REND. SEM. MAT. UNIVERS. PdLITECN. TORINO

Vol. 42°, 1 (1984)

Stanislaw Janeczko (')

UNIVERSAL CRITICAL EXPONENTS AND STABLE SINGULARITIES

Summary. The aim of this paper is to investigate some aspects of critical phenomena from the mathematical point of view. An interesting question in this field is the univers­ality (i.e. the independence of material and to some extend, the type of process) of the so-called critical exponents. We try to show that one can compute critical ex­ponents using structural stability arguments. The universality occurs then as a con­sequence of the similarity of models and structure of stable singularities. From the mathematical point of view we consider singularities which are stable when deformations are constrained. Then, we can obtain some information about the equation of state in the critical region of ferromagnet and compute the generic critical exponents.

1. Introduction.

In the last few years, the theory of universal critical exponents has become a relatively well established subject (cf. [15], [21]). One more ap­proach to this universality was proposed by R. Thorn in [16] and J. Komo-rowski in [8]. They have explained the Hypothesis of Universality by the stability type arguments, which were originated in the theory of smooth stable singularities (cf. [17]).

This paper is a direct continuation and extension of a preceding one ([7]) in which the new space of undercritical states of ferromagnet was introduced to describe the critical region phenomena. The motivation for

Classificazione per soggetto: AMS (MOS) Subject classification (1970), Primary 14B05, 58C25; Secondary 57D45, 57D70, 80A35.

(*) Institute for Mathematical Methods in Physics, The University of Warsaw, ul. Hoza 74, Warszawa, Poland. Permanent Address: Institute of Mathematics, Technical University of Warsaw, PI. Jednosci Robotniczej 1, 00 661 Warszawa, Poland.

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using singularity theory to derive critical exponents of equilibrium phase transitions was explained in [8].

Now we will sketch the main ideas of our approach, mathematical de­tails are to be found further on of the paper. The subject of our modelling is the experimentally known dependence of M (magnetization) on H (mag­netic field) and T (absolute temperature), for a sample of ferromagnetic material (see surface I in Fig. 1).

The mathematical model for a specific medium in a ferromagnetic phase consists of a physical space W (which is assumed to be the two-dimensional Euclidean space with the coordinates {x,y}:— parametric representation — in the neighbourhood of critical point), physical quantities, i.e. functions on W, and a notion of stability of these functions. We assume that the physical

1 1 LJ

quantities -=---=- , ~^r, s, M, E (Tc-critical temperature, ^-internal^energy,

s-density of entropy production) are represented by smooth functions: %(x,y), W(x,y), o(x,y), ~x,y (respectively), defined in an open neigh­bourhood of the origin of W = JR2 (for more rigorous details of the basic, concepts of this model see e.g. [7], [8], [9]). The transition from the formal to the physical category is done by the formulae

1 1 — H-= e(x,y)Z(x,y) , -^r = e(x,y)rj(xiy) , s = e{x,y)o(x,y) , T Tc

v ' -"*v '•" ' T

M = -e(x,y<)x , E = e(x,y)y ,

where

e(x,y):= 'sgnf(#,y) , %(x,y)=tp

<±1 , $(x,y) = 0

is the "function" introduced in [8] (cf. [7]) and called .the Rayleigh function (cf. [14]). This function distinguishes between two processes of maximal hysteresis loop as in Fig. 4.

The aim of the present paper is to classify normal forms (according to [1]) of the physically admissible "representants" t , i? ,a (precisely the normal forms of their germs at critical point) and derive some universal cri­tical exponents.

Natural relations observed between physical quantities, e.g.

a) symmetry of hysteresis (see Fig. 4), b) existence of Maxwell Identity, c) smootheness of the critical isotherm,

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d) divergence of the specific heat CM and susceptibility %T a t t n e critical point (and, of other response functions [15]),

e) positivity of CM (thermodynamic stability conditions)

define space of smooth representants of physical objects (Definition 3.1). Introducing the concept of stability (as in [17] and [8]) we can return to the physical level, where we assume that physical object may have only a smooth stable representant. As a consequence, for such stable quantities we obtain the following infinitesimal relations:

Mr(t)ccA\t\2/3 , XrW^Bltl-2'3 , for H = 0 ,

where A,5-real coefficients, t := -—- — , and Mr is the residual magneti­zation (see [3]).

ferromagnetic phase T<TC

paramagnetic phase T>TC

critical isotherm'

rem agneti2atio n do\v nward

- remagncti/.ation upward

Fig. 1

In other words, within our approach we have the following critical exponents

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p = 2/3 and 7' = 2/3 (2) .

An attempt to explain the merely partially satisfactory relation between our critical exponents and the mostly accepted experimental data (5 = 0,36-0.39, 7' = 1.2-1.36 [15], is given in Remarks 4.5 and 4.6.

Description presented in this paper revives in a sense Landau's method. But, while his idea had an ad hoc character, our model is based on structural stability - a central concept in the theory of singularities of differentiable mappings [17], [5]. It should be stressed that our description does not pretend to be a complete theory of phase transitions. Rather it is an attempt to present a new approach.

2. Preliminary ideas and definitions. /

Let us consider a ferromagnet in isothermal conditions. For each tem­perature T<TC we observe a hysteresis, i.e. two processes of remagneti­zation: the first called remagnetization upward, is realized by increase of H from -00 to +00, and second called remagnetization downward, is realized by decrease of H from +00 to -00. Because of the hysteresis phenomenon (see Fig. 4), one thermodynamic variable, e.g. M, does not suffice to deter­mine uniquely a state of the system. Such uniqueness is achieved, if we con­sider only the states obtainable, in the processes of remagnetization upward and downward. Presentation of our model starts with a new definition of undercritical thermodynamic space of a ferromagnet.

Let W be a real two-dimensional vector space and W0 an open half--plane in W. Then L := WQ H (~W0) is a line in W. We define

e(w) ::

it can be seen as a function on W, which is double-valued on L. The undercritical thermodynamic space of a ferromagnetic natural

stample (cf. [7]) is the pair (W,e). Physical interpretation of this space is given by two functions, magnetization M and internal energy E, defined as follows

1 , wew0 - 1 , we~w0

± 1 , w E L ,

(2) Derivation of the other critical exponents we leave to another paper.

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(1) M(x,y) = -e(x,y)x

E(x,y) = e(x,y)y ,

where we take a smooth (curve-linear) coordinate system {x,y} on <W, vanishing at zero and such that the line L is symmetric with respect to zero, i.e.

(2) (x,y)EL *=* (-x,~y)eL.

Obviously, M and E are double-valued on L; the coordinates x and y can be seen as their smooth representants (cf. [8]). The curve L in our model, represents the critical isotherm, i.e. L = {(x,y)EW\ T(x,y) =

= (-— ~-^r ){x,y) = 0}, where T is the absolute temperature (see Introduc­

tion) and r is defined by means of a smooth representant £ : W —• IR,

(3) r(x,y) = e(x,y)%(x,y) ; r > 0 , r(x,y) = 0 => (x,y) eL .

