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A geometrical interpretation of the scaling laws
Physica 64 (1973) 189-201 0 North-Holland Publishing Co.
A GEOMETRICAL
INTERPRETATION
OF THE SCALING
LAWS
L. BENGUIGUI Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel
Received 8 May 1972
Synopsis From the Widom-Schofield expression of the free energy A* in the neighbourhood of a critical point, we show that, in the plane (magnetic field-temperature) paths exist on which A* and its derivatives have the following asymptotic behaviour : A* (or one of its derivatives) z M”, A4 being the magnetization and p being independent of the chosen path. We call them equivalent paths, and their equation is H/IT/B8 = Cte. In particular, the critical isochore, the critical isotherm and the coexistence curve are among them. If we impose the existence of equivalent paths, we find the scaling-law equation of state, which shows that this procedure is equivalent to others already used as the homogeneity hypothesis of Widom. We investigated the linear model of Schofield et al., which gives a special role to three equivalent paths. It is possible to find it in using only the simplest functions that appear in this model.
1. Introduction. It has been proposed1,2) critical point has the form M = jTIOf(H/ITIfld).
that the equation
of state of the
(1)
We here adopt the magnetic language; H is the magnetic field, M the magnetization, T the temperature interval between the critical temperature and the actual temperature, and p and 6 are two of the critical indicesl). From this equation of state, it is possible to deduce the relations between the critical indices. In this paper, we propose a new approach to find eq. (1) and the scaling laws. The starting point is with the following remarks. We consider a system for which eq. (1) is valid, and we express some thermodynamic quantities, such as the freeenergy, the susceptibility aM/aH, the derivative aM/aT as functions of M, for (T < 0, H = 0) and for T = 0. For example, the susceptibility x = BM/aH has the asymp(We totic behaviour x cc M -Y”SforH=O,T
L. BENGUIGUI
190
paths equivalent paths, and ask whether other equivalent paths exist. From (l), it is possible to show that there is an infinite number of such paths. We can operate inversely. If we admit that the critical point is defined by the existence of equivalent paths, we find that the system has the equation of state (1). Our assumption is equivalent to the homogeneity hypothesis of Widomz) or to the cell model of Kadanoff et a1.l). The critical paths are a characteristic property of the critical point. We have developed the study of the critical paths in section 2 of this paper. In section 3 we investigate the relation between our results and the “linear model” of Schofield et aL4). As is shown below, this model gives a special role to three equivalent paths. We do not see any fundamental reason for selecting these particular equivalent paths and we show that the “linear model” can be formulated from another approach. We find that the model is, in fact, the simplest model on which it is possible to elaborate. 2. Equivalent critical paths. We do not start directly from expression (1) but use the transformation proposed by Schofield5). He writes H = rSah (0) 9
(2)
T = rt (e),
(3)
where h and t are some regular functions of 19only. So the thermodynamic variables are now functions of r and t9instead of H and T. We note that 0 is a function of H/jTI@ but not a single-valued one. Using (l), (2) and (3) one can show that the singular part A* of the free energy A is given by A* = #@+‘)p (0).
(4)
A* is related to A by6): A* = A - A, + TS,,
when SC is the value of the entropy at the critical point. Meanwhile we restrict ourselves to the case T < 0 and will later study the case T 2 0. From (2), (3) and (4) it is not difficult to show through straightforward calculations that A* and its successive derivatives have the following general property: each of these thermodynamic quantities can be written in the form F* = M”‘1, (H/ITI@).
(5)
I;; is the chosen quantity, ,ui is an exponent depending on fi and S. The functions 1, have finite values if H = 0 and/or T = 0. In table I, we give the values of pr for A*, its first derivatives S* = -aA*/aT, (for M = aA*/aH, ,I,+ is equal to 1) and its second derivatives aS*/aT, BM/dH
1-s y’ = /3 (6 - 1)
1 - (x’ = /I (1 + h)
2 - OL’= /9 (1 + S)
Scaling laws
2 - a’ = /I (1 + 6)
-W/B)
l+h
-@‘/IQ
1+s
l-6
cu4)
aH
aM
Calculated for T = 0, from definitions of critical indices
(1 - a’)/B
1 + 6 - (2/P)
6.4
aT
as*
(2 - a’)/B
Cf.4
-?j$
1 + 6 - (l/8)
St=
Calculated for H = 0, T = 0 from definitions of critical indices
011)
A*
1+s
quantities
Calculated from (2), (3) and (4)
Thermodynamic
OCS)
aH
as*
h + (l/b) = 6
l+h-6
1 - (f/8)
1 - (f/B)
aM -=-aT
Values of pi calculated in the three cases: (a) from A* given by (2), (3), (4); (b) from definitions of critical indices, for N = 0, T < 0; (c) identical with (b) but for T = 0. Since the paths defined by H = 0 and T = 0 are equivalent, the values of ,u~ must be equal, and this gives the scaling laws. The relations between critical indices obtained from the second derivatives are sufficient to get all the other relations between them.
