [4]. For systems with N > 2 phases without magnetic
1e-mail: [email protected]
2e-mail: [email protected]
1
field [4, 5] and for systems in magnetic field [6] the EMA cannot give an ex-
plicit simple formulas convenient for description of the experimental results in
a wide rangeofpartialparameters. Such formulascanbe obtainedonly inhigh
magnetic field [6, 7]
A situation is better in 2D systems, where a few exact results [8, 9, 10]
and approximate explicit expressions based on exact duality relations (DR)
have been obtained for isotropic self-dual heterophase systems without mag-
neticfield[5,11]. But,theexactresultsforinhomogeneoussystemsinmagnetic
field have been obtained only for binary systems at equal phase concentrations
x = x = 1/2 [12, 13] (see also [14] about systems with uniform σ ). In this
1 2 t
letterwewillshowthatthe exactresultsforbothdiagonal(σ )andtransverse
ed
(σ ) effective conductivities canbe obtainedfor isotropic self-dualheterophase
et
systems(randomaswellasregular)inmagneticfieldH andatarbitrarynumber
of phases N. They generalize the known results for binary systems with mag-
neticfield[12,13]andforarbitraryN-phasesystemswithoutmagneticfield[10].
Inparticular,itwillbeshownthatforN >2,contrarytotheN =2case,there
exist the whole hyperplanes in the space of concentrations and hypersurfaces
in the space of partial conductivities, on which σ does not depend on phase
e
concentrations. The full explicit transformation,connecting σ in systems with
e
a magnetic field and without it, is constructed.
2. Conserved currents, symmetries and duality transformations
in magnetic field
We start with a brief description of some general properties of classical
isotropicstationaryconducting systems in the externalperpendicular magnetic
field H. Such systems are described by the following differential equations for
a current j(r) and an electric field e(r) [1]
div j(r)=∂ j (r)=0, e(r)=ǫ ∂ e =0, i,k =1,2 (1)
i i ik i k
· ∇×
and the Ohm law
j(r)=σˆ(r)e(r), (2)
where
σˆ =σ =σ δ +σ ǫ , σ (H)=σ ( H), σ (H)= σ ( H), (3)
ik d ik t ik d d t t
− − −
All conductivities σˆ of the form (3) commute between themselves and can be
treated as ordinary functions. The effective conductivity σˆ of N-phase ran-
e
dom or regular symmetric systems with the partial conductivities σ , σ (i=
id it
1,2,...,N) (we assume that σ 0) and concentrations x (satisfying the nor-
id i
≥
malization condition N x = 1) must be a symmetric function of pairs of
i=1 i
arguments (σˆ ,x ) and a homogeneous (a degree 1) function of σ . For this
i i di,ti
P
reason it is invariant under all permutations
σˆ (σˆ ,x σˆ ,x ...σˆ ,x )=σˆ (P (σˆ ,x σˆ ,x ...σˆ ,x )), (4)
e 1 1 2 2 N N e ij 1 1 2 2 N N
| | | | | |
2
where P is the permutation of the i-th and j-th pairs of arguments, and can
ij
be represented in the next form
σˆ (σˆ ,x σˆ ,x ...σˆ ,x )=σ fˆ(z ,x z ,x ...z ,x ).
e 1 1 2 2 N N s 1d,1t 1 2d,2t 2 Nd,Nt N
| | | | | |
Here z = σ /σ , where σ is some normalizing conductivity. It is con-
id,it id,it s s
venient to choose σ in the form symmetrical relative σ , what conserves a
s id,it
symmetry property of f.
The effective conductivity of N-phase systems must satisfy, together with
the usual boundary conditions on boundaries between i-th and k-th phases
j =j , e =e , i,k=1,2,...,N, (5)
in kn i|| k||
the following natural reduction and limiting requirements:
1) σˆ of N-phase system with n (2 n N) equal partial conductivities
e
≤ ≤
must reduce to σˆ of system with N n+1 phases and the concentrations of
e
−
the phases with equal conductivities must add;
2) it has not depend on partial σ and must reduce to the effective con-
di,ti
ductivity of (N 1)-phase system, if the concentration of this phase x =0;
i
−
3) it must reduce to some partial σˆ , if x =1.
i i
The duality relations for σˆ in magnetic field have in general a more richer
e
and complicated structure, which generalizes the inversion structure of DR of
systems without magnetic field. It was established firstly by Dykhne in [12],
where an idea and a method to solvethis problemhave been proposed. We use
theseideaandmethod,refinedlater(see,forexample,[13,15,16]),inamodern
form. The main observation is that in the 2D conducting system, described by
equations (1,2), really exist 2 conserved currents and two curlfree vector fields.