Whathever the future definition of the temperature T will be, under-critical isotherms (level sets r(», •) = const. > 0) have to be situated somehow along L, e.g. as the dotted and dashed curves in Fig. 2. If we run over them

RliMAGNIiTIZATION DOWNWARD

€ = - 1

e = + l REMAGNKT1ZAT10N UF WARD

Fig. 2

from left to right then the magnetization M is increasing on the dotted

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isotherm and it is decreasing on the dashed one. This is why the domain where e = —1 (resp. e = +1) is called a region of remagnetization downward (resp. remagnetization upward). On the critical isotherm L, if we choose for the double-valued e the value - 1 (resp. +1) then the magnetization M decreases (resp. increases) from left to right. These two physical interpreta­tions of L correspond to two different experiments: the critical isothermal remagnetization downward and upward. It is easy to see that the symmetry (2) of L is a consequence of the introduced definitions (1) of M and E (cf. [8] §2, and [7]). By (1), magnetization and energy are not uniquely defined on L because e is double-valued on L. By the same reasons (cf / [3]) the external magnetic field H can be defined on L merely in a unique up to e way, i.e.

H > (4) ^ 9 ( x j ) - ^ y M : = e k y ) t i f e y ) ,

where 17 is a smooth function on W. In [7] the functions t and H are defined in the another way, namely:

(T- Tc)(x,y) = e(x,y) %(x,y)y H(x,y) = e(x,y)r\(x,y). In fact, the aim of our paper is to describe the critical region phenomena, near the Curie tem­perature Tc > 0. From the mathematical point of view, in the neighbourhood of Tct we can take for modelling bothe the quantity T and l/7\ Taking into account physical interpretation, the two approaches (i.e. contained in [7] and presented now) are admissible. However an advantage of our approach consist in convenient formulation for more detailed calculations (e.g. more consistent treatment of the Maxwell Identity).

The hysteresis phenomenon is related to "internal friction" between growing rotating and moving magnetic domains (cf. [3] and [10]). When a specimen is made to go through one complete cycle, the total work done on the specimen (by surroundings) is the hysteresis loss. This positive work appears as heat in the specimen. The heat produced in such cyclic isothermal process disperses into surroundings. Thus the hysteresis loss may be used as a measure of irreversibility of the processes of remagnetization downward and remagnetization upward. Let us consider one of the processes, Fig. 3,a. Ac­cording to experimental data (see [3] or [11]), various stages of the magneti­zation curve gives different contributions to the totale produced entropy. Magnetization can change as a result both of domain wall motion and domain rotation. At the stages I and III of the process rotation predominates, appro­ximately this is reversible part of the remagnetization curve (cf. Fig. 3,b). Irreversible wall motion is a main feature of the stage II (cf. [3]). Contribution

19

of the respective stages of the process, to the total produced entropy can be described by a function s-density of the entropy production. Qualitatively, a graph of this function is shown in Fig. 3,b. Total entropy, produced in the process of remagnetization downward (Fig. 3,a) is given by the integral

-Af f °sdM, where M0 is a saturation magnetization. M

a)

b)

Fig. 3

In thermodynamics the conservation of energy is called the first law. We will use it for the stage 7^ of the process (see Fig. 3), in the form

(5) dE

B~T~ L H , 1

7A

where Q B is a heat transferred to the system (from reservoir). Because the

process of remagnetization is not reversible, then by the second principle of

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thermodynamics

(6) SS^ = AS^B + A S ^ > 0 , A S ^ : = ^ f

B where AS B is an increase of entropy of surroundings in the y process.

Thus (5) one can write down in the following form

f dE f H r (7) / — - / — dM + 8SyB= dS,

k T kT 7A k where 5 is an equilibrium entropy function and 55 B is an entropy pro-

R A

duced in the TA process 6S B = / B sdM. ^A yA

Let a: W —^ IR be a smooth function on W. The same arguments as before (see [8]) force us to look for the function of entropy production s of the form (8) W 3 (x,y) —^ s(x,y) = e{x,y) ofay) .

On the basis of experimental observations the function 5 has a following properties _

(9) s>0, s(x,y) = 0 <=* fay) EL .

Let us notice that (7) connects temperature, magnetic field and function s by the Maxwell Identity. In other words the smooth representants of these functions £, T? , a are connected in the fallowing partial differential equation

dor _ ^ = JL ( T ,_ a ) .

We require from our model the following symmetry property: the state characterized by a triple (M, H, T) is attainable in an isothermal remagnetiza-tion upward process iff the state characterized by the triple (~M,-H, T) is attainable in the analogous downward process, see Fig. 4. Finally, the fun­damental symmetry of magnetic phenomena can be expressed in terms of our model as follows: For every pair Wi,w2€W such that

(i) riWi) = r(w2) (ii) v)uw2 are attainable in the opposite remagnetization processes,

e(wl) = -e(w2),

(iii) M(wl) = -M(w2)

then

(iv) Hiw^ = -H(w2) and s ^ ) ' = s(w2).

Fig. 4

The above symmetry condition is equivalent to the following impli­cation

/ i n w u , (v(x,y)~ri(x,y') = 0 (11) , Y^mJ(x>y) + ttx,y') = 0=*

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3. Smooth representants of physical quantities.

We want to describe the critical region phenomena near the Curie point (see [15]). Hence we are interested in the local properties of the physical quantities (i.e. functions M,E,T,H,s on W). Therefore, instead of the smooth functions £, 7?, a on W we will consider rather their germs (see [4]) at OGW (critical point). We use the following standard notation: &w-the ring of germs at zero, of C°°-functions on IRW, ^ w - the maximal ideal in &M, consisting of ail germs vanishing at 0 , ^* - the &-th power of Jin. If it is possible, to avoid inessential mathematical purity we speak mostly about functions instead of their germs.

We know from experiments that in the critical isothermal process H is monotone function of magnetization M and its derivative, i.e. the inverse of

susceptibility Xr = (~aw~) » vanishes at the critical point (see e.g. [15]). The

"simplest" example of such a function is cubic parabola. Here the term "the simplest" can be understood rigorously as follows: by the previous arguments, the germ at 0 of the function M—+H(M,TC) must lay in Jt\, where the cubic germs form a unique orbit of codimension zero (cf. [23]). This encour­ages us to assume in our model that along the critical isothermal process, the magnetic field is cubic (infinitesemally at the critical point) with respect to magnetization. Then, we require a stability of this feature (as an observed), namely the infinitesimal cubicity of H restricted to all curves obtained from L by small perturbations preserving the critical point. By virtue of this as­sumption, for a smooth function 17 we obtain

(12) r\^M\

as well as, by Maxwell Identity (10)

(13) %ZJtV , a EJt* , (see [8] and [7] as well).

Let f€LM\, we say that a germ / is finitely determined (equivalent to the polynomial germ, see [23] p. 273) if and only if dim^ J£2/A(f) <°°;

where A(/) := (-r— , -r—) is an ideal of &2 generated by the partial deriva­

tives of / . Following the spirit of the approach presented in [16] (see [17] and

[9]), we shall investigate physical consequences of our model taking into account only finite determined smooth representants £, 77, a. Some justifica­tion of this assumption is that the set of not finite-determined function germs

83

has an infinite codimension in &2 (cf. [20] and [4]). Summing up the above remarks we define the physically acceptable (for

modelling the concrete ferromagnets in critical region) triplets of the smooth representarits (£,f?,cr).

