Generalized hydrodynamics of binary fluids

Physica A 264 (1999) 15–39 Generalized hydrodynamics of binary uids Kunimasa Miyazakia;∗ , Kazuo Kitaharab a Department of Physical Chemistry, National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305-8565, Japan b Department of Applied Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8550, Japan Received 12 May 1998 Abstract The k-dependent hydrodynamic uctuations for binary classical uids are discussed. The aim of this paper is to give the microscopic basis for the phenomenological formulation of hydrodynamics of binary uids which was explored by the present author. The generalized Langevin equation formalism is used to derive closed equations for the uctuations of temperature, mass, relative velocity, etc. Explicit expressions for the k-dependent thermodynamic susceptibilities are also derived. Using these quantities, it is shown that the equations for uctuations reduce to the macroscopic equations in the long wavelength limit. c 1999 Elsevier Science B.V. All rights reserved. PACS: 05.70.Ln Keywords: Hydrodynamic uctuations; Binary uids 1. Introduction In the previous paper (hereafter to be referred to as paper I) [1], we have generalized conventional formalism of nonequilibrium thermodynamics of multicomponent uids to the shorter time scale where inertial e ect of the di usional ow of each component becomes important. There, the di usional ow is introduced as a new local extensive variable by de ning the internal energy density as the part of energy, which does not include the kinetic energy of macroscopic ows. The balance equation for an extensive variable ai (t) has a general form as follows: X @S @S @ai X Lij + ; (1.1) {ai ; aj } = @t @aj @aj j j Corresponding author. Present address: Theoretical Studies Division, Institute for Molecular Science, Myoudaiji, Okazaki 444-8585, Japan. Fax: +81-564-53-4660; e-mail: [email protected]. ∗ 0378-4371/99/$ – see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 4 4 6 - 4 16 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 where @S=@ai is the intensive parameter conjugate to ai and Lij is the generalized Onsager coecient which has an operator form. Here, we have made use of discretized variables rather than continuous eld variables for simpli cation. The rst term of the right-hand side is the reversible part and the second is the irreversible part. The coecient matrix {ai ; aj } has to be antisymmetric, namely {ai ; aj } = −{aj ; ai }, so that the reversible terms do not contribute to the entropy production of the whole system. The balance equation for the di usional ow which has not been known except for a dilute case was thus deduced from the known equations for the total energy, mass density, and momentum density by using the antisymmetric nature of {ai ; aj }. However, there still remains some arbitrariness in the division of balance equations into the reversible part and irreversible part. The Onsager coecients satisfy the well-known reciprocal relation Lij = i j Lji , where i = 1 if ai is the even function under time reversal transformation and i = −1 otherwise. More precisely, if the Onsager coecients depend on a parameter B such as the external magnetic eld or the velocity eld which are odd in the time reversion, we have Lij (B) = i j Lji (−B). If one variable is even and another is odd in the time reversion and Lij does not depend on B, the Onsager coecient becomes antisymmetric, i.e., Lij = −Lji . The coecient thus has the same symmetry as {ai ; aj } and the term with this symmetry does not contribute to the entropy production, i.e., X @S @S @S @S dS X = Lij = ; (1.2) Lij dt @a @a @a i j i @aj i; j i; j; i j =1 P where i; j; i j =1 denotes the summation over i; j with the same time reversal symmetry, i.e., i j = 1. This implies that one may divide the reversible part and irreversible part arbitrarily for the contribution from the variables with the di erent time reversal symmetry. This arbitrariness may be avoided by going beyond phenomenological level of description and resorting to microscopic description. According to the idea of linear response theory, the Onsager coecients are related to thermal uctuations of ows in the equilibrium state via the uctuation–dissipation theorem. One expects that this theorem will avoid such arbitrariness in the division of reversible part and irreversible part in the macroscopic evolution laws. The primary purpose of this paper is to give microscopic basis of the hydrodynamic equations derived in paper I which was based on a purely phenomenological level of description. For this purpose, we shall consider the uctuations of thermodynamic variables around equilibrium state. Wavelength-dependence of the uctuations is fully taken into account. We shall focus here on binary classical uids which consist of two types of atoms or molecules. The closed equations for the time-dependent correlation functions of the microscopic uctuations of the mass concentration ratio, mass density, temperature, barycentric velocity, and relative velocity of each component are derived by using the generalized Langevin equation formalism [2,3]. The equations thus derived are expressed in terms of the k-dependent static correlation functions and memory kernels. Another purpose of this paper is, though supplementary, to give the microscopic expressions for k-dependent thermodynamic susceptibilities for binary K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 