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Critical multiple correlations in the Landau model
Volume 71A, number 5,6
PHYSICS LETTERS
28 May 1979
CRITICAL MULTIPLE CORRELATIONS IN THE LANDAU MODEL LM. TKACHENKO’ Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA Received 1 March 1979
The investigation of three- and four-particle critical correlations in a scalar system is reported. The self-consistent field model is utilized. The differential equations and integral expressions for the corresponding correlation functions are obtained and partly analyzed.
Recently there has been much interest in the multiple molecular light scattering in fluids [1—6]. Since the intensity of the multiple scattering is connected with the multi-point correlation functions, and the observation of this effect far from critical points of -lifferent systems is essentially difficult, the problem investigating the multi-particle correlations near ~sof the second-order phase transition seems to o-date and interesting. present letter we wish to report the analysis ~a!multi-particle static correlations in spacesystems describable by the scalar order We utilize here the self-consistent field 1
characteristic of the multi-particle mi-invariants K,~5~(r 5~ using 0ther,11). static Imulants K~ fluctuation—dissipation
The self-consistent field approximation is applicable in the temperature range defined by the Levanyuk— Ginzburg inequality [9]
Gi~itI.
(3)
This provides us with the small parameter for the corre1!2 < I In eq. lation functions calculation: A (Gill ti) (3) Gi E T~b2/ac3is the Ginzburg number, t = (T— Tc)lTc is the dimensionless temperature, and T~ is the critical temperature. Parameters b, ~, and c are those of the Landau —Ginzburg hamiltonian: ~eq
{~}= ~o+ ~
f [c(V~)2
+ ai
t102 + ~b ~4] dr
.
(4)
Let us first consider the multiple correlations in the nonsymmetric phase, where = (ci~t~Ib)1/~ ~ 0 .
(5)
It is convenient now to introduce the dimensionless cumulants H 01 ,H012, ... by the relation ~sefunctions x0. inits. The connec-
H01 (~_1)(r0 r,11) = ~~11A K~(r0 mi)
.
(6)
‘e multi-point Then for the correlation functions we have: (2)
K2(r0,r)ç5~{1+AH01}, K3(r0,r1,r2)q5~1+A[H01J012+A2H012~,
(7)
Volume hA, number 5,6
PHYSICS LETTERS
etc., where [..j01...n denotes the sumof all different terms of the type given in the brackets with the permutation of points 0, 1 n; for example: [H01 ]012
=
H01 + H02 + 1112 = H01 + H20 + H21.
4, If we neglect now the corrections of the order A the behavioral information of the cumulants H012 and
H0123 will be enough for the calculation of all correlation functions K~with n ~ 4. Indeed, we then have: 01234 K5(r0,...,r4)Ø~{l+ [H01] + A2 [H 234+ A3 [H 01234 01H23]°’ 0123] +A3[H 1234} (8) 012(l+H34)]° .
Thus, our problem is reduced [see eqs.(1), (7) and (8)] to the investigation of the response functions x3 and X4 It can be shown [11] that in the presence of the infinitesimal harmonic external field h(r) = h(k) exp (ik. m), the field non-equilibrium value of the order parameter ~1~h is equal average to
E x~÷
(~(r0))h~
1(k1, ...,k~)
n =1
U h(k1)exp (ik1
with K being the inverse correlation length equal to
i~ =
112. Inserting the Fourier-transform of eq. (11) (2aItI/c) into theorem (1) and utilizing eq. (6), we obtain the integral expression for the cumulant H 012. If we differentiate this expression twice with respect to r0, we produce the following equation for H
2H ~H012
—
g
012:
2H 012 = 3K
01H02
(12)
,
where HQl(ro,rl)=(23I27rKIrl~_roI)_1exp(_KIrl_roI). The formal solution of eq. (12) is H 012
=
(13)
3(2_h/2)K3fdr3H03H13H23.