TABLE I
L. BENGUIGUI
192
and dM/iYT = - aS*/aH. Now consider the critical paths defined by H/ITIS” = Cte. Along such a path, A* and its derivatives have the asymptotic behaviour FL cc MPi, which is independent of the chosen path. We call them equivalent paths, since the asymptotic behaviour of each thermodynamic quantity is the same on all the paths, provided they are expressed as functions of M. The paths defined by H = 0 and T = 0 are particular cases of the family of the equivalent critical paths, namely those with the constant respectively equal to zero and to infinity. From the definitions of the critical exponents (o(, p, y’, S and h), which we mention in table II, we calculate A* and its derivatives as functions of M, for H = 0 and T = 0, using the theorems given by Griffiths6) on the asymptotic behaviour. Since the paths H = 0 and T = 0 are equivalent, the exponents of each thermodynamical quantity calculated for H = 0 and T = 0 must be equal and the same applied to the corresponding pi. In this manner, we obtain the scaling laws. The results are given in table I. From this point of view, the scaling laws are a consequence of the property of equivalent paths. The choice of M in (5) to express the thermodynamic quantities is arbitrary. We obtain equivalent results if we use S* instead of M. We can also use the inverse of the second derivatives, since these derivatives diverge at the critical point. Evidently the indices ,ui depend on the choice of the “independent variable” but we obtain the scaling laws in all the cases. TABLEII Definition of critical indices Thermodynamic
quantity
Asymptotic ITIs; H=
aiw x=-&y
as* aT
M s*a x
as* aT
Ma MB
a These definitions are given by Griffith#).
behavior 0, T<
0
ITI-Y’;H=O,T
0, T>
0
T-a; H = 0, T > 0 H’las;S= 0 9 T> 0 T’I”; S = 0, T > 0
A GEOMETRICAL
INTERPRETATION
OF THE SCALING LAWS
193
Now we want to extend our results to the case T > 0 (see also table II, for definitions of the indices). Since the expressions (l), (2) and (3) are valid for T 2 0 or T < 0, the expression (5) is also valid at T 2 0. But we must pay attention to the following. If we adopt M or S* as an “independent variable”, we see that there are two particular paths: one for which M = 0 (the path H = 0, T > 0) and the other for which S* = 0 (the critical isentrope). But the second derivatives can be used to compare the three equivalent paths a) H = 0, b) S* = 0, c) T = 0. In this manner we obtain
s = a,,
c = l//I.
(6)
We also find other relations between critical indices from which we deduce that a = 01’, y = y’ when they are combined with relations given in table I and with (6). From table I, we can deduce relations between the indices ,u,. The interesting point is that we obtain relations in which no product appears as in the scaling laws. For example, we can write
We see that each index can be expressed as a function of ,ul (or p4) and another index. Griffiths and Wheeler3) have insisted on the difference in the divergences along the paths H = 0 and T = 0. Our approach shows that the difference appears only if we adopt the intensive parameters (H, T) to measure the distance to the critical point. If we choose, as we have done, the extensive parameters (M, S*), there is no difference at all. Furthermore, we must point out an interesting property of the equivalent paths. Consider for example the susceptibility x = LJM/BHalong all the critical paths except the path T = 0. These paths are “asymptotically parallel” in the sense of Griffiths and Wheeler3) to the coexistence curve. Along these paths we have x cc ITImY.But now consider all the paths except the path H = 0. They are all asymptotically parallel except the path T = 0, which makes a right angle with the coexistence curve. Along all these paths we have x oc H”-6”6. We see that although these paths are not all of the same nature (asymptotically parallel or making an angle with the coexistence curve), the susceptibility has the same critical behaviour. These results are not in agreement with the suggestions of Griffiths and Wheeler. Now we want to operate inversely. Let us suppose that critical paths exist and let us try to find an equation of state. The critical paths are defined in saying that A* and its successive derivatives can be expressed as Fi = MYiZi(0)y
(7)
194
L. BENGUIGUI
8 being a function of H and T. We assume also that the functions I, (f3) are regular and that for (T = 0, H = 0), they have finite values. The equation of the critical paths is given by &’(H, T) = Cte. For the sake of definiteness, we choose M as the “independent variable”. We have to find the relation between M, H and T (the equation of state) and the equation of the equivalent paths. First, write that aA*/aH = M and suppose, from (7), that A* = M”Z, (0). We obtain
de
&p-l
d4 d0
aH
I y Myl-2
844 --II ^-dH
’
= 1.
(8)
From this expression we deduce _aM = __!_M+I-~) aH
L
VI
M _ ----.
VI
11
1 dl,
ae
I,
aH
de
(9)
But, from (7), we can write aM
-
aH
(e).
= MY3
We can see that expressions (9) and (10) are possible only if v3 = - (vr if M --= dz, ae de aH
M-(+)Q
(e) zl(e)
(11)
[e(e) being a function of 01. From (8) and (1 l), we conclude that Myl-’ and My’-’ (ae/aH) are functions of 8 only, and we write Mylm2 $-
= fi(e),
M”‘-
1
g
aM -
aT
=
m,
&f"-"z
(aM/BH)
(12)
=fi(e).
Now we proceed as above from the relation (aA*/aT) = -S* MY,-Y*-l
2) and
z
_
aT -
g,(R I
and we get (13)
when v2 is the exponent of M in (7) if Fz is the entropy S*. As we want to obtain partial-derivative equations giving T (M, e) and H (M, e), we calculate the derivatives aM/aH, aM/aT, ae/aH and ae/aT as functions of
A GEOMETRICAL
INTERPRETATION
derivatives of T (M, 0) and H(M,@.
a0
1
ae
aT
aH7aM
EC-0
aM
*
-= aH
-I-z
1 aT
OF THE SCALING LAWS
195
We get
i --
aM aH’
aM
--1 aH
-71 aT
ae’
(14)
where aH
L3 = -----* aH aT ae
aT
ahf ae
ah4
We substitute the expressions (14) in (12) and (13) and eliminating A, we arrive at the following equations : MdT aM
= de)
2
(cl = -F),
(15) MS
aM
= j(e) ff.