17 uids. It will be shown that the static correlation functions appearing in the generalized Langevin equations are expressed in terms of k-dependent thermodynamic susceptibilities. We shall show that, in the hydrodynamic limit, i.e., in the long wavelength and long time limit, the generalized Langevin equations have the identical form as those derived from hydrodynamical equations derived in paper I. The arbitrariness in phenomenological equations discussed above is, thus, clari ed by this procedure. Such a procedure was carried out for simple one-component uids [3,4]. Extrapolation of the hydrodynamic description into the short wavelength or k-dependent regime was referred to as the generalized hydrodynamics. The analysis in this paper is much along the line of these works for one-component uids. We shall also give formal expressions for the Onsager coecients. It is generally known that the generalized Langevin formalism gives the Onsager coecients in terms of the time correlation functions of the random forces whose dynamics are characterized by the “modi ed” Hamiltonian and therefore they are di erent from those which are given by the Green–Kubo formulae. The latter give the Onsager coecients in terms of the same type of time correlation functions of random forces whose dynamics are characterized by the total Hamiltonian. For one-component uids, the Onsager coecients derived from the generalized Langevin formalism reduce to those of the Green–Kubo integrands in the hydrodynamic limit since dynamic variables chosen are all conserved quantities (see for example Refs. [3]). In our problem, however, a non-conserved variable, the relative velocity, is incorporated as one of thermodynamic variables. This leads to discrepancy between correlation functions of the di erent formalisms. This paper is organized as follows. In the remainder of this section, we summarize the results of paper I. In Section 2, we introduce the proper microscopic expressions for the uctuations of the internal energy density, mass density, momentum, and the relative velocity and give their equations of motion. The k-dependent uctuations of temperature, pressure, and chemical potential are introduced here. This generalization is implemented such that they reduce to the macroscopic thermodynamic relations in the small-k limit. Using these generalized thermodynamic quantities, k-dependent thermodynamic susceptibilities such as the compressibility or speci c heat are derived in terms of the static correlation functions of uctuations of the thermodynamic variables. In Section 3, the equations for the thermodynamic variables are derived by using the generalized Langevin formalism. Explicit expressions for coecient matrices for the reversible parts and irreversible parts of the balance equations are shown. Section 4 is devoted to conclusions. The content of this paper was brie y reported in the Second Tohwa University International Meeting on Statistical Physics [5]. Let us summarize here results given in paper I [1]. We have shown that the equation for the set of local extensive variables, the total energy density e, mass density for the lth component l (l = 1; 2), total momentum, C ≡ 1 C1 + 2 C2 , where Cl is the velocity of the lth component, and di usional momentum for component 1, J1 ≡ 1 (C1 − C), 18 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 are given by     1 @ 2 (e) = −∇ · eC + P · C +  + (1 − 2c)! J1 + Jq + C ·  ; @t 2 (1.3) @ l + ∇ · (l Cl ) = 0; @t (1.4) (l = 1; 2) ; @ (C) + ∇ · (CC) = −∇ · P − ∇ ·  ; @t (1.5) @ J1 + C · ∇J1 + J1 ∇ · C + J1 · ∇C + (1 − c)∇ · {(1 − c)!J1 } − c∇ · {c!J 1 } @t       1 @J1  + ∇ + ; (1.6) =c(1 − c)T ∇ − T T @t irr where c ≡ 1 = is the mass concentration ratio for component 1 and ! = C1 − C2 is the relative velocity between the two components. In the above expressions, P ≡ p + c(1 − c)! ! is the generalized hydrostatic pressure, and  = 1 − 2 is the chemical potential difference between two components.  appearing in Eqs. (1.3) and (1.6) is an arbitrary quantity which is not speci ed within the phenomenological level of description. Jq ; ; and (@J1 =@t)irr are the irreversible parts of equations which are referred to as the irreversible heat ow, the viscous tensor, and the friction force, respectively. Note that the equation for another di usional momentum J2 is redundant since J1 = −J2 by its de nition. In the framework of linear thermodynamics, the irreversible parts are expressed as the linear combination of thermodynamic forces such as the temperature gradient, the velocity gradient, and the relative velocity. We may consider the following phenomenological expressions for the irreversible part as a typical example:   @v @v 2 + −  ∇ · C −  ∇ · C ;  = − @x @x 3 Jq = L00 ∇  @J1 @t     ! 1 + L01 − ; T T    ! 1 ; + L11 − = L10 ∇ T T irr (1.7) where  and  are the shear viscosity and the bulk viscosity, respectively. L00 is the coecient which is related to heat conduction and L11 is the friction coecient. L10 and L01 are the cross coecients which are related to thermal di usion. Onsager reciprocal relation and time-reversion symmetry yields L10 = −L01 . In view of Eq. (1.3), one K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 19 might expect that the heat ow carried by the mass di usion of constituents is given by 2 X {hl + 21 (Cl − C)2 }Jl = {h1 − h2 + 12 (1 − 2c)! 