- Email: [email protected]
An exact expression for the critical exponent η
PHYSICS LETTERS
Volume 42A, number 1
AN
6 November 1972
EXACT EXPRESSION FOR THE CRITICAL EXPONENT 7 MS. GREEN*
and J.D. GUNTON*
Temple University, Philadelphia, Pennsylvania 19122,
USA
Received 25 September 1972 An exact expression is derived for the exponent TJfor classical fluids in terms of an integral which involves only the two body potential, the critical point value of p/kgT and elements of the principal null vector.
In this paper an exact formula is derived for classical fluids for the exponent 7~which governs the long-range behavior of the pair correlation function at the critical point. The exponent is given in terms of an integral of a function which involves only the two body potential, the critical point value of fl p (where /3 = l/k, T and p is the pressure) and the second and third elements of the principal null vector A. This null vector has been shown previously to be significant in a number of ways in the theory of critical phenomena [ 1,2]. The expression for n provides in principle the basis for a systematic determination of its value, through approximation methods valid for short-range properties of the distribution functions. Since the expression explicitly involves the potential the question of the validity of universality [3] is raised. The basis of our analysis is to note that a certain kernel K( 1,l’) which is defined below and whose zeroth moment at the critical point directly yields n can be expressed in terms of the two body intermolecular potential V(i,j),Pp, and two elements of the principal null vector A( [n] ), where [n] signifies the coordinates of a set of n molecules. Our starting point is a set of equations derived by Schofield [4] for the equilibrium molecular distribution functions f( [n] ) which simply state that an infinitesimal scale change in the potential v(R)+ V(R( 1 +h)) = V(R) + dhR * V Vimplies the same scale change in the distribution functions. These equations can be written, assuming single body and pair potentials, in the compact matrix form 2 +Uml m=l m!
r([nl,[ml){~([ml)+A~([ml)}
where the only non-zero elements of the vector @ are a(i) = 1 and @ (9) = - Qo(g) = - $ P&i dv(Rii)/Wi. The quantity /3p can be expressed m terms of @ for a uniform system through the equation PP = P - $j-df2 f(12)QoU2h
(2)
where p is the density, with p =f(i). The vector A@ represents the effect of a small, external single particle potential, u(i), with the only non-zero element being A@(i) = J ri.Vi(-pU(i)). The matrix elements of r( [n] , [m] ) have been given in general in ref. [ 11. For our present discussion only those for n equals 1 and m equal 1 or 2 are needed. These are fYJ,2) l-‘(1,23)
= f(J2)
-f(l)f(2)
= f(123)
+f(1)6(1,2)
(3)
-f(l)f(23)+./(23)[6(1,3)+6(1,2)]
The critical point can be characterized by the statement that the vector @ is orthogonal to the principal null vector A, whose elements can be defined as [2] A( [n-l) = (af( [n] )/ap),, since it follows from eq. (2) that 1 - $WQoWW2
= (WdWp,
(4)
where the inverse isothermal compressibility p(ap/ap)p vanishes at the critical point. We now consider the change in the local potential, 6u(2), which corresponds to a small change in the local density, 6p( 1’). As is well known, this change is governed by the behavior of the direct correlation function f(2,l’) = (6 (-fl1((2))/Sp (l’))@ which is the inverse of r(i,2), i.e. dr, r(i,2)t(2,i’) = 6(1,1’). (9 s We consider the relation of this change to the corresponding change in the molecular distribution functions for the case n equals 1. Eq. (1) becomes
* Supported by N.S.F. Grant #550-301-01.
7
Volume
42A. number
PHYSICS
1
(5-v) dr2 r‘(1,2) - iJJdrZdr3 1 = 3
r1 .v,f(l)
+f(l>
r(1,23)QO(23) - sdr2
= $rl.vls(l,l’)
-fJdrZ
r(1,2)r2
(6)
rO,WW).
+6(1,1’) .V2t(2,1’)
(7)
where the kernel K( 1,l’) is given by the expression 2K(l,l’)
(8)
Eq. (7) can be put in a more convenient form by substituting eq. (5) for 6 (1,l’) in the term involving V, 6(1, l’), integrating by parts in the last term in eq. (7) and noting that V2r(1,2) = - V,r(1,2) since I’( 1,2) is a function only of the distance Ra = RI - R2. We obtain the relation = 2S(l,l’)+fj-d~2Rlz-V12r(l,2)r(2,1’) (9) or, in Fourier transform i(k)
= i(l-3
language,
(k.V&k))/i’(k)).
(10)
Near the critical point, for small values of the wave number k, the expected behavior of F(k) is I’(k) - k -(2-q)go(kt), where t is the correlation length and where g,$k’) approaches unity for k’large and approaches k’2--71 for k’ small. Thus i(k)
= t {(5 - q) - k . Vklog g,,(k.$)} .
(11)
The second term approaches zero in the limit t + m and k + 0 (in that order! *) and we may write lim lim i(k) k-0 [-+m
= k (5 -7)).