DEFINITION 3.1. Let (£, r?, a) E ®M\; we say that (£, rj, a) is an admissible

triplet if it satisfies the following conditions

1°. J - 1 (0) is a germ at zero of smooth curve (critical isotherm),

(x,y) by > 0 for (x,y) ¥= 0 2°. In some open neighbourhood of 0 E IR2,

(positivity of the specific heat CM > 0),

3°. £(0,y) = Ay3 4- 0(y4), where A =£ 0 (cf. [8]),

4°. The symmetry condition (11) is satisfied, i.e. there is a neighbourhood V of 0E1R3, that the following holds

(7i(x,y)-ri(x,y') = 0

(x,y,y')ev [ a(x,y) + a(x,y') = 0

5°. £ is a finitely determined germ.

The set of such admissible triplets we denote by ff. We are interested in the structure of the set Sf. Several next results are

based on the theory of singularities (3). Let lR[[#,y,y]] be the ring of formal power series, an element # E

G R p x , ) / , / ] ] is called reducible if there exists noninvertible elements quq2EJR[[x,y,y,]~] such that q=q1q2- L e t / E & 2 , b y (T0f)(x,y)_ we denote the formal Taylor series of / (at 0 E 1R2), (T0f)(x,y) E IR[[#,y]J.

PROPOSITION 3.2. Let (£, 77, o) E y and the formal power series (T0 %)(x,y) + + (T0 li)(x,y'}E:'JR[[x,y,y'\^ is not reducible, then there are f,gE<M2 such that

(14) r)(x,y)=f(x, t2(x,y)), o(x,y) = %(x,y)g(x, %2(x,y)).

Proof. By 2° of Definition 3.1 x and £ are smooth coordinates on fl\{0}, where 0 is an open neighbourhood of 0 E IR2. Hence, on this annular neigh­bourhood of zero, there are: a smooth f0 such that rj(xyy) = / 0 (# , £(*>.)/))

(3) We are not able to present here this theory; as main references we recommend [17], [l] ,[2],[23],[4f,[5],[18].

84

and a smooth go such that a(x,y) = g0(x, %(x,y)). Decomposing f0 and g0 into their even and odd parts

/o (* ,0=/ (* ,* 2 ) + */i(*,*2)

go(x,t) = tg(xit2)+gl(x)t

2)

and taking into account 4° of Definition 3.1 we get f\(x, %2(x,y)) = 0 and g\ (x, Z2(x,y)) = 0. Thus outside zero we have / such that

(15) n(x,y)=f(x,£2(x,y)) and g such that

(16) o(x,y) = ¥(x,y)g(x, ¥(x,y)).

Now the following lemma can be applied.

LEMMA 1. If there exists the formal power series Sx, S2 E IR[[#,j/]] such that

(T0v)(x,y) = Sl(x, (T0lt)2(x,y))

(T0 o)(x,y) = (TQ t)(x,y)S2(x, (T0 $f(x,y))

then / and g can be smoothly extended to zero. The detailed proof of this lemma is contained in [7] (see p. 25). So we have only to show the existence of such Sx and 5 2 . Let

F(x,y,y') •= Z(x,y) + £(*,/) (18) G(x,y,y')

P(x,y,y') = v(x,y)-v(x,y') = o(x,y) + o(x,y').'

The symmetry condition for £,7?,a, i.e. 4° of Definition 3.1, means that for any (x,y,y') E IR3 close to 0 E IR3

(19) F(x,y,y') = 0 => G(x,y,y') = 0 and P(x,y,y') = 0 .

Let us check the following analogous implication for their Taylor series.-for each formal curve z(X) E IR[[X]]3 close to 0E1R[[X]]3

*(20) (T0 F)(z(\)) = 0 ~ (T0 G)(z(X)) = 0 and (T0 />)(*(*)) = 0 . .

By BorePs theorem (see [4] Theorem 4.9) there exists a smooth curve X—• ->20(X)EIR3 such that (T0z0)(k) = z(k) and F(z0(-)) isflat (i.e. F(z0(-))£ E ^ ~ ) . Now, let us recall the Tougeron's generalized version of Implicit Function Theorem.

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By tkn+p we denote the ring of germs at 0 of C°°~functions on IRW x IR*\ 3 /

If fE&n+p then also If denotes the ideal of &n generated by -z—(*,0),

1 < i < p, where / : 1RW x JRP 3 (x,y) —+f{xty) G 1R.

THEOREM (Tougeron [19]). Let / G &»+/?, if J is a proper ideal of &w such that / ( • ,0) SJll then there exists germs »Pi,..., <Pp GJ/ / such that

/ o (id , <£i,... ,<A*) = 0 .

Using this theorem we get the existence of a smooth curve X —• z(X) G G 1R3 that 2 — z0 is flat and F(z(X)) = 0. But we have to check the hypo­thesis of the Theorem. To this end, let us notice that by the finite determinacy of £, an appropriate change of coordinates in IR3 allows us to assume analyticity of F. Then the components of its gradient satisfy the Lojasiewicz inequality

KUf^fjia ( # 1 , %2 t ^3/ ^ C\ 2J X} 1 ,\a

for some positive C and a; see [18], Definitions 4.1, 4.2, p. 102 and Corol­lary 1.6, p. 119. Thus

"3FV YbF\2 /bF\2~ at

for Cj, Ofj > 0; we consider only z(\)¥=0. Hence the ideal If, where

f(\ix,yJ):=F(z0(\) + (x,y,y')), generated by "3^" ( M 0 ) and j - (z0(')) =

= -jv-T (z0 (•)) is not flat and, in particular, M \ =jCxIf. Therefore F(z0 (•)) G

^Jt°[li and Tougeron's theorem ensures the existence of smooth z for which F(z(*)) = 0 and z - z 0 G.JCxIf -<M~x. Now by (19) we obtain the right-hand side of (20).

It is easily seen that in the light of our assumptions about £, (V70 ^)(x0(X), yo(fc))=£0 for almost all formal curves (x0(\)ty0(\))

G 1R[M] 2 . Now if we denote F := TQF, G :=T0G, P := T0P, the following lemma can be applied.

LEMMA 2. Let F,G,Pe !R\[xlt..., *„]], F be irreducible. If 3c(X) G JR[[X]]W ,

3c(0) = 0, is such that

86

F(x(\)) = 0 , (VF)(*(X))=£0

and for every j/(X) from a neighbourhood of x(k) in IR[[X]]W (endowed with the Krull topology [18])

*Xy(X)) = 0 -* G(j/(X)) = 0 and P(j?(X)) = 0

then

F divides G and F divides P.

Proof of this lemma is contained in [7] (page 28). Since G = F-Ql and P = F- Q2; d , Q2 € HUt* ,^ / ] ] , then we have

(21) (T0F)(2(X)) = 0 =* (T0G)(z(X)) = 0 and (T0P)(z(\)) = 0

for all complex formal curves z(X)GC[[X^]3 (now we look at the series FfG,P as elements of C[[^,y,y']]).

So we have pulled our problem to the complex domain and we are going to investigate T0 r? and T0 a on the set of complex formal curves being zeros of JT0£. The Puiseux decomposition (cf. [21] p. 97) for r 0 ? G EC[[#,y|] has the form

(To l)(x,y) = ofi II (y - <Pi(xl,Pi))a(x,y),

where ^ = 0 ,1 , . . . , <A(Z) €<C[[z]], # = 1,2,... , and a(x,y)e<L[[x,y]] is invertible, Vi(xvpi) are fractional formal power series (cf. [21]).