2 }J1 ; (1.8) l=1 where hl (l = 1; 2) is the molar enthalpy and, thus,  should be the di erence of the molar enthalpy for each component h1 − h2 as it was suggested by several authors [6,7] in the framework of conventional nonequilibrium thermodynamics. It is found that it is not the case when the faster time scales are concerned and the di usional ow is taken as a thermodynamic variable rather than an irreversible ow. This point will be discussed in Sections 3 and 4. For later use, we shall give the linearized equations of Eqs. (1.3) –(1.6) around equilibrium state. It is convenient to choose the mass concentration ratio, total mass density, temperature, longitudinal parts of barycentric velocity and relative velocity as independent variables and introduce the Laplace and Fourier transformed variables by Z ∞ Z dt dr e−zt eik·r A(r; t) : A(k; z) = 0 Then, we have a set of equations in the following form: (z − i + ) · a(k; z) = ã(k) ; (1.9) where a(k; z) = (c; ; T; k̂ · C; k̂ · !) with k̂ ≡ k=k. ã(k) = a(k; t = 0) is the initial value of the variable. We shall not consider the transverse parts of C and ! since they are independent of the dynamics of other variables. i · a stands for the reversible part of the equations whose coecient matrix is given by   0 0 0 0 i c!  0 0 0 i v 0     (1.10) i ≡ 0 0 0 i Tv i T!  ;  i vc i v i vT 0 0  0 0 i !c i ! i !T where each element is given by i c! = ikc(1 − c) ; i v = ik ; T H ; i T! = ikc(1 − c) ; cv T cv   1 @ ; i v = ik 2 ; i v T = ik ; i vc = ik @ T; c  T T     @ H @ ; i ! = ik ; i !T = ik ; i !c = ik @c T;  @ T; c T i Tv = ik (1.11) 20 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 where cv = (@u=@T ); c is the speci c heat at constant volume, = −−1 (@=@T )p; c is the thermal expansion coecient, T = −1 (@=@p)T; c is the isothermal compressibility and   @  2 + : (1.12) H ≡T @T T ; c The last term in the left-hand side of Eq. (1.9) is the irreversible part and its coecient matrix is given in terms of Onsager coecients by   0 0 0 0 0 0 0 0 0 0      (1.13) ≡  0 0 TT 0 T!  0 0 0 0  vv 0 0 !T 0 !! with TT = vv = !T where l L00 2 k ; cv T 2 lk 2 = −ik T! = ik L01 ; cv T ; L10 2 T c(1 − c) ; !! = L11 ; Tc(1 − c) (1.14) = (4=3 + )= is the longitudinal viscosity. 2. Thermodynamic derivatives In this section, we introduce microscopic uctuations of extensive variables around the equilibrium state. The uctuations of intensive parameters such as temperature, hydrostatic pressure are also properly introduced. Thermodynamic susceptibilities are shown to be expressed in terms of the proper combination of static correlation functions of these uctuating variables. 2.1. De nition of the microscopic quantities and their equations of motion We consider a system with volume V consisting of N1 and N2 particles of each component. We assume that an intermolecular potential of the system is given by a sum of pair potentials which depend only on the intermolecular distances: U (r1(1) ; : : : ; r1(2) ; : : :) = Nl X Nm 2 1 X X lm (|ri(l) − rj(m) |) ; 2 (2.1.1) l; m=1 i=1 j=1 where ri(l) denotes the position of the ith particle of the lth component. The microscopic de nition of mass density, momentum density of each component, and internal energy K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 21 density are de ned by l (r; t) = ml Nl X (r − ri(l) ); (l = 1; 2) ; i=1 Nl X ui(l) (r − ri(l) ); (l = 1; 2) ; (2.1.2) i=1   Nl  2 Nm 2 X  X 1 XX 1 ml ui(l)2 + lm (|ri(l) − rj(m) |) (r − ri(l) ) ; (u)(r; t) =  2 2 (l Cl )(r; t) = ml m=1 j=1 l=1 i=1 where ml and ui(l) = ṙi(l) are the mass and velocity of a particle of the lth component, respectively. The averages over the equilibrium ensemble are given by hl (r; t)i = l = ml nl ; hCl (r; t)i = 0 ; hu(r; t)i = Z 2 1 X 3 nkB T + nl nm dr glm (r)lm (r) ; 2 2 (2.1.3) l; m=1 where h· · ·i denotes the grand canonical ensemble average over the phase space (ri(l) ; ui(l) ). Here nl = Nl =V (l = 1; 2) is the number density of the lth component per unit volume, and n ≡ n1 + n2 : glm (r) (l; m = 1; 2), which appears in the last equation, is the radial pair distribution function de ned by *N ; N + m l 1 X (l) (m) ′ (r − ri + rj ) ; (l = 1; 2) ; (2.1.4) nl nm glm (r) = V i; j=1 where the prime on the sum indicates that terms with i=j; l=m are excluded. Hereafter, we shall consider the uctuations around the equilibrium values in the wavevector space, i.e., ak = ak − hai(2)3 (k). From the above de nition, we obtain the equations of motion which the uctuations obey. The total mass uctuation  ≡ 1 + 2 is governed by the continuity equation. @k = ik · Ck ; (2.1.5) @t where Ck ≡ cC1k + (1 − c)C2k is the uctuation of the barycentric ow given by ( ) N1 N2 X X 1 (1) ik·ri(1) (2) ik·ri(2) m1 ui e + m2 ui e : Ck =  i=1 i=1 The quantities without subscript k denote the average values, e.g.,  = hi. For the mass concentration ratio ck , using ck = −1 {(1 − c)1k − c2k }, we have @ck = c(1 − c)ik · !k ; @t where !k = C1k − C2k is the uctuation of the relative velocity given by !k = N1 N2 (1) (2) 1 X 1 X ui(1) eik·ri − ui(2) eik·ri : n1 n2 i=1 i=1 (2.1.6) 22 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 For the barycentric velocity, the equation of motion is given by @Ck = ik · k @t with the total pressure tensor;   (lm) (lm) Nl  Nm 2 X 2 X  X X r r 1 ij ′ ij Pk(lm) (rij(lm) ) eik · ri(l) ; ml ui(l) ui(l) −  k= (lm)2   2 rij  l=1 i=1 (2.1.7) (2.1.8) m=1 j=1 where the Greek indices are the Cartesian coordinates x; y; z. In this expression, we have introduced dlm (r) 1 − e−ik·r : Pk(lm) (r) ≡ r dr ik · r The equation for !k