The second partners for j and e are their conjugated fields, which satisfy the
corresponding equations
˜j =ǫ e , e˜ =ǫ j , ∂ ˜j =0, ǫ ∂ e˜ =0. (6)
i ik k i ik k i i ik i k
Asaresultonecanconsiderlinearcombinationsofconservedcurrentsandcurl-
free electric fields
j=aj′+b˜j′, e=ce′+de˜′, (7)
where the coefficients a,b,c,d R are arbitrary real. Futher it will be more
∈
convenient to use the complex representation for coordinates and vector fields
[1]
z =x+iy, j =j +ij , e=e +ie , σ =σ +iσ .
x y x y d t
The primed currents and electric field satisfy again the Ohm law
cσ ib
j′ =σ′e′, σ′ =T(σ)= − . (8)
idσ+a
−
It follows from (7),(8) that under a linear transformation (7) a conductivity
transforms under the corresponding linear fractional (LF) transformation T
3
from (8). In the absence of magnetic field σ is real (or, in a matrix repre-
sentation, diagonal) and a transformation (8) reduces to the inversion [8, 9]
σ2
σ′ = 0, σ2 =b/c, a=d=0 (9)
σ 0
Thus, instead of one-parametric inversion transformations in systems without
magnetic field, σ in magnetic field transforms under linear fractional transfor-
mation with 3 real parameters (since one of 4 parameters can be factored due
to the fractional structure of T). There are various ways to choose 3 parame-
ters. In our treatmentit will be convenientto factor d. This gives 3 parameters
a¯=a/d,¯b=b/d,c¯=c/d,determiningatransformationT.Thetransformations
T from (8) form a subgroup of a group of all linear fractional transformations
witharbitrarycomplexcoefficientsa,b,c,d,conservingtheimaginaryaxis. This
subgroup contains also the shift transformations for an imaginary part of σ,
which correspond to d=0, a=c.
Thetransformation(8)hasthefollowingformintermsofconductivitycom-
ponents σ and σ
d t
ac+bd a¯+¯b/c¯
σ′ =σ =c¯σ ,
d d(dσ )2+(a+dσ )2 d(σ )2+(a¯+σ )2
d t d t
cdσ2+(a+dσ )(cσ b) σ2+(a¯+σ )(σ b/c)
σ′ = d t t− =c¯ d t t− . (10)
t (dσ )2+(a+dσ )2 (σ )2+(a¯+σ )2
d t d t
They have rather interesting structure. Any general transformation gives non-
trivial transverse part σ′ = 0, even if it was absent (σ = 0), while σ′ remains
t 6 t d
always equal to 0, if σ =0 (or σ′ =0, if det T =0).
d d
This general consideration can be applied also to the effective conductiv-
ity of inhomogeneous systems. Then one obtains the following transformation
property for σ
e
σ ( σ , x )=T(σ (T( σ ), x )),
e i i e i i
{ } { } { } { }
T−1(σ ( σ , x ))=σ (T( σ ), x )). (11)
e i i e i i
{ } { } { } { }
The relation (11) means that σ transforms under conjugated representation
e
( T−1) of linear fractional transformations and its change as a function of
∼
its arguments canbe completely compensated by the same transformationof it
itself(thisformtakesplaceforthedualtransformationsconnectedwithsymme-
trytransformationssatisfyingtheconditionT T−1 andforthemisequivalent
∼
to the usual case from [12]). Note also that the transformations (11) does not
depend and does not act on concentrations. Following a tradition, we will call
such transformations the duality transformations (DT) and such relations the
duality relations (DR). An existence of the such exact duality relations is very
important property of two-dimensional inhomogeneous systems. For example,
they can be used for the obtaining the exact values of effective conductivity.