(j
= -2).
The general solutions of these equations are ITI = M%(e),
H = M*g(e).
(16)
The functions v and g are solutions of
Finally, to find the values of a and b, it is sufficient to write that M”‘- ’ (t%/aH) and My’-’ 1 lM/aT are independent of M. This is done, using (14) and (16), and we find a = y1 - v2 and b = v1 - 1. We identify v1 - v1 with l/p and v1 - 1 with 6. Thus the final results are H = Mdg(8),
ITI = M”%(e).
(17)
We deduce from (17): a) 0 is a function of H/ITI@ and as a consequence the equation of the critical paths is H/ITISd = Cte; b) the equation of state is identical with (1). We conclude that the critical paths are a characteristic property of the critical point and that we can use them as a starting point to find the equation of state and the scaling laws, as did Widomz) with the homogeneity hypothesis or Kadanoff et aZ.l) with their cell model.
196
L. BENGUIGUI
3. Critical paths and the “linear model”. Our approach to the properties of the critical point shows that there is some isotropy in the asymptotic behaviour of thermodynamic quantities, if expressed as functions of M or S*. The paths H = 0 and T = 0 are chosen to define the critical indices of table I but it is not certain that these paths are of special importance, except from the experimental point of view. In particular the “linear model” of Schofield et aL4) gives explicitly a particular role to these three paths. We show in this section that it is not necessary to give a particular role to these paths to get the “linear model”. When we use the transformation of Schofield defined by (2) and (3) H = rPah (e), we
T = rt (0),
also have s* = rs@+l)-ls (O),
M = Pm (e),
-aM =y -fica-“k
-as* = ,B@+l)-*,
(Q,
aH
(Q,
aM -
=
aT
aT
r@-ln (0).
The “linear modeP4) is defined by the following points: a) The functions h, t and m are h(0) = a0 (1 - ez),
t(e) = i - b*e*,
m(0) = ge.
The linearity of m(0) gives its name to the model. b) If m(0) is linear, Schofield et a/. showed that bd-3
C,,B’-’ D”
co c
=
(b2
_
2 = (b* - l)‘-’
(18)
l)Y-1’
1 [l - b* (1 -
28)] -
(19)
The definitions of the coefficients C, B, D are given in table III and we see that these definitions correspond to the paths T = 0 and H = 0 T < 0 or T > 0 (critical isotherm, coexistence curve and critical isochore). c) Schofield et. al. take the value of b, which gives minimum values to the right-hand sides of (18) and (19). Both minima occur if b*
=
6-3 (S - 1) (1 - 28) *
(20)
With this value of b* Schofield et al. compute the expressions (18) and (19) and compare the results with the experimental determinations of these expressions.
A GEOMETRICAL
INTERPRETATION
OF THE SCALING
LAWS
197
TABLEIII Definition of critical coefficients
CO, B, D and C*
H = 0, T > 0 (e = 0)
aM/aH = CoT-Y
T=
M = DH1ls
O(0 = f&-l)
H=O,T
+l)
M = BITIS aM/aH = C,ITI-Y’
The agreement is quite good, for some experimental situations, and for the threedimensional Ising model, but not for the two-dimensional Ising model. We note that a) the model gives special importance to the paths T = 0 and H = 0; b) if 6 and /3 have the classical values, 3 and 3, respectively, bZ is undetermined; c) the model is not applicable to the two-dimensional Ising model. We propose the following approach. The chosen functions h, t and m are the simplest that we can find, and we ask what the value of b is if .we state that the other functions s(e), k(8), a(0) and n(0) also have the simplest form. One can choose, as was done by Schofield et al., 8 values between 0 and 1. Thus the functions h, t, m, s have the following properties:
a>
h(0) = 0,
h(l)
b)
t(0) = 1,
t (l/b) = 0,
4
m(0) = 0,
4
+e)
= 0,
m&e>
= s(e),
h(-8)
= -h(B);
t(-e)
= t(e);
= -m(e);
s(c) = 0
with
c < l/b.
The simplest functions are the shortest polynomials tions. We then have h(e) = ae (1 - ez),
t(e) = 1 - bze2,
satisfying the above condi-
m(e) = ge,
s(e) = 2 - 82.
These four functions are not independent, which means that there is a relation between b and c (a and g are only scale factors). The procedure to find this relation is given in the appendix. We obtain c?=__- 1 /l’b2
6-l
2-/!?(S+l)
6-3
@(6+1)-l’
(21)
198
L. BENGUIGUI
We can verify that c < l/b with the common values of j3 and 6. [p’ = p (6 + 1) - Ql. The function k can be easily calculated, and we obtain /lmt ’ -
k=
/%ht
m’t
- h’t -
The numerator is a polynomial of degree two, and the denominator a polynomial of degree four. The simplest form is obtained if the numerator and the denominator have one or two common zeros. The numerator is equal to +g [l - 02b2 (1 - 2b)], and the denominator has the same two zeros that the numerator has if bZ = (6 - 3)/(6 - 1) (1 - 2/I), which is just expression (20). The function n deduced from (aMI&“) is equal to n(f3) =
/lShm’ - /lmh’ Bdht ’ - h’t
*
The numerator is of degree 3 and the denominator of degree 4. There are two common zeros if bZ also has the value given by (20). Now calculate the function a@) deduced from A.SIdT. We obtain a(e)
=
Bbhs’- B’sh’ ,i3dht’ - h’t ’
The numerator and the denominator are polynomials of degree four, but the terms in 8 and e3 do not exist. The complete calculation (see appendix) shows that
a(e) = pk*
re*- e:(BIB’, w*)i co*- 0; (B/P’, w)i [e*- el ut wi re*- 0%(it b*)i ’
e1 (x, y) and e2 (x, y) are two functions of the variables (x, JI) which have the values (PI/?‘, l/c*) and (/I, b*) for the numerator and the denominator respectively. The form obtained for a(0) cannot be simplified unless B/p’ = 1 and b* = l/c*, which is impossible, according to (21). Thus, we obtain the “linear model” by another approach, which requires the simplest functions of 8. We can call it the simplest model. This immediately gives the reason for the lack of complete agreement between the “linear model” and some experimental results; a better fit may be obtained if we use polynomials with more terms. On the other hand it is easy to verify that if we take the classical values @ = + and 6 = 3, we immediately obtain the simplest functions, whatever the values of b and c may be. This explains why the linear model does not give a definite value of b in this case.