is given by @!k = ik · k − Fk ; @t (2.1.9) where   (11) (11) N1  N1  X r r (1) 1 1 X ij ij (11) (11) ′ P (r ) eik·ri ui(1) ui(1) −  k= ij k (11)2   n1 i 2m1 rij j=1   (22) (22) N2  N2  X r r (2) 1 X 1 ij ij (22) (22) ′ − P (r ) eik·ri ui(2) ui(2) − ij k (22)2   n2 i 2m2 rij (2.1.10) j=1 and F k= N1 X N1 X N2 N2 rij(12) d12 (rij(12) ) ik·r(1) rij(12) d12 (rij(12) ) ik·r(2) 1 X 1 X i e e i : + 1 2 rij(12) drij(12) rij(12) drij(12) i=1 j=1 i=1 j=1 (2.1.11) For the internal energy density, since (u) = u + u, we have 1 1 uk = {(u)k − uk } = {(u)k − uk } ;   which can be written as   Nl  Nm 2 X 2 X  X X (l) 1 1 1 ml ui(l)2 + lm (rij(lm) ) − ml u eik·ri : uk = 2   2 l=1 i=1 (2.1.12) m=1 j=1 Then, the time derivative of this is given by  @uk = ik · qk ; @t (2.1.13) where qk = qk(1) + qk(2) with   Nl Nm 2 X 2 X  1 X X (l) 1 ml ui(l)2 + lm (rij(lm) ) − ml u eik·ri ui(l) qk(1) =  2 2 l=1 i=1 m=1 j=1 (2.1.14) K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 23 Nl X Nm 2 r (lm) rij(lm) (lm) (lm) ik·r(l) 1 X X (l) (m) ij ≡− (ui + uj ) (lm)2 Pk (rij )e i 4 rij (2.1.15) and q(2) k l; m=1 i=1 j=1 is the energy ow. 2.2. Temperature, pressure, and chemical potential We shall here generalize uctuations of temperature, hydrostatic pressure, and chemical potential to the k-dependent regime. These uctuations are given in terms of linear combination of the internal energy, mass density, and mass concentration ratio. The coecients in the linear combination are determined such that they reduce to the usual thermodynamic relation in the long wavelength limit. Using these variables, microscopic expression for thermodynamic derivatives (thermodynamic susceptibilities) are derived. The uctuation of temperature can be naturally introduced by using the fact that, in the long wavelength limit, it is statistically independent of both the mass density and mass concentration ratio (see Appendix A) [4,8]. We may write the k-dependent temperature uctuation as Tk = a1 (k){uk + a2 (k)k + a3 (k)ck } : (2.2.1) Coecients a2 (k) and a3 (k) are determined such that hTk −k i = hTk c−k i = 0. Therefore, we have a2 (k) = − 1 {(uk ; k )(|ck |2 ) − (uk ; ck )(k ; ck )} ; (k) a3 (k) = − 1 {(uk ; ck )(|k |2 ) − (uk ; k )(k ; ck )} (k) (2.2.2) with (k) ≡ (|k |2 )(|ck |2 ) − |(k ; ck )|2 : (2.2.3) Here we have de ned (ak ; bk ) ≡ V −1 hak b−k i and (|ak |2 ) ≡ (ak ; ak ). On the other hand, a1 (k) can be determined by comparing Eq. (2.2.1) with thermodynamic relation which is valid for the long wavelength limit;     1 @u 1 @u 1  − c : (2.2.4) T = u − cv cv @ T; c cv @c T;  Therefore, it is natural to introduce the k-dependent speci c heat at constant volume by a1 (k) ≡ 1=cv (k). The k-dependent speci c heat can be determined by using the formula for uctuations (see Appendix A). lim (|Tk | 2 ) = hT 2 i = k→0 kB T 2 ; cv (2.2.5) 24 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 where hT 2 i is the amplitude of the local temperature uctuation in the long wavelength limit de ned by hT (r)T (r′ )i ≡ hT 2 i(r − r′ ). Using Eqs. (2.2.1) and (2.2.5), we have  (|uk + a2 (k)k + a3 (k)ck |2 ) cv (k) = kB T 2 =  {(|uk |2 ) + a2 (k)(uk ; k ) + a3 (k)(uk ; ck )} : kB T 2 (2.2.6) By comparing Eq. (2.2.1) with Eq. (2.2.4), the following relations have to be satis ed.       @u @u 1 T −p ; lim a3 (k) = − = 2 : lim a2 (k) = − k→0 k→0 @ T; c  T @c T;  (2.2.7) As for the uctuation of the hydrostatic pressure, we may determine it from the de nition of the stress tensor which has been introduced in the derivation of the microscopic equation for the barycentric velocity in Eq. (2.1.8). We can separate the stress tensor into a “Thermodynamic part” and a “Viscous part” as [4] ′ ; lk = pk + lk (2.2.8) where lk = k̂·· k̂ is the longitudinal component of the stress tensor. pk is determined ′ is orthogonal to T; , and c. We may write it as such that the remainder lk pk = b1 (k)Tk + b2 (k)k + b3 (k)ck : (2.2.9) Then, each coecient is given by b1 (k) = (lk ; Tk ) ; (|Tk |2 ) b2 (k) = kB T (|ck2 |) 1 {(lk ; k )(|ck2 |) − (lk ; ck )(k ; ck )} = ; (k) (k) b3 (k) = 1 kB T (ck ; k ) {(lk ; ck )(|k |2 ) − (lk ; k )(ck ; k )} = − ; (k) (k) (2.2.10) where use has been made of identities: (lk ; k ) = kB T and (lk ; ck ) = 0 : In the long wavelength limit, by comparing with the thermodynamic relation, the coecients bi (k) are to reduce to     1 @p @p = ; = lim b2 (k) = ; lim b1 (k) = k→0 k→0 @T ; c T @ T; c T lim b3 (k) = k→0  @p @c  T;  : (2.2.11) K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 25 Using the relations exhibited above, one may derive the general expressions of thermodynamic derivatives of arbitrary quantities in terms of the static correlation functions of microscopic quantities. If one takes the temperature, mass density, and concentration ratio as independent variables, uctuations of a quantity A are written in the long wavelength limit as X  @A  ai : A = @ai a =T; ; c i Each coecient can be derived by taking the ensemble average after multiplying a = T; ; c from the left-hand side. Thus, we obtain   (Ak ; Tk ) @A : (2.2.12) = lim @T ; c k→0 (|Tk |2 ) This formula may be generalized as   (Ak ; Tk ) @A : = lim k→0 @B ; c (Bk ; Tk ) For other two coecients, we have   1 @A {(Ak ; k )(|ck |2 ) − (Ak ; ck )(k ; ck )} ; = lim k→0 (k) @ T; c   1 @A {(Ak ; ck )(|k |2 ) − (Ak ; k )(k ; ck )} = lim k→0 (k) @c T;  or more generally,   (Ak ; k )(|ck |2 ) − (Ak ; ck )(k ; ck ) @A ; = lim k→0 (Bk ; k )(|ck |2 ) − (Bk ; ck )(k ; ck ) @B  T; c (Ak ; ck )(|k |2 ) − (Ak ; k )(k ; ck ) @A : = lim k→0 (Bk ; ck )(|k |2 ) − (Bk ; k )(k ; ck ) @B T;  (2.2.13) (2.2.14) (2.2.15) Applying the above argument for the entropy, pressure, and concentration ratio, the adiabatic derivative is obtained;   (Ak ; pk ) (Ak ; lk ) @A = lim : (2.2.16) = lim @B s; c k→0 (Bk ; pk ) k→0 (Bk ; lk ) Likewise, the isothermal and isobaric derivative is given by choosing the temperature, pressure, and mass concentration ratio as independent variables,   (Ak ; ck ) @A : (2.2.17) = lim @B T;  k→0 (Bk ; ck ) If we take A = u and B =  in Eq. (2.2.16), then we obtain the virial expression of the hydrostatic pressure;   Z 2 1 X @u dlm (r) glm (r) : = nkB T − (2.2.18) nl nm dr r p = 2 @ s; c 6 dr l; m=1 26 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 On the other hand, Eq. (2.2.17) with A = u; −1 and the enthalpy per unit mass, h ≡ u + p−1 , etc. and B = c leads to the molar energy, molar speci c volume, and molar enthalpy, etc., respectively. The di erence between molar quantities per unit mass of each component is de ned by [6]   @A : (2.2.19) A1 − A2 = @c T;p Combining this equation with the relation; A = cA1 + (1 − c)A2 , one has     @A @A ; A2 = A − c : A1 = A + (1 − c) @c T;p @c T;p (2.2.20) For example, the partial speci c volume is given by putting A = −1 , 1 (k ; ck ) (1 − c) lim k→0 (|ck |2 ) 2 R 1 + n2 dr{g22 (r) − g12 (r)} R : = m1 [n + n1 n2 dr{g11 (r) − 2g12 (r) + g22 (r)}] v1 = −1 − (2.2.21) Speci c volume for component 2 is obtained by exchanging the indexes 1 with 2. Following the similar way, we may also derive the microscopic expressions for the thermal expansion coecients = −−1 (@=@T )p; c , the isothermal compressibility T = −1 (@=@p)T; c and we shall nd that they are in accordance with Eqs. (2.2.2) and (2.2.11). Finally, let us consider the uctuation of the chemical potential. Instead of , we shall consider =T for convenience;  = c1 (k)Tk + c2 (k)k + c3 (k)ck : (2.2.22)  T k ci (k) are determined such that Eq. (2.2.22) reduces in the long wavelength limit to         @    1 @ 1 @ = T +  + c  T @T T ; c T @ T;  T @c T; c 1 =− T  @u @c  1 T + 2  T T;   @p @c  1  + T T;   @ @c  c : T; c Comparing this relation with Eq. (2.2.4), we have c1 (k) = 1 a3 (k); T c2 (k) = 1 b3 (k) : 2 T (2.2.23) c3 (k) is derived from the relation in the long wavelength limit (see Appendix A);   @ kB T h2 i ; (2.2.24) =  @c T;  where  ≡  (k = 0). Therefore, one has c3 (k) = kB (|k |2 ) : (k) (2.2.25) K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 27 The relations derived in this subsection will be used in the next section to see the relevance of the microscopic equations for the uctuations to the phenomenological equations. 3. Generalized Langevin equation formalism In this section, we shall derive a set of closed equations which describes the dynamics of uctuations of variables ak = (ck ; k ; Tk ; vk ; !k ), where vk = k̂ · Ck and !k = k̂ · !k are the longitudinal part of the velocity elds. For this purpose, we shall introduce the projection operator P de ned by [3] PA(t) = (A(t); aik )(aik ; ajk )−1 ajk : (3.1) Following the conventional procedure which has been applied for one-component uids, one obtains the equation for the correlation function matrix, in Laplace transformed representation, Cij (k; z) = (aik (z); ajk ) ; which is given by [z − i (k) + (k; z)]il Clj (k; z) = C̃ ij (k) ; (3.2) where C̃ ij (k) ≡ Cij (k; t = 0) is the initial value of the correlation function. i ij (k) −1 ≡ (ȧik ; alk )C̃ lj (k) (3.3) is the coecient of the reversible part of Eq. (3.2) and it is expressed in terms of static correlation functions. On the other hand, (k; z) is the memory kernel which describes the irreversible processes and is given by ij (k; z) = (U ′ (z)fik ; flk )(alk ; ajk )−1 ; (i; j = T; v; !) ; (3.4) where fik ≡ Qȧik = ȧik − i ij (k)ajk (3.5) are the random forces and Q ≡ 1 − P is the projection operator which is orthogonal to P. U ′ (z) ≡ 1 z − QiLQ (3.6) is the time-evolution operator de ned by the modi ed Liouville operator iQLQ. It can be shown that one may add an arbitrary linear combination of aik to the random forces in the kernels [9]. The simplest choice is, of course, +i ij (k)ajk so that the random forces are given simply by fik = ȧik . 28 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 Explicit expression of i (k) is given by  0 0 0   0 0 0 i   i (k) =  0 0 0 i    i (k) i (k) i v vT (k)  vc i !c (k) i ! (k) i !T (k) where i c! (k) = ikc(1 − c) ; i v (k) = ik ; i Tv (k) = 0 i c! (k)       Tv (k) i T! (k)  ;   0 0  v (k) 0 0 0 T (Ṫ k ; vk ) = ik b1 (k) ; (|vk |2 ) cv (k) c(1 − c) (Ṫ k ; !k ) = ik [a3 (k) + h† (k)] ; (|!k |2 ) cv (k) ik kB T 1 (k ; ck ) = ik b3 (k) ; i vc (k) = − (k)  1 ik kB T (|ck |2 ) = ik b2 (k) ; i v (k) = (k)  i i T! (k) vT (k) (3.7) = =i Tv (k) (3.8) 1 (|vk |2 ) = ikb1 (k) ; (|Tk |2 )  ik kB T (|k |2 ) = ikTc3 (k) ; (k) 1 ik kB T (ck ; k ) = ikTc2 (k) = 2 ikb3 (k) ; i ! (k) = − (k)  i !c (k) = i !T (k) =i T! (k) 1 (|!k |2 ) = ik [a3 (k) + h† (k)] ; (|Tk |2 ) T where use has been made of (ċk ; !k ) = (!