Let us show the following facts

a) (TQri){x,^i(xvpi)) does not depend on i, and (T0r))(x,\pi(xy?i))G ejR[[x]]

b) (T0 o)(x, &(x1/pi)) = 0 for all i = 1,... ,>.

Let us denote p=pr ...'pk, Pil=P/Pily Pi2= P/Pi2 G N; * i ,z 2 e

• € { 1 , . . . , * } .

We take two complex formal curves yt (x) := ,- (^Al) £ £[[#]], .y2(tf) :=

= #2(*P|'2)G £[[*]] such that

(22) < r 0 * ) ( * ^ ( * ) ) = o , y = i , 2 .

Hence, T0 F vanishes on the formal curve (xp, yx (#)-, y2 (x)). By (21), we have

(To t?)(*', yi(*)) = (T0 T?)(?P , y*(*)) ,

i.e.

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(7-0*?)(*, M*"* 1 ) ) = (T07i)(x,yi(xyP)) = (T0r))(x,y2(x

l/p)) = = (T0r})(x,w2(x

l/*2)).

So the complex fractional power series

where (x,yi(x1/p))G(Cl[x1/p]]2 runs over all zeros of r0f, does not depend o n i, % = l9...,k.

Let e be any of the complex p-th roots of 1, i.e. ep = 1. If we substi­tute ex1/p into (22) in the place of x we see that (T0^)(xfyi(ex1/p)) = Q, i.e. the curve (x,yl(exl/p))E(C[[x1/P]]2 coincide with one of the curves (xfyi{xyP)\ i = 1,2,...,*. Thus Hx1/p) = (T0 !?)(*, yife*1*)) = = (T0ri)((ex1/p)p, yt(ex1/p)) = \l;(ex1/p).

Hence all terms of the series \jj{xl/p) containing fractional (non-natural) powers of x must vanish. In other words, (TQri)(x,\pi{xvpi))=i(T0n)(x, yi(x1/p))E<£,[[x]]. Since the formal series T0 £ has real coefficients, there

exists such i=l,•. . . ,& that the complex conjugate yy{xl/p)= yi(xyp). This and the fact that T0T? is a real formal series tells us that (T017) (x, <Pi(x1/pi)) G 1R[[#]]. This completes the proof of fact a).

The proof of fact b) is similar to the preceding one. Let us consider the curve (xp,yi(x)i yi(x)), yi(x) = &(xpi), z=l , . . . ,&. By (21) for every i = l , . . . , . * , (T0P)(xp

>yi(x)J yi(x)) = 0i thus (T0o)(xp, # (* ) ) = = -(T0o)(xp, yi{x)). Hence (70 a)(x, u(x1/pi)) = (T0 oHx,ydx1/p)) = 0, where (x,yi(x1/p)) E C [ [ x ^ ] ] 2 runs over all zeros of T0 £.

Let us denote ®(x,y) := (T0r})(x,y)- \p(x) G IRHX3/]]. Obviously 0 and T0 a vanish on all complex formal curves being zeros of (T0%)(x;y). It is easily seen that in the light of our assumptions about £ (see Definition 3.1), (V70i)(x0(\)}y0(K)) =£ 0 for almost all formal curves (x0(\), y0(\)) G EIR[[A]]2, and finite determinacy of £ implies that T0% has no multiple factors. Now, using the Lemma 2, for each irreducible factor of T0 £ we have the following fact: there exists T^ , ol G lR[[x,y]] such that

{TQ v)(x,y) = \p(x) + rji (x,y)(T0 i){xty)

(T0o)(x,y) = d1(x,y)(T0£)(x,y).

Let us define

Pi fay,?') := n\ (x,y) + Vi(x,y') G JR{[x,y,y']]

Gi (x,y,y') '•= 1 (x,y) ~ ox (x,y') G IRf[^y,y']] .

88

It follows from (21) and (23) that

(24) (T0F)(z(X)) = 0 =* Gx(z(X)) = 0 and Pj(z(X)) = 0

for all complex formal curves z(X) E C[[X]]3. It is easily seen that the part of our proof contained between (20) and

(23) can be repeated with T0 r\, T0 a, T0 G, T0P replaced by rJitatlGi, Pt

(respectively). Then (24) is an analog of (20) and as an analog of (23) we get

*h (x,y) = *i(pe) 4- r\2{x,y){TQ £)(x,y)

Oi(x,y) = Xi(x) + o2(x,y)(T0%)(x,y).

The infinite sequence of such steps gives us

(T0v)(x,y) = *(*) + S ^2k(x)(iXo it)(x,y))2k

k = l /

(r0 o)(X,yy= i x2*+i (*)((r0 Mx,y))2k+1 . k = 0

This completes the proof of existence of series SlfS2 as in the Lemma 1. Finally we put

oo

Si(x,y) = \jf(x) +-2 \ls2k(x)yk

oo

S2(x,y) = Xi(x)+ 2 X2k+iWyk

which completes the proof of our Proposition 3.2. Let & be the sub-group of the group of zero-preserving diffeomorphisms

(germs) of IR2; if ge&, g: (IR2,0)—• (IR2,0), then g(x,y) = (<p(x), 4>(x,y)), where ipE<J^ly \p Et$2- Let us consider the right action of # in ®JCl, defined as follows 3 2

^g(i,V^):=(^°gf V°g, oo-g) . •

PROPOSITION 3.3. tf is invariant with respect to the action <i>, i.e. for every gE<^, 4»g(y) C / . Thus the restriction of <i> to & defines an action on<^.

Proof. Let g G ^ , g(x,y) = (y(x), \p(x,y)) and let ( M ^ ) ^ ^ For (Z°g> V°g, o°g) we shall prove that conditions of Definition 3.1 are fulfilled. This is evident for points l°-3°, 5°. Now we check this for point 4°.

Let H: (IR3,0) —• (IR3,0) be a ferm of a diffeomorphism H(x,y,y') := = -(<P.(*), ^(xty)9'^(.xty'))t and C / ^ H ' ^ V O with V as in point 4° of

89

Definition 3.1. If (xfy,y() G U and (%°g)(x,y) + (t*g)(x,y') = 0 then (x±>yi>yi)'-= (fix), Mx,y), $(x,y'))GV and we have £(*r,^i) + + £(xi >yi) = 0- As a result of this and the assumption of Definition 3.1, we obtain following equations

v(xi,y1)-v(xl,y'i) = 0 •ff(*i ,<yi) + < 7 ( * i , / i ) = 0 ,

we can rearrange them to the form:

(T? og) (x,y) - (T? og) (#,</) = 0

(oog)(x,y) + (a°g-)(*,/) = 0 for (xtyty') G £/.

This completes the proof of the Proposition 3.3. We say that (£, r?, a), (^ , rjj , ol) G ^ are equivalent if there exists g

such that (?i,T/,,a1) = 4>g(^,T?,a).

PROPOSITION 3.4 (on the normal form). Orbits of action of <£ on the setcS^ are parametrised by triplets of the form:

(25) (y3 +a(x)y + b(x),ri(xfy), o{x,y)) where a(')SJt2lt b(')eJt] .