For this one needs the transformations, which action on the set of partial con-
ductivities, can be compensated by symmetries of the effective conductivity as
4
a function of its partialconductivities and phase concentrations. It means that
the set of all partial parameters and concentrations must be invariant under
some joint action of the DT and symmetry operations, i.e. it must be their
fixed point (FP). At this set of partial parameters ( σ , x ) one obtains
i i
{ ∗ } { ∗ }
for σ the exact equation
e
σ ( σ , x )=T(σ ( σ , x )), (12)
e i i e i i
{ ∗ } { ∗ } { ∗ } { ∗ }
where T is the corresponding DT. This equation means also that σ at the FP
e
is a fixed point (or aneigenvectorwith the eigenvalue equalto 1) ofT. Inorder
to exclude a misunderstanding of different fixed points, we will call the fixed
pointsofT asitseigenvectors. Asisknown,theLFtransformationsT canhave
only one or two eigenvectors in dependence on value of its discriminant. For T
from (8) one has for such eigenvectors z
1,2
a c (a c)2+4bd
T(z )=z , z = i − − − (13)
1,2 1,2 1,2 − d ± 4d2
r
It follows from (13), that for (a c)2 < 4bd the corresponding z will have a
1
−
positive real part, what gives a physical conductivity. For (a c)2 > 4bd both
−
eigenvectors have only imaginary part, i.e. they can correspond only to σ with
σ =0, σ =0.Inclassicalandusualsystemsthiscaseisunphysicalandcannot
d t
6
berealized(though,itcanserveforaphenomenologicalapproximatedescription
of the low-temperature systems with quantum Hall effect [15]). It follows from
(13) that one can always construct for a given physical σ the transformation
T , which has σ as its eigenvector,
σ
a c
σ =σ +iσ , 2σ = ( − ), b/d=σ2+σ2, (14)
d t t − d d t
As one can see from (14), this σ does not define the corresponding T unam-
σ
biguously, since there are 3 real parameters in T. The third parameter can be
determined if T has some additional constraint. There is a very important
case of transformations T, satisfying the condition T2(z) = z, (i.e. T2 I or
∼
T−1 T, it includes all T acting as permutations). For these transformations
∼
a= c and, consequently, one has
−
σ = a/d, b/d=σ2+σ2. (15)
t − d t
For transformations with c = a and b/d 0 the eigenvectors can be only real
≥
z = b/d.
1,2
±
Thus,weseethatthoughanumberofparametersof”dual”transformations
p
T becames larger (3 instead of 1), but, since a number of partial conductivities
doubled, they are still not enough to make the DR essentially richer, than in a
casewithH =0(netherless,somenewpossibilitiesappear,seebelowsection4).
Fortunately, in any case one can define the transformation T , interchanging
12
arbitrary σ and σ ,
1 2
T σ =σ , T σ =σ (16)
12 1 2 12 2 1
5
In components (16) has the form
c¯σ (a¯+¯b/c¯) σ2 +(a¯+σ )(σ b/c)
σ = 1d , σ =c¯ 1d 1t 1t−
2d (σ )2+(a¯+σ )2 2t σ2 +(a¯+σ )2
1d 1t 1d 1t
cσ (a¯+b/c) σ2 +(a¯+σ )(σ b/c)
σ = 2d , σ =c¯ 2d 2t 2t− . (17)
1d (σ )2+(a¯+σ )2 1t (σ )2+(a¯+σ )2
2d 2t 2d 2t
Though, formally, this transformation relates 4 independent parameters and
satisfies the condition T2 I, it has one-parametric general solution of the
∼
form
σ σ +σ σ
1d 2t 2d 1t
a= d , c= a,
− σ +σ −
1d 2d
σ σ2 +σ σ2 +σ σ (σ +σ )
b=d 1d 2t 2d 1t 1d 2d 1d 2d , (18)
σ +σ
1d 2d
wheredisincludedasanarbitrarycommonfactor. Thisfactreflectsanexistence
of some hidden symmetry, wich effectively reduces a number of independent
equations (see also discussions in the Sections 3 and 4).
3. Fixed points and exact values of the effective conductivity.
Havingallmathematicalmachineryready,letusconsiderthenontrivialfixed
points of the DT. We begin with the known results for binary case N =2.
(1). A binary case N =2.