A GEOMETRICAL
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199
OF THE SCALING LAWS
4. Conclusion. We have shown from the scaling-law equation of state that critical paths exist with a particular property. Along those paths, the free energy and its derivatives have the same asymptotic behaviour, if one of them is chosen to express the others. We took M = aA*/aH as the “independent variable”, and all the thermodynamic quantities Fi have along the critical paths the asymptotic behaviour I;; cc M”‘(,u~depending on the quantity F,). The equation of these equivalent paths is H/ITl@ = Cte, and the critical isochore, the critical isotherm and the coexistence curve are among them. From this, we obtained the scaling laws. Inversely, if we assume that the critical point is characterized by the existence of equivalent paths, we obtain the scaling-law equation of state. This shows that this assumption is equivalent to the others made to obtain this equation of state (homogeneity hypothesis of Widom or cell model of Kadanoff). We have discussed the “linear model” of Schofield et ~2.~) and have shown that to get it, it is sufficient to impose the simplest functions that appear in the model. Thus it is not necessary, in order to find it, to give special importance to some equivalent paths, as was done by Schofield et al.
APPENDIX
Relation between b and c, and calculations of the functions of the “linear modell”. We start from the expressions (2), (3) and (4) and we calculate M = aA*/aH, S* = -aA*/aT, BM/BH, aM/aT, aS*/aT. For example we have
M
_
&._
aA* _ p* aH
-\
aT
ar ae
afi*q ae
I[ aH aT
ar ]/ \ ar ae
aH q ae
ar /
or
(
M = r,, b (8 + ‘)@ - Pit @ht’ - h’t ’
t’ =
dt
-ii-’
p’ =
-9,
et,.>.
In this manner we obtain m(e) = B (6 + l) Pt’ - Pft /3Ght’ - h’t ’
s(e) =
,6Shp’ - ,3 (6 + 1) h’p ,6Ght’ - h’t ’
-
(A.11 (A.21
In order to obtain the relation between h(B), t(e), m(e), s(e), we eliminate p and p’ between (A. 1) and (A.2). First we calculate p and p’ from (A. 1) and (A.2).
200
L. BENGUIGUI
1
[/36mh + st],
p = /3 (6 + 1) p’ = t’s + h’m.
64)
Then we differentiate (A.3) and the results equal (A.4). Thus we get the desired relation : P’st’ - s’t = j3dm’h - Bh’m,
[fi’ = j3 (6 + 1) -
11.
(A.5)
Now we substitute in (A.5) the chosen functions for h(O), t(O), m(O) and s(0) and arrive at
2 r1 - b2c2@‘] = /egu (6 - l),
(A.6)
2b2 (1 - B’) = pga (6 - 3),
from which we deduce 6-l -
--
6-3
l-/Y -.
j3’
We calculate the functions k(B), n(O) and a(O) as above. We obtain
k(B) =
-g [82b2 (1 - 2/!?) - l] 1 + (2b2@ - 3 - b2) O2 - b2 (286 - 3) O4 ’ -
8
-
83
n(e) = 1 + (2b2,%(1- 3 6) - b2) + O2 (6 - b2 3) (2/3S - 3) e4 ’ a(e) = pc2
The numerator can write
1 + O2 [(2/9/c”#I’)6 - 3 - b2] - (l/c’) (286 - 3) O4 1 + O2(2b2/?6 - 3 - b2) - b2 (286 - 3) O4
-
coefficients are analogous to those of the denominator,
v - e:(it
ei
fez- (mt w)i b2)1[e2 - e$u, wi
a(e) = pc2 v2 - eZ(p/it l/a
and we
’
Acknowledgement. The author wishes to express his gratitude to Professor J. de Boer of the University of Amsterdam and to Professor M. Revzen of the Technion for their encouragement, and to Professor P. Saphar of the Department of Mathematics, Technion, for his interest in this work.
A GEOMETRICAL
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LAWS
REFERENCES 1) Kadanoff,
L.P., et al., Rev. mod. Phys. 39 (1967) 395.
Fisher, M.E., Rep. Progr. Phys. 30 (1967) 615. 2) Widom, B., J. them. Phys. 43 (1965) 3898. 3) Griffiths, R.B. and Wheeler, J., Phys. Rev. A3 (1970) 1047. 4) Schofield, P., Litster, J.D. and Ho, J.T., Phys. Rev. Letters 23 (1969) 1098. 5) Schofield, P., Phys. Rev. Letters 22 (1969) 606. 6) Griffiths, R.B., J. them. Phys. 43 (1965) 1958.