Proof. We recall some necessary facts from the theory of universal unfolding of singularities (see [1], [4], [23]).

We denote by &(n,p) the set of germs at 0 G IRW of smooth mappings from IRW to R*. If p = 1 we shall write &n for &(»,1), and Jin for the set of germs / : 1RW —• IR at 0 such that f(0) = 0. We define the category of unfoldings of /, for fixed f^Jin.

An unfolding of / is a pair (F,/) where FEZkn+r and Fi = / (by • IR

&n+r we denote the ring of germs at zero of C°°-functions on Rw x IRr). Let Fj E l „ + n F2 G &n+s be unfoldings of fS^n. A morphism from

Fi to F2 is a pair ($,<p), where 4> G &(n + r, n + s) and <£>G&(r,s), satisfying the following conditions a ) 4> | = id I ,

IIRW I I R » '

b) # = (\p ,ip)> where ^ G &(ra + r, w), c) for all (^,W)G1RW x IRr

i Fj(^,w) = F2(\l/(x,u),<p(u)) .

Let Fj %and F be unfoldings of /. We say Fi is induced from F if

90

there is a morphism from Ft to F. Let F be an unfolding of /, F is said to be universal unfolding of /'

(cf. [4]) if every unfolding of./ can be induced from F. ' In [ 1 ] Arnold has classified the so-called simple singularities ([ 1 ] theorem

7.1) i.e. germs of smooth functions f^M^, with their G-orbits (G-group of germs of diffeomorphisms preserving 0 G IRW acting in Mn by means of superposition J%n x G B (f,g)—yf°g^^n) being simple (cf. [13]). The following theorem describes universal unfoldings of these singularities.

THEOREM l (cf. Corollary 8.4 in [1]). Universal unfoldings of the simple singularities of types A^ (±xk+l, k > 1), D^ {x\x2 ±x*~l, & > 3), E^ (x\ ±x2), E7 (x\ + xxx\), Es (x\ + x5

2) can be of the form:

A%: F(x, ult..., Uk) = ± xk+1 + ukxk~l 4- nk-lx

k~2 4- ... 4- u2x 4- ux , 4~ O b 1 b —0

Dk: F(xlt x2, ul7..., Uk) = xtx2 ± x2 4- ukxx + u^-ix\ + ... + u2x2+ul ,

£JQ '. F \Xi, X2 f Uj , . . . , U§) = = Xi i X2 i UftXi X2 "r U§Xi X2 +" U$X2 T" H$X2 T

T" U2X} T* Wj ,

t X2 f U^ f ... f Uy ) — X i ~r~ X }X2 ~T UyX }X2 ~r UfiX2 v U$X2 T" %l^X2 T"

4- w3#2 4* w2^i 4- Ui ,

£ 8 : F(* l , X2, Ui, ... ,14%) — Xi 4" # 2 4" W g t f ! ^ 4 \ W 7 # i # 2 "^ ^ 6 ^ 1 ^-2 "^

U^X2 ~r ^ 3 ^ 2 "+ M2X\ T MJ .

As a consequence of Definition 3.1, 3° and a structure of ^ , J can be treated as an one-parameter unfolding of the germ £(0,y) = Ay3 4- 0(y4) =

doc = a3(y), where a E < 4 , , - 7 - (0) #'.0 ( ' ^ " is the parameter of this unfold­ing). According to Arnold's classification the germ {'(0,") is of type A2 . The above quoted theorem states that the universal unfolding of the germ of this type has the form:

y3 4 u2y 4- ux .

Thus we have a morphism (<!>,</?), $ 6 ^ ( 1 + 1,1 + 2), y?G&(l,2), 4>(x,y) = = (\l/(x,y); &(#), #(#)), <P(#) = (£(#), *(#))» such that

(26) £(*,y) = ^3(*,.y) 4- a(x)\p(x,y) 4- £(*), i

where i//(0,y) = a(y). Taking (x, \p(x}y)) as a new coordinates and expressing (J, 17, a) in

them we arrive at (25), which completes the proof.

91

Smooth representants of physical quantities are connected by the Maxwell Identity. At the next step of our approach we shall study properties of (£, 17, a) provided that the equation (10) is fulfilled. Let us denote

(27) Mi= {tt,T?,a)Gee4rj; (10) is fulfilled} .

PROPOSITION 3.5. For every (£,r?,a)e yc\ M there exist ii,fx,f2G.Jl2

such that t(x,y) = *P3(x,y) + a(x) \jj(x,y) r)(x,y)=f1(x,\lj2(x,y)) o(x,y) = \p(x,y)f2(x, \l^2(x,y)),

where — - (0) =£ 0, a(')eJV2. oy

First we prove the following lemma.

LEMMA 3. For every (£, rj, a) E yc\ Ji the formal power series (TQ %)(x,y) + + (T0%)(x,y*) is reducible in JR[[x,y,y']].

Proof. Suppose (T0 £)(x,y) + (T0 %)(x,y') is not reducible in JR[[x,y,y']]. Proposition 3.2 implies that there are f,g€Jt2

such that rj(x,y) =f(x, £2(x,y}), o(x,y)=i(x,y)g(x, $2(x,y)).

We observe that (17 - a)(x,y) = h(x, %(x,y))y where h E.M2. Since (£, 77,(7)6:^, then we have

3£ db 3£

By this equation the Jacobi ideal A({) has a following form

• ~ \ dx ' 3y / \ 32 ' 1 / 3y ~ \ dy'

It is easily to seen that the real vector space tJ£2/A(J;) = JK2l\-r-/ n a s

not finite dimension (it suffice to show that xk & A(£) for k = 1,2,... , then the classes [xk] ^J62l&(£) are linearly independent elements of the vector space <^2/A(i;)) hut this contradicts the assumption 5° of Definition 3.1.

REMARK 3.6. The set y C\ M is not empty. It is easy to check that {&{x,y), V(x,y), a(ix,y)) = ((y-x)(x2 + (y~x)2), x(x2 + (y~x)2), (y~x)(x2 +

92

+ 2.(x^y)2)).e&f\Jl, namely; the points 1°, 3° of Definition-3.1 are obvious. Condition (y — x)(x2 + (y - x)2) + (y' - x)(x2 +(y' - x)2) = - (y + yf ~ 2x)(x2 + (y-x)2 ~(y-x)(y' -x) + (y''.- x)2) = 0 implies the subsequent condition

y+y'-2x = 0,

by this, we have also

x(x2 + (y~x)2)-x(x2 + (y'-x)2) = 0 and (y-aOOc 2+ 2(x~y)2) +

+ (y' - x){x2 + 2(* - / ) 2 ) = 0 ,

thus we have verified 4° of Definition 3.1. The point 2° results from the following evaluation -r— {(y~x)(x2 +

+ (y - x)2) = x2 + 3(y- x)2 > 0 for (x,y) =£0. By assumption £ is equiv­alent to the germ yx2 + j / 3 , which is a finite determined singularity oftype D\ (according to [1], see also [23]), so £ is finite determined too. Moreover by simple calculations we can check the equation (10).