The same procedure can be carried out in the 4-point casetogive: 110123 2H— K~H0123 = 3~ 01H0~H03+ H 112)K3 0123 = 3(2
2 3K
[H
123 ,
(14)
012H03] (15)
Xfdr4{H04H14H24H34+H04[H412H43]123}.
a
X
28 May 1979
r0),
(9)
The analysis of these expressions is a complicated
problem, and we report here only results of most interest for the light scattering theory [7]
where the nth order susceptibility x~(ki, k,~_i) is the Fourier-transform of the corresponding response function x~(ro,r1, r0_j). On the other hand,the ...,
...,
equilibrium configuration of the order parameter field in the presence of the infinitesimal field h (r) can be
obtained by the minimization of the functional t~~} (eq. 4) with the external-field term —fhcbdr added:
(i) jf r2 = 0, ~r1 ~ 1, r1 ~ r0, ~r0 1, the main term of the asymptotic of the 2) form exp(—ic r cumulant H012 is —(3 ln(3)/l6ir 0)/K r0 (16) ~-
(ii) in the case that ~r0 ~ 1, r2 c~r0, r1 — m0 I ~ r0, = 0, the main term of the cumulant H0123 asymptotic expansion is proportional to
Kr1 ~ 1, r3
2 exp (—2Km
—fh~dr~
~eq{ø} =
+
—
o~tlØ+
—
h = 0.
(Kr)
(10)
Substituting the expansion (9) into the Euler equation (10), we express the quadratic susceptibility x3(k1, k2) in terms of the linear susceptibility ~2(k): —3b00x2(k1)x2(k2) x2(k0) (1U where = —(k1 + k2). It can be easily found by the linearization of eq. (10), that [8] k2)
=
2+ =
c’(K
k2)1
0) [c + ln(4Kr0)]
where c is Euler’s constant. In the temperature range T~’T~the equilibrium value of the order parameter is øo = 0 (and K = 0), which makes all odd semi-invariants equal to zero. Examination of eqs. (7), (l4)—(l7) shows that the cumulant K~S)is not equal to zero on both sides of the transition point; therefore, the gaussian approximation is inapplicable
even in the symmetric phase. We hope that our results will prove to be useful for the theory of the multiple light scattering in liquids. 439
Volume 71A, number 5,6
PHYSICS LETTERS
It is a pleasure to acknowledge helpful discussions with Professor I.Z. Fisher who originally suggested this line of research to the author. The author would also like to thank the University of Maryland for its hospitality.
References Lii H.L. Frish and J. McKenna, Phys. Rev. 139 (1965) A68. [2] Ye.L. Lakoza and A.V. Chaly, Zh. Eksp. Teor. Fiz. 72 (1977) 875 (Soy. Phys. JETP 45 (1978) No. 3). [31 D.W. Oxtoby and W.M. Gelbart, Phys. Rev. AlO (1974) 738; J. Chem. Phys. 60 (1974) 3359. [4] V.L. Kuzmin, Opt. Spektrosk. 39(1975)546 (Opt. Spectrosc. 39 (1975) 306).
440
28 May 1979
[5] H.H.J. Boots et al., Physica 79A (1975) 397; 84A (1976) 217; 87A (1977) 185. [6] N.J. Trappenierset al., Chem. Phys. Lett. 34 (1975) 192; 48 (1977) 31. [7] (1979) I.M. Tkachenko andJETP I.Z. 50 Fisher, Zh. Eksp. Fiz. 77 (Soy. Phys. (1980)), to be Teor. published. [8] A.Z. Patashinsky and V.L. Pokrovsky, Fluktuatsionnaya teoriya fazovykh perekhodov (Fluctuation theory of phase transitions) (Nauka, 1975). [9] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, 1978) p. 1. [101 V.L. Leonov and A.N. Shiryaev, Teor. Veroyatn. Ee Primen. 4 (1959) 342 (Theory Probab. Its Appl. 4 (1959) 319). [111 I.M. Tkachenko and I.Z. Fisher, Zh. Eksp. Teor. Fiz. 69 (1975) 1092 (Soy. Phys. JETP 42 (1976) 556).