201
A GEOMETRICAL
INTERPRETATION
OF THE SCALING
LAWS
L. BENGUIGUI Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel
Received 8 May 1972
Synopsis From the Widom-Schofield expression of the free energy A* in the neighbourhood of a critical point, we show that, in the plane (magnetic field-temperature) paths exist on which A* and its derivatives have the following asymptotic behaviour : A* (or one of its derivatives) z M”, A4 being the magnetization and p being independent of the chosen path. We call them equivalent paths, and their equation is H/IT/B8 = Cte. In particular, the critical isochore, the critical isotherm and the coexistence curve are among them. If we impose the existence of equivalent paths, we find the scaling-law equation of state, which shows that this procedure is equivalent to others already used as the homogeneity hypothesis of Widom. We investigated the linear model of Schofield et al., which gives a special role to three equivalent paths. It is possible to find it in using only the simplest functions that appear in this model.
1. Introduction. It has been proposed1,2) critical point has the form M = jTIOf(H/ITIfld).
that the equation
of state of the
(1)
We here adopt the magnetic language; H is the magnetic field, M the magnetization, T the temperature interval between the critical temperature and the actual temperature, and p and 6 are two of the critical indicesl). From this equation of state, it is possible to deduce the relations between the critical indices. In this paper, we propose a new approach to find eq. (1) and the scaling laws. The starting point is with the following remarks. We consider a system for which eq. (1) is valid, and we express some thermodynamic quantities, such as the freeenergy, the susceptibility aM/aH, the derivative aM/aT as functions of M, for (T < 0, H = 0) and for T = 0. For example, the susceptibility x = BM/aH has the asymp(We totic behaviour x cc M -Y”SforH=O,T
L. BENGUIGUI
190
paths equivalent paths, and ask whether other equivalent paths exist. From (l), it is possible to show that there is an infinite number of such paths. We can operate inversely. If we admit that the critical point is defined by the existence of equivalent paths, we find that the system has the equation of state (1). Our assumption is equivalent to the homogeneity hypothesis of Widomz) or to the cell model of Kadanoff et a1.l). The critical paths are a characteristic property of the critical point. We have developed the study of the critical paths in section 2 of this paper. In section 3 we investigate the relation between our results and the “linear model” of Schofield et aL4). As is shown below, this model gives a special role to three equivalent paths. We do not see any fundamental reason for selecting these particular equivalent paths and we show that the “linear model” can be formulated from another approach. We find that the model is, in fact, the simplest model on which it is possible to elaborate. 2. Equivalent critical paths. We do not start directly from expression (1) but use the transformation proposed by Schofield5). He writes H = rSah (0) 9
(2)
T = rt (e),
(3)
where h and t are some regular functions of 19only. So the thermodynamic variables are now functions of r and t9instead of H and T. We note that 0 is a function of H/jTI@ but not a single-valued one. Using (l), (2) and (3) one can show that the singular part A* of the free energy A is given by A* = #@+‘)p (0).
(4)
A* is related to A by6): A* = A - A, + TS,,
when SC is the value of the entropy at the critical point. Meanwhile we restrict ourselves to the case T < 0 and will later study the case T 2 0. From (2), (3) and (4) it is not difficult to show through straightforward calculations that A* and its successive derivatives have the following general property: each of these thermodynamic quantities can be written in the form F* = M”‘1, (H/ITI@).
(5)
I;; is the chosen quantity, ,ui is an exponent depending on fi and S. The functions 1, have finite values if H = 0 and/or T = 0. In table I, we give the values of pr for A*, its first derivatives S* = -aA*/aT, (for M = aA*/aH, ,I,+ is equal to 1) and its second derivatives aS*/aT, BM/dH
1-s y’ = /3 (6 - 1)
1 - (x’ = /I (1 + h)
2 - OL’= /9 (1 + S)
Scaling laws
2 - a’ = /I (1 + 6)
-W/B)
l+h
-@‘/IQ
1+s
l-6
cu4)
aH
aM
Calculated for T = 0, from definitions of critical indices
(1 - a’)/B
1 + 6 - (2/P)
6.4
aT
as*
(2 - a’)/B
Cf.4
-?j$
1 + 6 - (l/8)
St=
Calculated for H = 0, T = 0 from definitions of critical indices
011)
A*
1+s
quantities
Calculated from (2), (3) and (4)
Thermodynamic
OCS)
aH
as*
h + (l/b) = 6
l+h-6
1 - (f/8)
1 - (f/B)
aM -=-aT
Values of pi calculated in the three cases: (a) from A* given by (2), (3), (4); (b) from definitions of critical indices, for N = 0, T < 0; (c) identical with (b) but for T = 0. Since the paths defined by H = 0 and T = 0 are equivalent, the values of ,u~ must be equal, and this gives the scaling laws. The relations between critical indices obtained from the second derivatives are sufficient to get all the other relations between them.