If (£,T?,CF)E Sf then on the basis of Proposition 3.4 (see (26)), \ can be write down in the following form

%{x,y) = \p3(x,y) + a{x) \p(x,y) + b(x), »\ i «——-

where \peJZ2i -^ - (0)=£0, a(-)eJZ2 b(-)<Ejt\. oy

s

LEMMA 4. If (TQi)(x,y) + (TQi)(x,y') is reducible in JR[[x,y,y']] then

(T0b)(x) = 0.

Proof. Now, without restriction of generality, we can take f. in the normal form (25), then the formal power series P := (T0l;)(x,y) + (7Y£)(#,y) = = y3 +y(T0a)(x) +y'2> +y'(TQa)(x) + 2(T0b)(x) is a distinguished poly­nomial (see [4]), P E IR[[^,y']][y]. By assumption P is reducible in IR[[#,j/,y]] hence P is not prime in IR.[[tf,j/,y]]. On the basis of Lemma 3.10 (version for formal power series which states that: if /*£ IR[[#,y]][y] is a distinguished, irreducible polynomial then P is prime in IR[[#,.y,y]]) and Remark 3.11 in [12] P decomposes onto distinguished polynomials

P' = (y + Xj (x,y'))(y2 + y\2(*,/) + X3(x,y')),

where X t-eiR[[^,y]], X,(0) = 0, i = 1,2,3. On the other hand, taking into account the definition of P we have

93

y3 +y(T0a)(x)+y'* +y'(T0a)(x) + 2(T0 b)(x) =

=y +y(x2(*y)>+x1(*y))4-^ + X1(* f /)X3 ( * , / ) ,

i.e. the following equation is fulfilled

y'3 +y'(T0a)(x) + 2(T0b)(x) = XiOcJWoaKx) + Xf ( * , / ) .

Let us rewrite this equation for the new series p (x,y') := y - Xj ( # y )

<•> p(*y)(3ya - 3p(*,/>/+p2(*,y) + (r0«x*))=- 2<T0*x*), where p(0,y') = 0.

Let us suppose that there exists ^ G N , & > 3 such that (T0b)(x) = oo

= 2 a(Xl and a^i^O. Thenby(*) i=k

p{x,y') = xkpl(x,y') , P i G R [ [ * y ] ] .

Dividing the both sides of equation (*) by xk we obtain

pi(^y)((^o^)W + P2(^y)-3p(xy)y+ 3y2) = -2 Jo**+,-*':,

but this is impossible unless tf* = 0 because of (T0a)(x) 4- p2(x,y')-oo

— 3p'(x,y'}y'+ 3y'2 and 2 a^+iX1 is noninvertible element of IR[[#y]J

and invertible respectively. This completes the proof.

Proof of the Proposition 3.5. From the lemmas 3,4 and Proposition 3.4

it follows that there exists i// ^Jt2, ( ~ (0) ¥= o V a eJ£2 and b ^M~ such that *

i{x,y)= ^{x,y)-\-a{x)^(x,y)-\-b{x).

Now we shall prove that there exists \jj E<M2, such that — — (0) =£ 0 and dy

(29) S(x,y) = ^3(x,y) + a(x) \jj(x,y).

Let us consider the following equation

F(x,y-, z) := y3 + a(x)y - (y,+ z)3 - #(#)()/ + z) + /?(#) = 0 .

bF Note that for every representative of the germ — - (x,y; 0) = - 3y2 - a(x)

oz

94

there exists Kr := {(x,y) G JR2; x2 + y2 < r 2 }, ^ G N (such that k k S (

i - i * 2 ( .) r2k 2 > 2) and a positive constant C G IR such that

(30) bF ^-(* ,y;0)

2 ^ * >C(x2+y2)

for every (x,y)EKr. From Proposition 3.4 and Definition 3.1 (points 2°, 5°) it follows that

a(x) = a# 2* 4- 0(#2*+1)> for some a: > 0 and k > 0. One can choose a new 3F

coordinate {x} such that -JT- (x,y; 0) =?= - 3y2 - # (see page 17). For

this case the constant C one can choose in the form C = 3/r2k~2 2 ( .) + 1. 3F 2 •,'"1 *

By (30) the ideal I2F := <-r— (x,y; 0)y is a Lojasiewicz ideal of &2

(see [18], def. V.4.1, p. 102). Thus on the basis of Proposition 4.3 (see [18], p. 102) we have that

(31) F ( - , - ; 0 ) = £(•) G ^ ~ / 2 (since fc(-)e-^7).

We see, by (31), that the Tougeron's Theorem can be applied (cf. page 12), and using this we have that there exists ip G ^ ~ such that the following equation is fulfilled ._,

F(m,-, *<v)) = °-

Taking the new function 4/(x,y) := \jj(x,y) + y(x, \l/(x,y)) we obtain £ in the desired form (29).

_Let us_consider the following ^decomposition i(x,y)+\(xty') = = (Hx,y) + #(#, / ) ) (# 2 (x ,y) - Hx,y) l;(x,y') 4- i//2(#,/)j4- <*(*)). Locally, in_an appropriate coordinates {3c,y,z}, the function \p2(x,y)-- \p(x,y) \p(x,y') + \p2(x,y') + a(x) has the following form

y2+z2+x2k\

Hence in a sufficiently small neighbourhood U of 0 E 1R3 we have that {(*,y, / ) GX/; £(x,y) + $ (* , / ) = 0} = {(*,.)/,/) G £/; jfcfoy) + * ( * , / ) = 0}.

Let x : (x,y)—* (x, ty(x,y)) be a new local coordinates. Now we define ^{x,y) := Tjox_1(x,y), 0 ( # , y ) : = ^°X_1(^,y)- L e t UCU be a symmetrical neighbourhood of 0 G 1R2 (i.e. (x,y) G [/ =» (x, -y) G £7) then the condition 4° of Definition 3.1 for the functions rj,d we can write as follows

for every (x,y) G £7 77( ,3/) = rj(x, -y) and a(tf,y) = - a(x, -y).

95

In fact this property implies (see § 6 and § 7 of [4]) that there exists fXi f2 E such that

n(x,y) = (r? °x)(*,y) = /i (*, 2(x,y))

o(x,y) = (oox)(x,y) = ji(x,y)f2(x, 4>2(x,y)).

This completes the proof of the Proposition 3.5.

COROLLARY 3.6. Let ty,atK are such that i / / E ^ 2 , --—(0) =£ 0, a^M\,

K EL<M\ and for £(x,y) := \p3(x,y) + a(x) \p(x,y) the points 1°, 2° of Defi­nition 3.1 are satisfied, then (£, 17, a) E y n «^ may be uniquely defined by i//,#, K, i.e. one can mapp the above introduced triplets (i//,tf, K) onto the set y n ^ surjectively.

Proof. Let us rearrange the equation (10) to the form

(32) -^Z(x,y) = —f(x,Mx,y)) , feJl2t

Functions flt f2 as in the Proposition 3.5 are defined by / as follows

1 1 fi(x,z):=-^(f(x,z)+f(x,-z)) , f2(x,z):=-j(f(x,-z)-f(x,z)).

By (32) after a simple calculations we obtain

where ^(#,y) := (#, 4>(x,y)) is a local diffeomorphism. Hence, for a representative of the germ / in an appropriately chosen

neighbourhood of 0 E IR2 we have

f(x,z)=l G^a(x,s)ds + K(x). 0

This completes the proof of our corollary.