In this case the effective conductivity σ depends on two sets of partial
e
parameters σ ,σ ,x ,i = 1,2. As it was shown in the previous section, one
id it i
can always find the DT T , interchanging partial conductivities of two phases.
12
Then, using (11) and a symmetry of σ one obtains the following DR
e
σ (σ ,x σ ,x )=T (σ (σ ,x σ ,x )), (19)
e 1 1 2 2 12 e 1 2 2 1
| |
At the point of equal phase concentrations one obtains the exact equation for
σ
e(12)
σ (σ ,1/2σ ,1/2)=T (σ (σ ,1/2σ ,1/2)), (20)
e 1 2 12 e 1 2
| |
whichmeans thatthis σ is a fixed point (or aneigenvector)ofT transforma-
e 12
tion. From (15) and (18) one gets the exact value for σ
e
σ σ 2 1/2
σ (σ ,1/2σ ,1/2)= a¯2+¯b=√σ σ 1+ 1t− 2t ,
ed 1d 2d 1d 2d
| − (cid:18)σ1d+σ2d(cid:19) !
p
σ σ +σ σ
1d 2t 2d 1t
σ (σ ,1/2σ ,1/2)= a¯= , (21)
et 1 2
| − σ +σ
1d 2d
which firstly has been obtained in this form in [13]. It is easy to see that the
change H H changes a sign of σ only, in accordance with the Onsager
et
→ −
symmetry (3). The exact formulas (21) are very interesting result. It shows
that the change of the exact result for σ at H = 0 is completely determined
ed
6
bya differenceofthe partialtransverseconductivities σ σ . Whenthey are
1t 2t
−
equal,thediagonalpartoftheeffectiveconductivitycoincides(initsform)with
that without magnetic field and the transverse part of σ remains unchanged.
e
Later it was shown that this property takes place for all concentrations, what
allows to describe the systems with uniform transverse conductivity in terms
of systems without a magnetic field [14]. At the same time, when σ = σ ,
1d 2d
the effective transverse conductivity is simply equal to the average of partial
transverse conductivities σ =(σ +σ )/2.
et 1t 2t
(2). 3 phase self-dual systems.
For 3-phase systems without magnetic field the FPs ( σ , x ) of the
{ ∗} { ∗}
DT exist, when two any phases have equal concentrations, for example, x =
1
x = x,(but otherwise they are arbitrary, except the normalization condition
2
2x+x =1) and their partial conductivities satisfy the condition [9, 10]
3
x =x , 2x+x =1; σ =√σ σ . (22)
1 2 3 3 1 2
Onecantryto findthe FP ofthe sameformfor3-phasesystemswith magnetic
field. Then,the interchangingtransformationT canagainbecompensatedby
12
the permutation symmetry due to the equality x = x , and one obtains that
1 2
the FP of this type can exist if
T σ =σ , T σ =σ , T σ =σ . (23)
12 1 2 12 2 1 12 3 3
The last condition means that σ must be also an eigenvector of T transfor-
3 12
mation. It is possible only if σ coincides with the effective conductivity for
3
two phase systems at equal phase concentrations σ(N=2) from (21) (since this
e
σ(N=2) is also the eigenvector of T )
e 12
σ σ 2 1/2 σ σ +σ σ
1t 2t 1d 2t 2d 1t
σ =√σ σ 1+ − , σ = . (24)
3d 1d 2d 3t
σ +σ σ +σ
(cid:18) 1d 2d(cid:19) ! 1d 2d
Theequalities(24)generalizethecorrespondingequalities(22)forsystemswith
H = 0 and reduce to them when σ = 0, i = 1,2,3. Thus, we obtain that the
it
effective conductivity of 3-phase self-dual systems for the set of partial param-
eters ( σ , x ), satisfying (24) and x = x ,, is also the eigenvector of T
1 2 12
{ ∗} { ∗}
and, consequently, coincides with two phase effective conductivity from (21)
σ(N=3)( σ , x )=σ =σ(N=2). (25)
e { ∗} { ∗} 3 e(12)
The setofpartialparameters,satisfyingthe FPconditions,has the same struc-
ture as the corresponding sets for 3-phase systems without magnetic field and
generalizesit. In this caseσ does notdependoncommonconcentrationoftwo
e
first phases and system behaves as a homogeneous one.