(12)
Upon integration over the coordinate 1’ in eq. (8) using the definitions for r, eq. (3), and for A, and the relationship eq. (2) valid at the critical point we obtain * The limit taken in the opposite to be equal to %.
+j-drl(Ac(l2)
t
order easily can be shown
1972
(13) -
f~dqQ,,(WA,(W
-P&.
Eq. (13) is the expression for q in terms of the interaction potential and the principal null vector, where all the terms in eq. (13) are the critical point values. We note that the integrand in eq. (13) approaches zero for Ru large. This can be shown from an identity for the asymptotic behavior of A,( 123) when particle 1 is far from particle 2 and 3, namely A,(l23)
=Jdr,@r(1,2)/Sp(l’)
- ; j-d+'(1,2W~(~'))Q&W~.
2K(l,l’)
= 3(-l
=
Taking the functional derivative (6/6p( l’))p of eq. (6) and setting u(2) equal to zero we obtain the relation 2K(l,l’)
6 November
LETTERS
=f,(l)A,(23)+
+A,(l)f,(23)
+(A,(l2)
- 2f,(l))A,(23),
(14)
which can be derived ** using the concepts of operator algebra [5]. We can conclude upon performing the integral over coordinate 3 in eq. (13), using eqs. (14), (2) and (4) that the integrand approaches zero, as it must. We believe that the principle significance of our expression eq. (13) is that it relates the exponent 17to short-range properties of the null vector and therefore of the distribution functions. Since the critical singularities are mainly manifest in the long-range behavior of these functions one might hope that simple approximation methods applied to the short-range part will yield a useful result for rl. We also note that the formula explicity contains the intermolecular potential as well as potential sensitive parts of the distribution function. It thus raises the question as to whether exponents depend on the potential, i.e. the question of the validity of universality. ** This formula can be derived by applying
the concepts of operator algebra first to the product ~1~2~3 and then successively to the products p2p3 and PIPI’, where 4 is the instantaneous value of the local density at a point i and 1’ is a point in the neighborhood of 2 and 3. Only terms involving the principal null vector are kept.
References
[l] MS. Green, J. Math. Phys. 9 (1968) 875. [2] J.D. Gunton and M.S. Green, Phys. Rev. A4 (1971) 1282. [3] L.P. Kadanoff, Proc. Enrico Fermi Summer School of Physics Varenna, 1970, Course on Critical phenomena (Academic Press, New York, to be published). [4] P. Schofield, Proc. Phys. Sot. 88 (1966) 149. [S] L.P. Kadanoff,
Phys. Rev. Lett. 23 (1969)
1430.
Volume 42A, number 1
AN
6 November 1972
EXACT EXPRESSION FOR THE CRITICAL EXPONENT 7 MS. GREEN*
and J.D. GUNTON*
Temple University, Philadelphia, Pennsylvania 19122,
USA
Received 25 September 1972 An exact expression is derived for the exponent TJfor classical fluids in terms of an integral which involves only the two body potential, the critical point value of p/kgT and elements of the principal null vector.
In this paper an exact formula is derived for classical fluids for the exponent 7~which governs the long-range behavior of the pair correlation function at the critical point. The exponent is given in terms of an integral of a function which involves only the two body potential, the critical point value of fl p (where /3 = l/k, T and p is the pressure) and the second and third elements of the principal null vector A. This null vector has been shown previously to be significant in a number of ways in the theory of critical phenomena [ 1,2]. The expression for n provides in principle the basis for a systematic determination of its value, through approximation methods valid for short-range properties of the distribution functions. Since the expression explicitly involves the potential the question of the validity of universality [3] is raised. The basis of our analysis is to note that a certain kernel K( 1,l’) which is defined below and whose zeroth moment at the critical point directly yields n can be expressed in terms of the two body intermolecular potential V(i,j),Pp, and two elements of the principal null vector A( [n] ), where [n] signifies the coordinates of a set of n molecules. Our starting point is a set of equations derived by Schofield [4] for the equilibrium molecular distribution functions f( [n] ) which simply state that an infinitesimal scale change in the potential v(R)+ V(R( 1 +h)) = V(R) + dhR * V Vimplies the same scale change in the distribution functions. These equations can be written, assuming single body and pair potentials, in the compact matrix form 2 +Uml m=l m!