96

4. Critical exponents.

On the begining of the § 3 we were speaking about some kind of stability. As every stability (it was a stability of... with respect to ...). Namely, we were interested in the stability (persistence) of the cubic character of a function restricted to a curve, with respect to perturbations of that curve. Now we are going to discuss another kind of stability. The notion we want to speak about concerns smooth functions (our triplets) - more precisely, their germs. There-

H fore, however interested in (physical quantities) r, — and s (see §2), we

* ^

deal with their smooth representants £,17 and o which, as triplets (J, 17, a) belong to yriJf.C 3

2*

DEFINITION 4.1. Let (£, 77, a) E <f C\M\ we' say that triplet (£,77, a) is stable, if there exists such an open neighbourhood U of (£, 77, a), in 5^0 M (endowed with the topology induced from ®M\), that for every (f, 77, a)E U

there exists a triplet (glt g2, £3) of zero preserving diffeomorphisms (germs) gi; : IR2, 0 —> IR2,0 such that

tt,^f,of) = (f0#r.^0^2.orog3).

The above definition can be expressed in other words. Let G' be the group of triplets of zero-preserving diffeomorphisms of IR2 (more precisely, their germs at 0 E R 2 ) . G' acts iri ®Jl\ by superposition. The stability of

an element ( J , i ? , a ) E y o ^ means that the intersection of the G'-orbit in ®M\ with 'Sf'nJl is open in

Now the problem is to find out all open orbits of G'-action in tfnJt. By virtue of Corollary 3.6 we can reduce this problem to the well known clas­sification (see [1]) of the open G-orbits in M\, G 2£Xs'vs\<M\ by super­position (see page 17). It is known that there are only two open G-orbits in <M\, which, following Arnold [1] are denoted by D4. To be more explicit, / E D 4 iff for appropriately chosen coordinates '«, w on IR2

/ = u(u2 ± w2).

One of differences between D4 and D4 is exhibited by the zero-level sets of their representatives.

The form of our functions %, t\ and 0 is restricted to some extend by Proposition 3.5 and Corollary 3.6. But, in order to be physically acceptable, the critical isotherm J_ 1(0), the level set o~l(0) and as well as 77_1(0) con-

97

sisting of two components of the undercritical part of the residual magnetiza­tion curve, should be smooth curves.

The set of stable and physically acceptable triplets (£, 17, a) is defined in the next proposition (4.2).

Let us introduce the following expansions for the functions \jj, a,flf f2

(see Proposition 3.5)

$(x,y) = oc1y + a2x(modJ%l) (4) ,

a(x) = 0! x2 (modJZ]),

/1 (x>y) — x3(b 4- higher order terms in x) + yib^x 4- b2x2 + b3y + b4xy +

+ b$y2 + higher order terms in x and y),

f2(x>y) = x2(cl + higher order terms in x) + y(c2 + higher order terms in x and y).

Taking advantage of the Definition 3.1 and inequality r(x,y) = e(x,y) £ (x,y) > 0, for physically acceptable £ one can derive the following inequalities (for the definition of e see Fig. 2)

(33) • at < 0 , Pi>0.

According to (observed) cubical dependence of magnetic field on magne­tization (see § 2) for the critical isotherm we have

(34) b<0

PROPOSITION 4.2. For physically acceptable and stable triplets (£,17, a) E E £^n M the following inequalities holds true

«! < 0 , 0! > 0 , b < 0 , OL2 -h 0 .

If for £, 17, a the above inequalities are fulfilled then (£, 77, o) E © D4 n

Proof. Let us notice that if (£, T? , q) E 5^ H ^ then taking into account Corollary 3.6 we obtain

> i = 0 i M , cl=-ela2/ali C2=-OL2IOLX .

Among acceptable triplets (£, 77, a) defined by the inequalities (33) and (34) an open subset of the set £f fl Ji is formed by these which satisfy the fol-

i

(4) It means that the functions on both sides are equal up to a function the germ (at zero) of which belongs to the ideal »M \.

98

lowing conditions: 0i ^ 0, Q!2 =£0. We shall prove that this set is contained in fD*.

The discriminants of the cubic forms J3,T?3,a3 of two variables, standing on the right-hand side of the following expansions —"

%(x>y) = &ly3- + Zoi\oi2y2x + oi\{$i + Za\)yx2 +

+ OL2(OL\ + fix)x2 (mod Jl\)

i7(#,y) = (& + J—a\)x* + 2$la2x2y + pxaixy2 (mod^t)

o(x,y) = -a2a\yz -la\aly2x~a2($1 + Za\)yx2 -

-(a22lay){a2+$x)x*(modM\),

are

A ' « 3 ) = t f / 2 7 a 5 , A'(i?3) = p f a ? ^ / 2 7 ^ + | ^ ' a i ) , A'(a3) = j3?/27a? ,

and for c^ < 0, jSj > 0, & < 0, a2 =£ 0 we have

A ' ( £ 3 ) > 0 , A'( i? 3 )>0, ' A ' ( a 3 ) > 0 .

Therefore, the cubic forms J 3 , i? 3 ,a 3 belongs to Dj and their zero-sets are single lines. It is known from the theory of singularities (see [1]) that £3,773 and a3 are 3-determined, i.e. for any triplet (£,r?, o) of functions (germs) of two variables, which has at zero the same 3-jets as f3, T?3 and a3

(respectively), there are diffeomorphlsms gi.,g2, #3 €=G s u c n t n a t £°gi =

= ?3» Wogi = Wz* a°g3 = °3- Thus the proof of our proposition is com­pleted.

Let us switch off the external magnetic field, i.e. we pass to H = 0. The remaining magnetization Mr is called a residual magnetization. By (4), H= 0 is equivalent to T? = 0. This is why we are interested in the set r?"1(0).

COROLLARY 4.3. The set T?~r(0) is the curve

(36) u -^ p{u)'.= irib2a\l^)u2 + 0(u3),u)E]R2 .

Proof. Because of 1? E D^ then there exists a diffeomorphism IR2,0 —• —^IR2,0; (x,y)—*(X(x,y)yY(x,y)) such that

n{x,y) = X(x,y)(X2(x,y) + Y2^)) (cf. [2]).

Let us recall the form of 77, namely

99

,n(x}y) - bx3 + biXiotiy + a2x)2 + b2x2(otly + a2x)2 +

+ b3(a1y + a2xf + dx4(modJ£s2),

thus we have

X(x,y) = x(b1/3 4- higher order terms in x and y) + (^3ai/j31)y2(mod^2) •

By ordinary Implicit Functibn Theorem (see e.g. [5]) applied to the equation

X(p(-) , ' ) = 0

we obtain (36). Hence the proof is completed. Now we want to calculate at the critical point x = 0 = y the infinite­

simal dependence of the residual magnetization Mr(t) and the inverse of the isothermal magnetic susceptibility (shortly i.m.s.) l/x(h,t) as functions of temperature (cf. [15]). Let be fixed a magnetic field h and a temperature t, What does correspond in our approach to l/x(b, t) which in thermodynamics

is introduced as I ~Trr) ? Let tp(x,t) be defined by the equation

e(x, <p(x, t)) %(x, y(x, t)) ~ t = 0

H Since -=- (x}y) = e(x,y) r)(x,y) and M(x,y) = ~e(x,y)x, the inverse of

Tc

i.m.s. is the derivative of the mapping assigning —e(x, ip(x, t)) rj(x, #(x,t)) to each ~e{x, \p(x,t))x.