The similar FPs exist, when phase concentrations of other pairs of phases
are equal:
1) x =x , σ =σ(2) , 2) x =x , σ =σ(2) , (26)
1 3 2 e(13) 2 3 1 e(23)
7
No any other nontrivialFPs appear. For example, a new FP can appear, when
all concentrations are equal x = x = x = 1/3. In this case, in principle, it
1 2 3
is possible that the phases are permutated along the shemes (nontrivial cycles)
”1” ”2” ”3” ”1” (or ”1” ”3” ”2” ”1”). It means that a
→ → → → → →
transformation T must exist such that
T(σ )=σ , T(σ )=σ , T(σ )=σ , T3 I. (27)
1 2 2 3 3 1
∼
Let us consider the corresponding transformations of the diagonal components
c¯σ (a¯+b/c) c¯σ (a¯+b/c)
1d 2d
σ = , σ = ,
2d (σ )2+(a¯+σ )2 3d σ2 +(a¯+σ )2
1d 1t 2d 2t
c¯σ (a¯+b/c)
3d
σ = . (28)
1d (σ )2+(a¯+σ )2
3d 3t
It appears that the condition T3 I determines T up to one parameter, say a
∼
a=c, b/d=3a2/d2. (29)
Then, one obtains from(28)fora 3 relations,which contain6 parameters(par-
tial conductivities). Another 3 independent relations can be obtained analo-
gouslyfromexpressionsfor transversecomponents (they aremorecomplicated,
for this reason we do not write them here). As a results one has 6 various ex-
pressions for a. They give 15 nonlinear relations for 6 parameters. In principle,
thisoverdeterminedsystemofrelationscanhavenontrivialsolutions,evenpara-
metric,ifthereissomehiddensymmetry,whichreducesnumberofindependent
relations. Such situation usually takes place for exactly solvable models [17].
If any solutions of this system of relations exist, they will give only diagonal
eigenvectors, since, due to a¯ = c¯, the corresponding eigenvectors of this T are
z = √3a¯.ThismeansthatatthisFPσ =0.Suchunusualresult,though
1,2 et
± | |
it seems an impossible one, needs a separate consideration.
(3). General N-phase case
We show now that for heterophase systems with arbitrary number N of
phases in magnetic field the duality relation(4) admits also the FPs, which are
a generalization of the FP of the N-phase systems with H =0.
Ofcourse,forsystemswithN >3,therearethefixedpointswithastructure
similartotheN =3case,whenforsomepairofphases(i,j)theirconcentrations
areequalx =x andallotherσ (l =i,j)areequalbetweenthemselvesandare
i j l
6
equal to σ of these two phases. But this case is trivial, since it reduces to the
e
3-phase case. Another possibilities appear for N > 3. It will be convenient to
considerthecaseswithevenandoddN separately. Letusconsiderfirstlyacase
whenN isevenN =2M.Thenanewfixedpointispossible,whichcorresponds
to M pairsofphases withequalconcentrationsandto the same transformation
8
T from (16). But now, T must act as the interchanging operator on all M
12 12
pairs. For example, the fixed point with the phase concentrations
x =x =y , y =1/2, (i=1,...,M) (30)
2i−1 2i i i
X
is possible, if
T σ =σ , T σ =σ , (i=1,...,M) (31)
12 2i−1 2i 12 2i 2i−1
For this one needs that all elements a,b,c,d of T must have the same depen-
12
dence on the partial parameters of all pairs of phases
σ σ +σ σ
(2i−1)d 2it 2id (2i−1)t
a= d , c= a,
− σ +σ −
(2i−1)d 2id
σ σ2 +σ σ2 +σ σ (σ +σ )
(2i−1)d 2it 2id (2i−1)t (2i−1)d 2id (2i−1)d 2id
b=d , i=1,2...M.