PHYSICS LETTERS
28 May 1979
CRITICAL MULTIPLE CORRELATIONS IN THE LANDAU MODEL LM. TKACHENKO’ Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA Received 1 March 1979
The investigation of three- and four-particle critical correlations in a scalar system is reported. The self-consistent field model is utilized. The differential equations and integral expressions for the corresponding correlation functions are obtained and partly analyzed.
Recently there has been much interest in the multiple molecular light scattering in fluids [1—6]. Since the intensity of the multiple scattering is connected with the multi-point correlation functions, and the observation of this effect far from critical points of -lifferent systems is essentially difficult, the problem investigating the multi-particle correlations near ~sof the second-order phase transition seems to o-date and interesting. present letter we wish to report the analysis ~a!multi-particle static correlations in spacesystems describable by the scalar order We utilize here the self-consistent field 1
characteristic of the multi-particle mi-invariants K,~5~(r 5~ using 0ther,11). static Imulants K~ fluctuation—dissipation
The self-consistent field approximation is applicable in the temperature range defined by the Levanyuk— Ginzburg inequality [9]
Gi~itI.
(3)
This provides us with the small parameter for the corre1!2 < I In eq. lation functions calculation: A (Gill ti) (3) Gi E T~b2/ac3is the Ginzburg number, t = (T— Tc)lTc is the dimensionless temperature, and T~ is the critical temperature. Parameters b, ~, and c are those of the Landau —Ginzburg hamiltonian: ~eq
{~}= ~o+ ~
f [c(V~)2
+ ai
t102 + ~b ~4] dr
.
(4)
Let us first consider the multiple correlations in the nonsymmetric phase, where = (ci~t~Ib)1/~ ~ 0 .
(5)
It is convenient now to introduce the dimensionless cumulants H 01 ,H012, ... by the relation ~sefunctions x0. inits. The connec-
H01 (~_1)(r0 r,11) = ~~11A K~(r0 mi)
.
(6)
‘e multi-point Then for the correlation functions we have: (2)
K2(r0,r)ç5~{1+AH01}, K3(r0,r1,r2)q5~1+A[H01J012+A2H012~,
(7)
Volume hA, number 5,6
PHYSICS LETTERS
etc., where [..j01...n denotes the sumof all different terms of the type given in the brackets with the permutation of points 0, 1 n; for example: [H01 ]012
=
H01 + H02 + 1112 = H01 + H20 + H21.
4, If we neglect now the corrections of the order A the behavioral information of the cumulants H012 and
H0123 will be enough for the calculation of all correlation functions K~with n ~ 4. Indeed, we then have: 01234 K5(r0,...,r4)Ø~{l+ [H01] + A2 [H 234+ A3 [H 01234 01H23]°’ 0123] +A3[H 1234} (8) 012(l+H34)]° .
Thus, our problem is reduced [see eqs.(1), (7) and (8)] to the investigation of the response functions x3 and X4 It can be shown [11] that in the presence of the infinitesimal harmonic external field h(r) = h(k) exp (ik. m), the field non-equilibrium value of the order parameter ~1~h is equal average to
E x~÷
(~(r0))h~
1(k1, ...,k~)
n =1
U h(k1)exp (ik1
with K being the inverse correlation length equal to
i~ =
112. Inserting the Fourier-transform of eq. (11) (2aItI/c) into theorem (1) and utilizing eq. (6), we obtain the integral expression for the cumulant H 012. If we differentiate this expression twice with respect to r0, we produce the following equation for H
2H ~H012
—
g
012:
2H 012 = 3K
01H02
(12)
,
where HQl(ro,rl)=(23I27rKIrl~_roI)_1exp(_KIrl_roI). The formal solution of eq. (12) is H 012
=
(13)
3(2_h/2)K3fdr3H03H13H23.