Tc This results in differentiating of the mapping x—• - -—••—- 77 (x,

<p(x,t)). Thus we define cV

(37) Vxibj)^--J^7±ri(.M.tt))i i - t - i c r ax \x. ii(X,y(x,t))=h

As concerns the residual magnetization, we are interested only.in its absolute value and therefore we put

Mr(t) := M(x,v(x,t)) X: H(X, <p(x,t)) = 0

By the infinitesimal dependence of the above two functions on temperature we mean the asymptotic proportionalities

/ l 1 \0

T T. c •

/ 1 1 V 1 i/x(0,O« T TcJ ' T Tc

for TtTr.

100

PROPOSITION 4.4. For our (£.,17, a) as in Proposition 4.2, we have

(38)

(39)

Mr(t) = ft t2/3 + 0(f ) ,

i / X (o ,0 = - r ^ 2 g l *2/3 + o(t) . Q!

Proof. For a given £ the corresponding parameter ut on the curve (36) is given by the equation

t = a\uf + 0(«*) ( 5) .

Thus ut = \/oix tl/3 + 0(£2/3) and

A4.(0 = Af(p(uJ) = - ^ t 2 ? 3 + 0(t). Pi

Passing to the other critical exponent we see that, by (37)

(40) l / x ( 0 , 0 = " i + r c* bx

(x,y(x,t))-

by

It is easy to see that

bv

3T? 3£ .^ 3£ (x, #(x, t))~r~ (wix, t))f ~-- (x,v(x, t)) by

X : H(x, <{>(x,t)) = 0

by

K bx

(x,v(x,t)) X: H(x,\p(x,t)) = 0

(x, ip(x, t))/ — {x, y(x, t))

01 t + 0(£4/3)

and br}

bx

by

(x,v(x,t))

= — + 0 ( t 1 / 3 ) , (Xi x.H(x,\p(x,t)) = 0 l

^a2

X: H(x,lp(x,t)) = 0

Inserting these into (40) we obtain (39).

«? t2/3 + 0(f)

REMARK 4.5. The shape and the size of hysteresis loop of a given ferroma­gnetic sample at fixed T<TC, may vary while the sample is subjected, for

(5) Without loss of generality, we restrict our considerations to W0 C W, where e = + 1.

101

instance, to a plastic treatment (cf. [3]). Such or other technological treat­ments can modify friction between magnetic domains as well as influence the facility of their growth. From the macroscopic point of view this enables us to get a new sample the hysteresis loop of which is muche narrower and almost rectangular.

Besides a single ferromagnet we are going to consider a family of samples labelled by a parameter z, made of the same material but with different properties of magnetic domain structure. Namely decrease of 2 corresponds to narrower and narrower hysteresis loops. Thus for a fixed T<Tcy the coercivity Hc and the residual magnetization Mr are functions of 2 and Hc—• 0, Mr—>MS as 2 \0 , where Ms is the spontaneous magnetization of the ferromagnet. Assuming that the properties of this family of ferro-magnets depends continously on parameter 2 we have the two critical ex­ponents (cf. [15]) 0 = 2/3, 7' = 2/3.

REMARK 4.6. Measurements of the critical exponents consists in fitting the / 1 1Y*P

experimentally obtained points by a "monomial" a0\ ~T~~T~) •» where

tf0,a0EIR. The fitting is done on an interval [T1,T2] laying strictly below Tc, i.e. T2 < 7\ . Such a method assumes implicitly that the coefficients a^ in a rational power expansion

- / 1 1 V* (41) ha\f-T) of the thermodynamical function the critical exponent of which we want to

/ 1 l \ f t o measure, are such that the first term ao\lF~~~f~) dominates on the

interval [Tif T2]. No doubts that it must not be true in general. Therefore we propose another point of view mentioned already in [8]. According to it

1°. This is only a theory which is able to provide a type of the expansion (41), i.e. the exponents a,-, f = 1,2,....

2°. Fitting of experimental data is a source of information about coefficients ai9 i= 1,2,... .

102

Our point of view is represented in the following diagram

theory

"qualitative properties" e.g. symmetry, pheno­

menology of phases, etc.

^ critical

exponents coefficients

of (41)

X y

/ /

quantitative measurements

Acknowledgments. I am very grateful to Tadeusz Mostowski for his help, stimulating advices and interest in my work. I would like also express my deep thanks to Krzysztof Maurin for intellectual stimulation and sympathetic criticism during preparation of this paper. Finally I would like to thank to Piotr Jaworski and Andrzej -Meihofer for helpful comments and communi­cations.

R E F E R E N C E S

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[2] Arnold V.I., Local normal forms of functions, Inv. Math. 35, (1976) p. 87-109.

[3] Bozorth R.M., Ferromagnetism, Toronto-New York-London (1951).

[4] Brocker Th., Lander L., Differentiable germs and catastrophes, London Math. Soc. Lecture Notes 17, (1975).

[5] Golubitsky M., Guillemin V., Stable mappings and their singularities, Graduate Texts in Math. 14, Springer Berlin, (1973).

[6] Giittinger W., Eikemeier H. (eds.), Structural Stability in Physics (1979), second printing 1980, Heidelberg, Springer).

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7] J aneczko S, Komorowski J., Mostowski T., Phase transitions in ferromagnetis and singularities, Preprint IHES/M/80/24, to appears in Rep. on Math. Phys..

8] Komorowski J., Gas-liquid phase transitions and singularities, Rep. on Math. Phys. 19 (1981) p. 174-212.

9] Komorowski J., On Thorn's idea concerning Guggenheim's one-third law in phase transitions, preprint, Math. Inst. Univ. of Warwick, may 1977.

10] Kondorski E., On the nature of coercive force and irreversible changes in magneti­zation, Phys, Z. Sowjetunion, 11, (1973).

11] Landau L.D., Lifshitz E.M., Electrodynamics of continuous media, Pergamon (1963).

12] Malgrange B., Ideals of differentiable functions, Oxford Univ. Press (1966).

13] Rand D., Arnol'd's classification of simple singularities of smooth functions, Du­plicated notes, Math. Inst., Univ. of Warwick (1977).

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15] Stanley H.E., Introduction to phase transitions and critical phenomena, Clarendon Press (1971).

16] Thorn R., Phase transitions and catastrophes; in "Statistical mechanics: new con­cepts, new problems, new applications" ed. S.A. Rice, K.F. Freed and J.C. Light, Univ. of Chicago Press (1972), p. 93-105.

17] Thorn R., Structural stability and morphogenesis, transl. by D.H. Fowler, Benjamin (1975).

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22] Wilson K.G., KogutJ., The renormalization group and the e-expansion, Phys. Reports 12, (1974) p. 75-199.

[23] Zeeman E.C., Trotman D.J.A., The classification of elementary catastrophes of codimension < 5 , in Lecture Notes in Math. 525, Springer 1976, p. 263-327 and in E.C. Zeeman, Catastrophe Theory selected papers 1972-1977, Addison-Wesley 1977, p. 497-561.

Lavoro pervenuto in redazione il 25/VI/1982