σ +σ
(2i−1)d 2id
(32)
Herepartialconductivitiesofallphasepairsmustbedifferent. Thentheequali-
ties(32)define surfacesinthe4-dimensionalspacesofpartialparametersofthe
phase pairs. The effective conductivity σ of N-phase system has in this case
e
the exact value
σ ( σ , x )=σ(2) =σ(2), (i=1,...,M), (33)
e { } { } e ei
whereσ(2) istheeffectiveconductivityofthei-thpair,whichallareequal. Note
ei
that again the equal concentrations of the phase pairs y (a= 1,...,M) can be
a
arbitrary,exceptthe normalizingcondition M 2y =1.Theexactvalue(31)
a=1 a
does not depend on y and ensures again the correct values for σ in the limits
a e
P
y 0, 1(a=1,...,M).Ofcourse,the similarfixedpointsexistforotherways
a
→
ofthepartitionofN phasesintothepairswithequalconcentrations. Anumber
ofsuchpointsisequal# =(2M 1)!!.Theexactvaluesofσ inthesepoints
2M e
−
have always the same general form
σ(N)( σ , x )=σ(2) (i<j, i,j =1,...,N),
e { ∗} { ∗} e(ij)
where possible pairs (ij) correspondto the phases with equal concentrations in
the corresponding fixed points.
NowweconsideracasewhenN isoddN =2M+1.Inthiscaseanewfixed
point is possible if the above conditions of the even case (29) are supplemented
by a condition on σ analogous to the condition for N =3 case
2M+1
T σ =σ , (34)
12 2M+1 2M+1
which ensures again the equality of σ with σ of the whole system. Thus,
2M+1 e
σ is again equal to the exact value (21), coinciding with the exact value of
e
system with N = 2M. Of course, the other fixed points related with various
9
ways of choice of the phase pairs with equal concentrations are possible. Their
number is equal to # =(2M +1)!!. A general form of the exact values of
2M+1
σ remains the same as in the even case with an additional equality
e
σ(N)( σ , x )=σ(2) =σ , i=j =k, i<j (i,j,k =1,...,N). (35)
e { ∗} { ∗} e(ij) k 6 6
where the pairs ij correspond to the phases with equal concentrations and k
corresponds to the unpaired phases. Note that the structure of FPs is in an
accordance with the symmetry.
It follows also from analysis of the case N = 3 that the exact values of σ
e
at the point, where all phase concentrations are equal, are not universal, when
partial conductivities have general values. An existence of new FP for N > 3
connected with higher nontrivial cycles is also improbable, since all equations
of the form Tn I, (n>3) require a=c and the corresponding FPs can give
∼
only diagonal σ (but they also need an additional consideration).
e
4. Transformationbetween systems with H=0 and H=0
6
AmorericherstructureoftheDTinsystemswithmagneticfieldallowsalso
to construct various transformations, connecting effective conductivities of the
initial system and other systems, related with the initial one. For example, the
transformations, connecting effective conductivities of the systems in opposite
magnetic fields, were constructed [12, 15]. Moreover, the transformation, con-
necting effective conductivities of two-phase systems with magnetic field and
without it was constructed [16]. Strictly speaking, it connects the conductiv-
ities of systems with nonzero transverse component and systems with a pure
diagonal structure. Such transformation allows to find the effective conductiv-
ity of inhomogeneous systems in a magnetic field, when it is known for these
systems without a magnetic field. But, a derivation of this transformation in
[16] was incomplete (in particular, it was not checked for known exact values
(21)). Under its derivation two simplifying conjectures have been used: (1) the
transformation, symmetrizing partial conductivities σ and σ , will simultane-
1 2
ously symmetrize the effective conductivity too, (2) one parameter, say c, is
effectively factorized and, since the conductivities ”that differ only by a pro-
portionality constant are obviously similar”, can be taken equal to 1. These
conjectures correspond, instead of (11), to the following transformation of σ
e
(see also schematic diagrams of different transformations in fig.1)
z ib′
T (σ ( σ , x ))=σ (T ( σ ), x ), T (z)= − . (11′)
m e i i e m i i m
{ } { } { } { } iz+a
−
Since it differs from (11), we will construct an analogous transformation in
an accordance with the transformation rules (11) (fig.1b), taking into account
all3parameters. Theparametersofsuchtransformation(letuscallitT )must
h
satisfytwofollowingequations,denoting acancellationofimaginarypartsofσ′
i
(i.e. ImT(σ )=0)
i
σ2 +(a¯+σ )(σ b/c)
σ′ =c¯ id it it− =0, (i=1,2). (36)
it (σ )2+(a¯+σ )2
id it
10