TABLE I
L. BENGUIGUI
192
and dM/iYT = - aS*/aH. Now consider the critical paths defined by H/ITIS” = Cte. Along such a path, A* and its derivatives have the asymptotic behaviour FL cc MPi, which is independent of the chosen path. We call them equivalent paths, since the asymptotic behaviour of each thermodynamic quantity is the same on all the paths, provided they are expressed as functions of M. The paths defined by H = 0 and T = 0 are particular cases of the family of the equivalent critical paths, namely those with the constant respectively equal to zero and to infinity. From the definitions of the critical exponents (o(, p, y’, S and h), which we mention in table II, we calculate A* and its derivatives as functions of M, for H = 0 and T = 0, using the theorems given by Griffiths6) on the asymptotic behaviour. Since the paths H = 0 and T = 0 are equivalent, the exponents of each thermodynamical quantity calculated for H = 0 and T = 0 must be equal and the same applied to the corresponding pi. In this manner, we obtain the scaling laws. The results are given in table I. From this point of view, the scaling laws are a consequence of the property of equivalent paths. The choice of M in (5) to express the thermodynamic quantities is arbitrary. We obtain equivalent results if we use S* instead of M. We can also use the inverse of the second derivatives, since these derivatives diverge at the critical point. Evidently the indices ,ui depend on the choice of the “independent variable” but we obtain the scaling laws in all the cases. TABLEII Definition of critical indices Thermodynamic
quantity
Asymptotic ITIs; H=
aiw x=-&y
as* aT
M s*a x
as* aT
Ma MB
a These definitions are given by Griffith#).
behavior 0, T<
0
ITI-Y’;H=O,T
0, T>
0
T-a; H = 0, T > 0 H’las;S= 0 9 T> 0 T’I”; S = 0, T > 0
A GEOMETRICAL
INTERPRETATION
OF THE SCALING LAWS
193
Now we want to extend our results to the case T > 0 (see also table II, for definitions of the indices). Since the expressions (l), (2) and (3) are valid for T 2 0 or T < 0, the expression (5) is also valid at T 2 0. But we must pay attention to the following. If we adopt M or S* as an “independent variable”, we see that there are two particular paths: one for which M = 0 (the path H = 0, T > 0) and the other for which S* = 0 (the critical isentrope). But the second derivatives can be used to compare the three equivalent paths a) H = 0, b) S* = 0, c) T = 0. In this manner we obtain
s = a,,
c = l//I.
(6)
We also find other relations between critical indices from which we deduce that a = 01’, y = y’ when they are combined with relations given in table I and with (6). From table I, we can deduce relations between the indices ,u,. The interesting point is that we obtain relations in which no product appears as in the scaling laws. For example, we can write
We see that each index can be expressed as a function of ,ul (or p4) and another index. Griffiths and Wheeler3) have insisted on the difference in the divergences along the paths H = 0 and T = 0. Our approach shows that the difference appears only if we adopt the intensive parameters (H, T) to measure the distance to the critical point. If we choose, as we have done, the extensive parameters (M, S*), there is no difference at all. Furthermore, we must point out an interesting property of the equivalent paths. Consider for example the susceptibility x = LJM/BHalong all the critical paths except the path T = 0. These paths are “asymptotically parallel” in the sense of Griffiths and Wheeler3) to the coexistence curve. Along these paths we have x cc ITImY.But now consider all the paths except the path H = 0. They are all asymptotically parallel except the path T = 0, which makes a right angle with the coexistence curve. Along all these paths we have x oc H”-6”6. We see that although these paths are not all of the same nature (asymptotically parallel or making an angle with the coexistence curve), the susceptibility has the same critical behaviour. These results are not in agreement with the suggestions of Griffiths and Wheeler. Now we want to operate inversely. Let us suppose that critical paths exist and let us try to find an equation of state. The critical paths are defined in saying that A* and its successive derivatives can be expressed as Fi = MYiZi(0)y
(7)
194
L. BENGUIGUI
8 being a function of H and T. We assume also that the functions I, (f3) are regular and that for (T = 0, H = 0), they have finite values. The equation of the critical paths is given by &’(H, T) = Cte. For the sake of definiteness, we choose M as the “independent variable”. We have to find the relation between M, H and T (the equation of state) and the equation of the equivalent paths. First, write that aA*/aH = M and suppose, from (7), that A* = M”Z, (0). We obtain
de
&p-l
d4 d0
aH
I y Myl-2
844 --II ^-dH
’
= 1.
(8)
From this expression we deduce _aM = __!_M+I-~) aH
L
VI
M _ ----.
VI
11
1 dl,
ae
I,
aH
de
(9)
But, from (7), we can write aM
-
aH
(e).
= MY3
We can see that expressions (9) and (10) are possible only if v3 = - (vr if M --= dz, ae de aH
M-(+)Q
(e) zl(e)
(11)
[e(e) being a function of 01. From (8) and (1 l), we conclude that Myl-’ and My’-’ (ae/aH) are functions of 8 only, and we write Mylm2 $-
= fi(e),
M”‘-
1
g
aM -
aT
=
m,
&f"-"z
(aM/BH)
(12)
=fi(e).
Now we proceed as above from the relation (aA*/aT) = -S* MY,-Y*-l
2) and
z
_
aT -
g,(R I
and we get (13)
when v2 is the exponent of M in (7) if Fz is the entropy S*. As we want to obtain partial-derivative equations giving T (M, e) and H (M, e), we calculate the derivatives aM/aH, aM/aT, ae/aH and ae/aT as functions of
A GEOMETRICAL
INTERPRETATION
derivatives of T (M, 0) and H(M,@.
a0
1
ae
aT
aH7aM
EC-0
aM
*
-= aH
-I-z
1 aT
OF THE SCALING LAWS
195
We get
i --
aM aH’
aM
--1 aH
-71 aT
ae’
(14)
where aH
L3 = -----* aH aT ae
aT
ahf ae
ah4
We substitute the expressions (14) in (12) and (13) and eliminating A, we arrive at the following equations : MdT aM
= de)
2
(cl = -F),
(15) MS
aM
= j(e) ff.