˙ k ; ck ) = ik kB T  and (˙k ; vk ) = (v̇k ; k ) = ik kB T : Derivation of i T! (k) and i !T (k) is elucidated in Appendix B. In the above equations, we have de ned h† (k) = h†1 (k) − h†2 (k) with the modi ed molar enthalpy h†l (k) = ul† (k) + 1 † p (k); l l (l = 1; 2) of the lth component. Here, ul† (k) is the modi ed molar energy de ned by ) ( Z 2 1 3 1X † nl kB T + nl nm drlm (r)glm (r) ; (l = 1; 2) ul (k) = l 2 2 m=1 (3.9) (3.10) 29 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 and pl† (k) is the partial pressure de ned by pl† (k) = nl kB T − 2 X m=1 nl nm Z dr(k̂ · r̂)2 Re[Pk(lm) (r)]glm (r); (l = 1; 2) ; (3.11) where r̂ = r=r. Note that in the long wavelength limit, sum over two components of these quantities satis es u = cu1† (k = 0) + (1 − c)u2† (k = 0); p = cp1† (k = 0) + (1 − c)p2† (k = 0) ; (3.12) where u is the average of the internal energy density given by the last equation of Eq. (2.1.3) and p is the virial expression of the hydrostatic pressure given by Eq. (2.2.18). Therefore, we have h = u + p−1 = ch†1 (k = 0) + (1 − c)h†2 (k = 0). Here it should be emphasized that the modi ed molar enthalpy is di erent from the molar enthalpy de ned in the equilibrium thermodynamics, though both lead to the same value if summed up over components. According to Eq. (2.2.19), conventional de nition of the molar enthalpy is given by [6]     @h @h ; h2 = h − c : (3.13) h1 = h + (1 − c) @c T;p @c T;p Making use of Eq. (2.2.17), the microscopic expression for Eq. (3.13) is given by (hk ; ck ) ; k→0 (|ck |2 ) (hk ; ck ) ; k→0 (|ck |2 ) h1 = h + (1 − c) lim h2 = h − c lim (3.14) where hk = uk − −2 pk + −1 pk is the uctuation of the enthalpy density. After laborious manipulation, one arrives at h1 = u1 + pv1 , where v1 is the molar speci c volume given by Eq. (2.2.21) and u1 is the molar energy given by u1 = u + (1 − c) lim k→0 = 1 3 kB T + 2 m1 2m1 A +n2 2 X (uk ; ck ) (|ck |2 ) " 2 Z X n drnl 1l (r)g1l (r) l=1 (−1)k+1 nl nm k; l; m=1 Z dr Z  dr′ lm (r){glmk (r; r′ ) − glm (r)gk2 (r ′ )} ; (3.15) where A ≡ n + n1 n2 Z dr{g11 (r) − 2g12 (r) + g22 (r)} : The expression for component 2 is given by exchanging subindices 1 and 2 in the above expression. The derivation of Eq. (3.15) is elucidated in Appendix D. In the 30 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 above expression, glmk (r; r′ ) is the three-particle distribution function de ned by *N N N + k l m X X 1 X (l) (m) (l) (k) ′ ′ ′ ′ ′ (r − ri + rj )(r − ri + rh ) ; nl nm nk glmk (r; r ) = V h=1 i=1 j=1 (k; l; m = 1; 2): (3.16) It is obvious that h1 di ers from Eq. (3.9). Both quantities are identical only in the dilute limit. In this limit, one may neglect correlations between particles and obtains ul† = ul = 3kB T=2ml and h†l = hl = 5kB T=2ml . Using the results in the previous section, it is easy to check that the matrix i (k) reduces to i in Eq. (1.10) in the long wavelength limit if we put  = limk→0 h† (k) = limk→0 {h†1 (k) − h†2 (k)}. Thus, we may conclude that a proper heat ow due to the di usional ow is given in terms of the modi ed molar enthalpy not of the molar enthalpy in a usual sense. On the other hand, the dissipative term has the following structure:   0 0 0 0 0     0 0 0  0 0     : (3.17) (k; z) =  (k; z) (k; z) (k; z) 0 0 TT Tv T!       0 0 vT (k; z) vv (k; z) v! (k; z)    0 0 !T (k; z) !v (k; z) !! (k; z) Comparing this equation with Eq. (1.14), we obtain the expressions for the Onsager coecients   Z ∞ 1 4 1 ∗ + = lim dthU ′ (t)lk · lk i; T 3 3kB V k→0 0 1 lim 3kB V k→0 Z ∞ 1 lim L01 = 3kB V k→0 Z ∞ 1 lim 3kB V k→0 Z ∞ L00 = L11 = dthU ′ (t)qk · qk∗ i ; 0 (3.18) ′ dthU (t)qk · ∗ J˙ 1k i = −L10 ; 0 ∗ dthU ′ (t)J˙ 1k · J˙ 1k i ; 0 where U ′ (t) = exp[iQLQt] is the modi ed time evolution operator and J1k = c(1 − c) !k is the di usional momentum of the component 1. One may also obtain similar expressions for LTv , Lv! but it is generally known that these coecients are negligible for simple uids 1 [10]. 1 As discussed in paper I, Lv! becomes important in such systems like colloidal suspensions, liquid 4 He under -point. 31 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 Note that the transport coecients Eq. (3.4) are di erent from the Green–Kubo integrands which have the form of ij (k; z) ≡ (U (z)ȧik ; ȧjk ) ; (|ajk |2 ) (i; j = T; v; !) ; (3.19) where U (z) = (z − iL)−1 is the Liouville operator. Both (k; z) and (k; z) are related to each other by [3] −1  1 · {(k; z) − i (k)} (3.20) (k; z) − i (k) = 1 − {(k; z) − i (k)} z or equivalently, −1  1 : (k; z) − i (k) = { (k; z) − i (k)} · 1 + { (k; z) − i (k)} z (3.21) lim lim (k; z) = lim lim (k; z) : (3.22) The proof of these identities is shown in Appendix C. For one component systems, these coecients become identical in the hydrodynamic limit, i.e., z→0 k→0 z→0 k→0 This is because all thermodynamic variables ak = (k ; Tk ; vk ) are conserved variables and therefore ȧk is proportional to k. Thus, ; are found to be ∼ O(k 2 ) for small-k limit and these quantities in the denominators in Eqs. (3.20) and (3.21) are neglected in this limit. For two component systems, however, since !k is not a conserved variable, the above argument for equivalence between Green–Kubo integrands and memory kernels does not hold. In order to see the di erence, let us expand the inverse matrix in Eq. (3.20) up to the second order in k. −1  1 ′ 1 −  (k; z) z (  "   −1 −1 −1 1 ′ 1 1 1 1 1 ′ 1 − 0 1 − ′0 · 1 + · ′2 + 2 1 − ′0 = 1+ z z z z z z ·′1  1 · 1 − ′0 z −1 · ′1 )#  where ′ ≡  − i and ′i is the ith obtain −1   1 ′ ′ ′ · 0 + 1 − = 1 − 0 z 1 + z ( 1 1 − ′0 z −1 · ′1 1 1 − ′0 z −1 + O(k 3 ) ; (3.23) order term in k of ′ . After some algebra, we  −1 )2  −1 1 ′ · 1 − 0 z as k → 0 : 1 ′  z 0 −1 · ′1 1 · 1 − ′0 z + ′2 (3.24) 32 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 Each coecient of the above expression is given by cv 1 cv 1 |T! − i T! |2 − |i Tv |2 ; z − !! c(1 − c)T zT 1 (T! − i T! )!v ; Tv = Tv + z − !!   