r([nl,[ml){~([ml)+A~([ml)}
where the only non-zero elements of the vector @ are a(i) = 1 and @ (9) = - Qo(g) = - $ P&i dv(Rii)/Wi. The quantity /3p can be expressed m terms of @ for a uniform system through the equation PP = P - $j-df2 f(12)QoU2h
(2)
where p is the density, with p =f(i). The vector A@ represents the effect of a small, external single particle potential, u(i), with the only non-zero element being A@(i) = J ri.Vi(-pU(i)). The matrix elements of r( [n] , [m] ) have been given in general in ref. [ 11. For our present discussion only those for n equals 1 and m equal 1 or 2 are needed. These are fYJ,2) l-‘(1,23)
= f(J2)
-f(l)f(2)
= f(123)
+f(1)6(1,2)
(3)
-f(l)f(23)+./(23)[6(1,3)+6(1,2)]
The critical point can be characterized by the statement that the vector @ is orthogonal to the principal null vector A, whose elements can be defined as [2] A( [n-l) = (af( [n] )/ap),, since it follows from eq. (2) that 1 - $WQoWW2
= (WdWp,
(4)
where the inverse isothermal compressibility p(ap/ap)p vanishes at the critical point. We now consider the change in the local potential, 6u(2), which corresponds to a small change in the local density, 6p( 1’). As is well known, this change is governed by the behavior of the direct correlation function f(2,l’) = (6 (-fl1((2))/Sp (l’))@ which is the inverse of r(i,2), i.e. dr, r(i,2)t(2,i’) = 6(1,1’). (9 s We consider the relation of this change to the corresponding change in the molecular distribution functions for the case n equals 1. Eq. (1) becomes
* Supported by N.S.F. Grant #550-301-01.
7
Volume
42A. number
PHYSICS
1
(5-v) dr2 r‘(1,2) - iJJdrZdr3 1 = 3
r1 .v,f(l)
+f(l>
r(1,23)QO(23) - sdr2
= $rl.vls(l,l’)
-fJdrZ
r(1,2)r2
(6)
rO,WW).
+6(1,1’) .V2t(2,1’)
(7)
where the kernel K( 1,l’) is given by the expression 2K(l,l’)
(8)
Eq. (7) can be put in a more convenient form by substituting eq. (5) for 6 (1,l’) in the term involving V, 6(1, l’), integrating by parts in the last term in eq. (7) and noting that V2r(1,2) = - V,r(1,2) since I’( 1,2) is a function only of the distance Ra = RI - R2. We obtain the relation = 2S(l,l’)+fj-d~2Rlz-V12r(l,2)r(2,1’) (9) or, in Fourier transform i(k)
= i(l-3
language,
(k.V&k))/i’(k)).
(10)
Near the critical point, for small values of the wave number k, the expected behavior of F(k) is I’(k) - k -(2-q)go(kt), where t is the correlation length and where g,$k’) approaches unity for k’large and approaches k’2--71 for k’ small. Thus i(k)
= t {(5 - q) - k . Vklog g,,(k.$)} .
(11)
The second term approaches zero in the limit t + m and k + 0 (in that order! *) and we may write lim lim i(k) k-0 [-+m
= k (5 -7)).
(12)
Upon integration over the coordinate 1’ in eq. (8) using the definitions for r, eq. (3), and for A, and the relationship eq. (2) valid at the critical point we obtain * The limit taken in the opposite to be equal to %.
+j-drl(Ac(l2)
t
order easily can be shown
1972
(13) -
f~dqQ,,(WA,(W
-P&.
Eq. (13) is the expression for q in terms of the interaction potential and the principal null vector, where all the terms in eq. (13) are the critical point values. We note that the integrand in eq. (13) approaches zero for Ru large. This can be shown from an identity for the asymptotic behavior of A,( 123) when particle 1 is far from particle 2 and 3, namely A,(l23)
=Jdr,@r(1,2)/Sp(l’)
- ; j-d+'(1,2W~(~'))Q&W~.
2K(l,l’)
= 3(-l
=
Taking the functional derivative (6/6p( l’))p of eq. (6) and setting u(2) equal to zero we obtain the relation 2K(l,l’)
6 November
LETTERS
=f,(l)A,(23)+
+A,(l)f,(23)
+(A,(l2)
- 2f,(l))A,(23),
(14)
which can be derived ** using the concepts of operator algebra [5]. We can conclude upon performing the integral over coordinate 3 in eq. (13), using eqs. (14), (2) and (4) that the integrand approaches zero, as it must. We believe that the principle significance of our expression eq. (13) is that it relates the exponent 17to short-range properties of the null vector and therefore of the distribution functions. Since the critical singularities are mainly manifest in the long-range behavior of these functions one might hope that simple approximation methods applied to the short-range part will yield a useful result for rl. We also note that the formula explicity contains the intermolecular potential as well as potential sensitive parts of the distribution function. It thus raises the question as to whether exponents depend on the potential, i.e. the question of the validity of universality. ** This formula can be derived by applying
the concepts of operator algebra first to the product ~1~2~3 and then successively to the products p2p3 and PIPI’, where 4 is the instantaneous value of the local density at a point i and 1’ is a point in the neighborhood of 2 and 3. Only terms involving the principal null vector are kept.
References
[l] MS. Green, J. Math. Phys. 9 (1968) 875. [2] J.D. Gunton and M.S. Green, Phys. Rev. A4 (1971) 1282. [3] L.P. Kadanoff, Proc. Enrico Fermi Summer School of Physics Varenna, 1970, Course on Critical phenomena (Academic Press, New York, to be published). [4] P. Schofield, Proc. Phys. Sot. 88 (1966) 149. [S] L.P. Kadanoff,
Phys. Rev. Lett. 23 (1969)
1430.