(j
= -2).
The general solutions of these equations are ITI = M%(e),
H = M*g(e).
(16)
The functions v and g are solutions of
Finally, to find the values of a and b, it is sufficient to write that M”‘- ’ (t%/aH) and My’-’ 1 lM/aT are independent of M. This is done, using (14) and (16), and we find a = y1 - v2 and b = v1 - 1. We identify v1 - v1 with l/p and v1 - 1 with 6. Thus the final results are H = Mdg(8),
ITI = M”%(e).
(17)
We deduce from (17): a) 0 is a function of H/ITI@ and as a consequence the equation of the critical paths is H/ITISd = Cte; b) the equation of state is identical with (1). We conclude that the critical paths are a characteristic property of the critical point and that we can use them as a starting point to find the equation of state and the scaling laws, as did Widomz) with the homogeneity hypothesis or Kadanoff et aZ.l) with their cell model.
196
L. BENGUIGUI
3. Critical paths and the “linear model”. Our approach to the properties of the critical point shows that there is some isotropy in the asymptotic behaviour of thermodynamic quantities, if expressed as functions of M or S*. The paths H = 0 and T = 0 are chosen to define the critical indices of table I but it is not certain that these paths are of special importance, except from the experimental point of view. In particular the “linear model” of Schofield et aL4) gives explicitly a particular role to these three paths. We show in this section that it is not necessary to give a particular role to these paths to get the “linear model”. When we use the transformation of Schofield defined by (2) and (3) H = rPah (e), we
T = rt (0),
also have s* = rs@+l)-ls (O),
M = Pm (e),
-aM =y -fica-“k
-as* = ,B@+l)-*,
(Q,
aH
(Q,
aM -
=
aT
aT
r@-ln (0).
The “linear modeP4) is defined by the following points: a) The functions h, t and m are h(0) = a0 (1 - ez),
t(e) = i - b*e*,
m(0) = ge.
The linearity of m(0) gives its name to the model. b) If m(0) is linear, Schofield et a/. showed that bd-3
C,,B’-’ D”
co c
=
(b2
_
2 = (b* - l)‘-’
(18)
l)Y-1’
1 [l - b* (1 -
28)] -
(19)
The definitions of the coefficients C, B, D are given in table III and we see that these definitions correspond to the paths T = 0 and H = 0 T < 0 or T > 0 (critical isotherm, coexistence curve and critical isochore). c) Schofield et. al. take the value of b, which gives minimum values to the right-hand sides of (18) and (19). Both minima occur if b*
=
6-3 (S - 1) (1 - 28) *
(20)
With this value of b* Schofield et al. compute the expressions (18) and (19) and compare the results with the experimental determinations of these expressions.
A GEOMETRICAL
INTERPRETATION
OF THE SCALING
LAWS
197
TABLEIII Definition of critical coefficients
CO, B, D and C*
H = 0, T > 0 (e = 0)
aM/aH = CoT-Y
T=
M = DH1ls
O(0 = f&-l)
H=O,T
+l)
M = BITIS aM/aH = C,ITI-Y’
The agreement is quite good, for some experimental situations, and for the threedimensional Ising model, but not for the two-dimensional Ising model. We note that a) the model gives special importance to the paths T = 0 and H = 0; b) if 6 and /3 have the classical values, 3 and 3, respectively, bZ is undetermined; c) the model is not applicable to the two-dimensional Ising model. We propose the following approach. The chosen functions h, t and m are the simplest that we can find, and we ask what the value of b is if .we state that the other functions s(e), k(8), a(0) and n(0) also have the simplest form. One can choose, as was done by Schofield et al., 8 values between 0 and 1. Thus the functions h, t, m, s have the following properties:
a>
h(0) = 0,
h(l)
b)
t(0) = 1,
t (l/b) = 0,
4
m(0) = 0,
4
+e)
= 0,
m&e>
= s(e),
h(-8)
= -h(B);
t(-e)
= t(e);
= -m(e);
s(c) = 0
with
c < l/b.
The simplest functions are the shortest polynomials tions. We then have h(e) = ae (1 - ez),
t(e) = 1 - bze2,
satisfying the above condi-
m(e) = ge,
s(e) = 2 - 82.
These four functions are not independent, which means that there is a relation between b and c (a and g are only scale factors). The procedure to find this relation is given in the appendix. We obtain c?=__- 1 /l’b2
6-l
2-/!?(S+l)
6-3
@(6+1)-l’
(21)
198
L. BENGUIGUI
We can verify that c < l/b with the common values of j3 and 6. [p’ = p (6 + 1) - Ql. The function k can be easily calculated, and we obtain /lmt ’ -
k=
/%ht
m’t
- h’t -
The numerator is a polynomial of degree two, and the denominator a polynomial of degree four. The simplest form is obtained if the numerator and the denominator have one or two common zeros. The numerator is equal to +g [l - 02b2 (1 - 2b)], and the denominator has the same two zeros that the numerator has if bZ = (6 - 3)/(6 - 1) (1 - 2/I), which is just expression (20). The function n deduced from (aMI&“) is equal to n(f3) =
/lShm’ - /lmh’ Bdht ’ - h’t
*
The numerator is of degree 3 and the denominator of degree 4. There are two common zeros if bZ also has the value given by (20). Now calculate the function a@) deduced from A.SIdT. We obtain a(e)
=
Bbhs’- B’sh’ ,i3dht’ - h’t ’
The numerator and the denominator are polynomials of degree four, but the terms in 8 and e3 do not exist. The complete calculation (see appendix) shows that
a(e) = pk*
re*- e:(BIB’, w*)i co*- 0; (B/P’, w)i [e*- el ut wi re*- 0%(it b*)i ’
e1 (x, y) and e2 (x, y) are two functions of the variables (x, JI) which have the values (PI/?‘, l/c*) and (/I, b*) for the numerator and the denominator respectively. The form obtained for a(0) cannot be simplified unless B/p’ = 1 and b* = l/c*, which is impossible, according to (21). Thus, we obtain the “linear model” by another approach, which requires the simplest functions of 8. We can call it the simplest model. This immediately gives the reason for the lack of complete agreement between the “linear model” and some experimental results; a better fit may be obtained if we use polynomials with more terms. On the other hand it is easy to verify that if we take the classical values @ = + and 6 = 3, we immediately obtain the simplest functions, whatever the values of b and c may be. This explains why the linear model does not give a definite value of b in this case.