z !! i T! ; T! − T! = z − !! z z v! ; as k → 0 ; v! = z − !!   c(1 − c) 1 cv 1 2 2 2 =  + | | − |i ; | + |i | vv vv !v Tv v z − !! z T 3 T cv cv ∗ ∗ ; ; !T = − vT = − T Tv c(1 − c)T T! z 1 ∗ !! ; ; !! = !v = c(1 − c) v! z − !! TT = T T − (3.25) where ij∗ denotes the complex conjugate of ij . The denominator z − !! appearing in the above expressions has a singularity at z = 0, since !! ∼ z as z → 0. This singularity leads to a constant value for transport coecients in the hydrodynamic limit, while some of ij vanishes in this limit. The last terms of the equations for T T and vv with singularities at z = 0 cancel out the similar contributions from T T and vv , respectively. 4. Conclusions In this paper, we have formulated the generalized hydrodynamics for binary uids. Using the generalized Langevin equation formalism, the equations for the timedependent correlation functions of thermodynamic variables are expressed in terms of k-dependent thermodynamic susceptibilities and memory kernels. In the hydrodynamic limit, these equations becomes identical to the linearized equations which were derived from the phenomenological level of description. Arbitrariness in the paper I was clari ed by analyzing microscopic uctuations and it was found that a free parameter  in Eqs. (1.3) and (1.6) is given in terms of the modi ed enthalpy limk→0 h† (k) = limk→0 {h†1 (k) − h†2 (k)} de ned by Eq. (3.9) contrary to the conventional choice of the usual molar enthalpy h = h1 − h2 . On the phenomenological level,  cannot be determined. Only if we go into the microscopic level of description, we can nd that  is actually h† . However, if one concerns longer time scales in which the inertial e ect of di usional ow is neglected, we may choose  arbitrarily. As shown below in detail, corresponding to each choice of , we can de ne the proper random forces whose time correlation functions give the transport coecients. In such long time limit, the di usional ow K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 33 plays the role of one of the irreversible currents rather than a thermodynamic variable. As was pointed out by Bearman et al. [7], it becomes arbitrary to divide the irreversible energy ow into the heat ow and the heat carried by the di usion from the statistical mechanical standpoint. To see this, let us consider the linear relation between the irreversible ow and thermodynamic forces. Neglecting the inertial terms in Eq. (1.6) and using Eqs. (1.3) and (1.7), one obtains     1 + L̃01 ∇ − ; J̃ q = L̃00 ∇ T T     1 + L̃11 ∇ − ; (4.1) J1 = L̃10 ∇ T T where L̃ij are Onsager coecients which are used in the conventional nonequilibrium thermodynamics and J̃ q ≡ Jq + h† J1 is the total irreversible ow of energy density. The heat conductivity, thermal di usion ratio, and the di usion coecient are given in terms of these coecients by   !     2 L̃11 @ @c @c L̃ L̃01 1 T;p ; −h ; D=  = 2 L̃00 − 10 ; kT = T @ T;p L̃11 T L̃11 (4.2) respectively [6]. Here use has been made of a thermodynamic relation:     @h @  2 : = −T h = h1 − h2 = @c T;p @T T p; c L̃ij are related to Lij by L̃00 = L00 + 1 {L10 + c(1 − c)Th† }2 ; L11 L̃01 = L̃10 = c(1 − c)T {L10 + c(1 − c)Th† }; L11 L̃11 = {c(1 − c)T }2 : L11 L̃ij can be expressed in the form of Green–Kubo formula as Z ∞ 1 lim dthU (t)qk · qk∗ i ; L̃00 = 3kB V k→0 0 1 lim 3kB V k→0 Z ∞ 1 lim L̃11 = 3kB V k→0 Z ∞ L̃01 = ∗ dthU (t)qk · J1k i; (4.3) 0 ∗ dthU (t)J1k · J1k i; 0 where U (t) = exp[iLt] is the time evolution operator rather than the modi ed operator U ′ (t), since both the energy density and concentration ratio are the conserved quantities. 34 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 Eq. (4.1) is the simplest way to de ne the heat ow. There is possibility of de ning the heat ow in di erent ways keeping the Green–Kubo formulae Eq. (4.3) invariant under this rede nition. If one divides the heat ow as ′ J̃ q = J̃ q + ′ J1 ; (4.4) where ′ is an arbitrary constant, then, the linear relation Eq. (4.1) is rewritten as       1 1  ′ ′ ′ ′ + ∇ + L̃01 ∇ − ; J̃ q = L̃00 ∇ T T T       1 1  ′ + L̃11 ∇ − + ′ ∇ ; (4.5) J1 = L̃10 ∇ T T T ′ where L̃ij is the Green–Kubo integral given by replacing the microscopic heat ow qk in Eq. (4.3) by qk′ = qk − ′ J1k : (4.6) A convenient choice of the parameter is ′ = h1 − h2 . Then, Eq. (4.5) becomes   1 ′ 1 ′ ′ − L̃01 ∇T  ; J̃ q = L̃00 ∇ T T (4.7)   1 ′ 1 ′ − L̃11 ∇T  ; J1 = L̃10 ∇ T T where ∇T indicates the di erentiation keeping the temperature constant. This is the choice adopted in most of the literatures [6,7]. Such arbitrariness does not appear when the friction force rather than the di usional ow is considered as an irreversible part of balance equations (1.3)–(1.6). In the generalized Langevin equation formalism, incorporation of the di usional ow as a thermodynamic variable corresponds to going down to the second level in the hierarchy of the continued fraction structure of the equation [2] for the concentration ratio. In this paper, any explicit analysis of the transport coecients was not shown. As was discussed in paper I, the formalism given here can a ord explaining fast processes [11] or phenomena where di usion couples to convection in uids [12]. Acknowledgements The authors are grateful to Prof. D. Bedeaux for illuminating discussions and useful advice. This work was nancially supported by the National Institute Postdoctoral Fellowship from the Research Development Corporation