The same procedure can be carried out in the 4-point casetogive: 110123 2H— K~H0123 = 3~ 01H0~H03+ H 112)K3 0123 = 3(2
2 3K
[H
123 ,
(14)
012H03] (15)
Xfdr4{H04H14H24H34+H04[H412H43]123}.
a
X
28 May 1979
r0),
(9)
The analysis of these expressions is a complicated
problem, and we report here only results of most interest for the light scattering theory [7]
where the nth order susceptibility x~(ki, k,~_i) is the Fourier-transform of the corresponding response function x~(ro,r1, r0_j). On the other hand,the ...,
...,
equilibrium configuration of the order parameter field in the presence of the infinitesimal field h (r) can be
obtained by the minimization of the functional t~~} (eq. 4) with the external-field term —fhcbdr added:
(i) jf r2 = 0, ~r1 ~ 1, r1 ~ r0, ~r0 1, the main term of the asymptotic of the 2) form exp(—ic r cumulant H012 is —(3 ln(3)/l6ir 0)/K r0 (16) ~-
(ii) in the case that ~r0 ~ 1, r2 c~r0, r1 — m0 I ~ r0, = 0, the main term of the cumulant H0123 asymptotic expansion is proportional to
Kr1 ~ 1, r3
2 exp (—2Km
—fh~dr~
~eq{ø} =
+
—
o~tlØ+
—
h = 0.
(Kr)
(10)
Substituting the expansion (9) into the Euler equation (10), we express the quadratic susceptibility x3(k1, k2) in terms of the linear susceptibility ~2(k): —3b00x2(k1)x2(k2) x2(k0) (1U where = —(k1 + k2). It can be easily found by the linearization of eq. (10), that [8] k2)
=
2+ =
c’(K
k2)1
0) [c + ln(4Kr0)]
where c is Euler’s constant. In the temperature range T~’T~the equilibrium value of the order parameter is øo = 0 (and K = 0), which makes all odd semi-invariants equal to zero. Examination of eqs. (7), (l4)—(l7) shows that the cumulant K~S)is not equal to zero on both sides of the transition point; therefore, the gaussian approximation is inapplicable
even in the symmetric phase. We hope that our results will prove to be useful for the theory of the multiple light scattering in liquids. 439
Volume 71A, number 5,6
PHYSICS LETTERS
It is a pleasure to acknowledge helpful discussions with Professor I.Z. Fisher who originally suggested this line of research to the author. The author would also like to thank the University of Maryland for its hospitality.
References Lii H.L. Frish and J. McKenna, Phys. Rev. 139 (1965) A68. [2] Ye.L. Lakoza and A.V. Chaly, Zh. Eksp. Teor. Fiz. 72 (1977) 875 (Soy. Phys. JETP 45 (1978) No. 3). [31 D.W. Oxtoby and W.M. Gelbart, Phys. Rev. AlO (1974) 738; J. Chem. Phys. 60 (1974) 3359. [4] V.L. Kuzmin, Opt. Spektrosk. 39(1975)546 (Opt. Spectrosc. 39 (1975) 306).
440
28 May 1979
[5] H.H.J. Boots et al., Physica 79A (1975) 397; 84A (1976) 217; 87A (1977) 185. [6] N.J. Trappenierset al., Chem. Phys. Lett. 34 (1975) 192; 48 (1977) 31. [7] (1979) I.M. Tkachenko andJETP I.Z. 50 Fisher, Zh. Eksp. Fiz. 77 (Soy. Phys. (1980)), to be Teor. published. [8] A.Z. Patashinsky and V.L. Pokrovsky, Fluktuatsionnaya teoriya fazovykh perekhodov (Fluctuation theory of phase transitions) (Nauka, 1975). [9] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, 1978) p. 1. [101 V.L. Leonov and A.N. Shiryaev, Teor. Veroyatn. Ee Primen. 4 (1959) 342 (Theory Probab. Its Appl. 4 (1959) 319). [111 I.M. Tkachenko and I.Z. Fisher, Zh. Eksp. Teor. Fiz. 69 (1975) 1092 (Soy. Phys. JETP 42 (1976) 556).