A GEOMETRICAL
INTERPRETATION
199
OF THE SCALING LAWS
4. Conclusion. We have shown from the scaling-law equation of state that critical paths exist with a particular property. Along those paths, the free energy and its derivatives have the same asymptotic behaviour, if one of them is chosen to express the others. We took M = aA*/aH as the “independent variable”, and all the thermodynamic quantities Fi have along the critical paths the asymptotic behaviour I;; cc M”‘(,u~depending on the quantity F,). The equation of these equivalent paths is H/ITl@ = Cte, and the critical isochore, the critical isotherm and the coexistence curve are among them. From this, we obtained the scaling laws. Inversely, if we assume that the critical point is characterized by the existence of equivalent paths, we obtain the scaling-law equation of state. This shows that this assumption is equivalent to the others made to obtain this equation of state (homogeneity hypothesis of Widom or cell model of Kadanoff). We have discussed the “linear model” of Schofield et ~2.~) and have shown that to get it, it is sufficient to impose the simplest functions that appear in the model. Thus it is not necessary, in order to find it, to give special importance to some equivalent paths, as was done by Schofield et al.
APPENDIX
Relation between b and c, and calculations of the functions of the “linear modell”. We start from the expressions (2), (3) and (4) and we calculate M = aA*/aH, S* = -aA*/aT, BM/BH, aM/aT, aS*/aT. For example we have
M
_
&._
aA* _ p* aH
-\
aT
ar ae
afi*q ae
I[ aH aT
ar ]/ \ ar ae
aH q ae
ar /
or
(
M = r,, b (8 + ‘)@ - Pit @ht’ - h’t ’
t’ =
dt
-ii-’
p’ =
-9,
et,.>.
In this manner we obtain m(e) = B (6 + l) Pt’ - Pft /3Ght’ - h’t ’
s(e) =
,6Shp’ - ,3 (6 + 1) h’p ,6Ght’ - h’t ’
-
(A.11 (A.21
In order to obtain the relation between h(B), t(e), m(e), s(e), we eliminate p and p’ between (A. 1) and (A.2). First we calculate p and p’ from (A. 1) and (A.2).
200
L. BENGUIGUI
1
[/36mh + st],
p = /3 (6 + 1) p’ = t’s + h’m.
64)
Then we differentiate (A.3) and the results equal (A.4). Thus we get the desired relation : P’st’ - s’t = j3dm’h - Bh’m,
[fi’ = j3 (6 + 1) -
11.
(A.5)
Now we substitute in (A.5) the chosen functions for h(O), t(O), m(O) and s(0) and arrive at
2 r1 - b2c2@‘] = /egu (6 - l),
(A.6)
2b2 (1 - B’) = pga (6 - 3),
from which we deduce 6-l -
--
6-3
l-/Y -.
j3’
We calculate the functions k(B), n(O) and a(O) as above. We obtain
k(B) =
-g [82b2 (1 - 2/!?) - l] 1 + (2b2@ - 3 - b2) O2 - b2 (286 - 3) O4 ’ -
8
-
83
n(e) = 1 + (2b2,%(1- 3 6) - b2) + O2 (6 - b2 3) (2/3S - 3) e4 ’ a(e) = pc2
The numerator can write
1 + O2 [(2/9/c”#I’)6 - 3 - b2] - (l/c’) (286 - 3) O4 1 + O2(2b2/?6 - 3 - b2) - b2 (286 - 3) O4
-
coefficients are analogous to those of the denominator,
v - e:(it
ei
fez- (mt w)i b2)1[e2 - e$u, wi
a(e) = pc2 v2 - eZ(p/it l/a
and we
’
Acknowledgement. The author wishes to express his gratitude to Professor J. de Boer of the University of Amsterdam and to Professor M. Revzen of the Technion for their encouragement, and to Professor P. Saphar of the Department of Mathematics, Technion, for his interest in this work.
A GEOMETRICAL
INTERPRETATION
OF THE SCALING
LAWS
REFERENCES 1) Kadanoff,
L.P., et al., Rev. mod. Phys. 39 (1967) 395.
Fisher, M.E., Rep. Progr. Phys. 30 (1967) 615. 2) Widom, B., J. them. Phys. 43 (1965) 3898. 3) Griffiths, R.B. and Wheeler, J., Phys. Rev. A3 (1970) 1047. 4) Schofield, P., Litster, J.D. and Ho, J.T., Phys. Rev. Letters 23 (1969) 1098. 5) Schofield, P., Phys. Rev. Letters 22 (1969) 606. 6) Griffiths, R.B., J. them. Phys. 43 (1965) 1958.
201