of Japan (JRDC), Grant-in-Aid for Scienti c Research (05245103), and Grant-in-Aid for Scienti c Research on Priority Area (07240106) from Ministry of Education, Science and Culture, Japanese Government. K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 35 Appendix A According to the thermodynamic theory of uctuations [8], the probability density for the uctuations of thermodynamic variables is proportional to   Z  dr{T (r)s(r) − p(r)v(r) + (r)c(r)} ; (A.1) exp − 2kB T where v ≡ −1 is the speci c volume per unit mass. If we choose (T; ; c) as independent variables, we have the following quadratic form: Z dr{T (r)s(r) − p(r)v(r) + (r)c(r)} Z dr +2  =  @ @ 1 cv 2 T (r) + 3 2 (r) T  T  (r)c(r) + T; c  @ @c  T;  2 c (r) ) : (A.2) This implies that the uctuations are completely local and its static correlation functions have a general form as hai (r)aj (r′ )i = hai aj i(r − r′ ) : (A.3) The amplitudes of the variances of all of these variables are given by   kB T @ kB T kB T 2 2 2 ; ; h i = ; hc2 i = 4 hT i = cv 
@c T;   T
kB T hci = − 
 @ @  ;  @ @c hTi = 0; hTci = 0 (A.4) T; c where 1
= 3  T  @ @c  T;  − 2 : T; c Or equivalently, we may have 1 kB T hc2 i kB T 2 ; ; = 2 3 hT i  T      @ @ kB T hci kB T h2 i ; =− = @ T;   @c T;   cv = with  ≡ h2 ihc2 i − hci2 : As can be seen from Eq. (2.2.3),  is the long wavelength limit of (k). (A.5) 36 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 Appendix B T! (k) In this appendix, derivation of i Eq. (2.2.1), we have and i !T (k) in Eq. (3.8) are given. Using (!˙ k ; Tk ) = a1 (k){(!˙ k ; uk ) + a2 (k)(!˙ k ; k ) + a3 (k)(!˙ k ; ck )}    ik kB T a1 (k) (!˙ k ; uk ) + a3 (k) ; =  i k kB T (B.1) where use has been made of (!˙ k ; k ) = 0 and (!˙ k ; ck ) = ik kB T=. The rst term in the braces on the right-hand side is   (!˙ k ; uk ) = − (!k ; u̇ k ) i k kB T i k kB T =− 1 ˆ ˆ k k ((v1 kB T (2) − v2 k ); (q(1) k + q k )) : k (B.2) Each term is calculated as "  2 c kB T m1 1 (1) hv1 k q −k i = ×5 N1 V Vn1 2 ml kB T +V m1 kB T = 1 " ( 2 1X n1 nl 2 l=1 Z dr1l (r)g1l (r) − 1 u 2 1X 5 n1 kB T + n1 nl 2 2 l=1 Z )#  # dr1l (r)g1l (r) − 1 u  (B.3) and 1 1 hv1 k q(2) −k i = − V 4Vn1 × * N1 X rij(lm)2 (ui(l) + uj(m) ) l; m=1 i=1 j=1 i′ =1 rij(lm) rij(lm) Nl X Nm 2 X X (1) ik·ri′ ui(1) ′ e (lm) (lm) −ik·ri(l) P−k (rij )e + + " * (lm) (lm) Nl X Nm N1 X 2 X ′ rij rij 1 X (lm) (lm) ik·r (l′ l) (1) (l) hui′ ui i P−k (rij )e i i =− 4Vn1 ′ rij(lm)2 i =1 l; m=1 i=1 j=1 (m) i + hui(1) ′ uj * rij(lm) rij(lm) rij(lm)2 ′ (lm) (lm) ik·r(l′ l) P−k (rij )e i i +# * (lm) (lm) " + Nl X Nm N1 X 2 X rij rij kB T X (lm) (lm) ik·r (1l) ′ P−k (rij )e i i i′ i 1l  =− 4Vn1 ′ rij(lm)2 i =1 l; m=1 i=1 j=1 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 * + i′ j 1m  rij(lm) rij(lm) rij(lm)2 Nm N1 X 2 kB T X X =− 4Vn1 l=1 i=1 j=1 + =− * rij(1l) rij(1l) rij(1l)2 "* (lm) (lm) ik·r (1l) P−k (rij )e i′ i rij(1l) rij(1l) rij(1l)2 (1l) (1l) −ik·rji(1l) P−k (rij )e 37 +# + (1l) (1l) P−k (rij ) +# Z 2 r r kB T X dr 2 Re[Pk(1l) (r)]g1l (r) : n1 nm 2n1 r (B.4) l=1 −1 hv2 k q(2) V −1 hv2 k q(1) −k i and V −k i can also be evaluated in the same way. Summing up all of these terms, we arrive at − 1 ˆ ˆ k k ((v1 kB T k (2) † − v2 k ); (q(1) k + q k )) = u1 (k) + 1 † 1 p (k) − u2† (k) − p2† (k) 1 1 2 (B.5) with the modi ed molar internal energy de ned by 1 ul† (k) = l ( 2 3 1X nl kB T + nl nm 2 2 Z ) drlm (r)glm (r) (B.6) (k̂ · r)2 Re[Pk(lm) (r)]glm (r) : r2 (B.7) m=1 and the modi ed partial pressure pl† (k) = nl kB T − 2 X m=1 nl nm Z dr This is the result shown in Eq. (3.9). Appendix C In this appendix, derivation of Eqs. (3.20) and(3.21) is shown. For the stationary system, we have an identity: ij (k; t) ≡ (ȧik (t); ȧlk )(alk ; ajk )−1 = (aik (t); ajk )(alk ; ajk )−1 = C ij (k; t)Clj−1 (k; t = 0) : (C.1) 38 K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 Laplace transforming both sides of this identity, we obtain Z ∞ Z ∞  t) · C−1 (k; t = 0) dt e−zt (k; t) = − dt e−zt C(k; (k; z) = 0 0 = [Ċ(k; t = 0) + zC(k; t = 0) − z 2 C(k; z)] · C−1 (k; t = 0) = i (k) + z − z 2 C(k; z) · C−1 (k; t = 0) : (C.2) Comparing this with Eq. (3.2), we arrive at Eqs. (3.20) and (3.21). Appendix D In this appendix, derivation of Eq. (3.15) is shown. Using Eq. (2.2.20), one has u1 − u2 = lim k→0 = lim k→0 (uk ; ck ) (|ck |2 ) 1 {(1 − c)((u)k ; 1k ) − c((u)k ; 2k ) − u(k ; ck )} : 2 (|ck |2 ) (D.1) We shall consider the rst term in the braces of the last line of the above equation. Let us divide this into the contribution from the kinetic energy and from the potential term as ((u)k ; 1k ) = ((u)k ; 1k )k + (( u)k ; 1k )p : (D.2) The kinetic term is written as ((u)k ; 1k )k = Nl 2 X X ml l=1 i=1 2 (l) ui(l)2 eik·ri ; m1 ik·ri(l) e i=1 l=1 ik·ri(1) ′ e i′ =1 Nl X 2 3m1 kB T X = 2 N1 X ; N1 X e ik·ri(1) ′ i′ =1 ! !   Z Z 31 kB T ik·r ik·r 1 + n1 dre g12 (r) : dre g11 (r) + n2 = 2 As for the potential term, we have   N1 2 NX l ; Nm X X (1) (l) 1 eik·ri′  lm (rij(lm) )eik·ri ; m1 (( u)k ; 1k )p =  2 ′ l; m=1 i; j=1  2 m1  1 X = 2 V m=1 i =1 *N ;N m l X i; j=1 + lm (rij(1m) ) (D.3) K. Miyazaki, K. Kitahara / Physica A 264 (1999) 15–39 2 1 X + V l; m=1 *N ;N N m X l 1 X ′ ′ i; j=1 lm (rij(lm) )e ik·rii(1l) ′ i′ =1 " 2 Z 1 X nl dr1l (r)g1l (r) = 2 39 +  l=1 + 2 X l; m=1 nl nm Z dr Z ′  dr′ lm (r)eik·r glm1 (r; r′ ) : (D.4) (( u)k ; 2k ) can also be evaluated in the same manner. (|ck |2 ) and (k ; ck ) in Eq. (D.1) was already derived in Eq. (2.2.21). 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